[{"data":1,"prerenderedAt":21084},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-08-04-the-representations-in-llm":3,"i-ci:tag":21078,"i-mingcute:time-line":21082},{"post":4,"nextPost":4813,"prevPost":4982},{"id":5,"title":6,"body":7,"description":4798,"draft":4799,"enableComment":4800,"extension":4801,"image":4787,"important":4799,"location":4802,"meta":4803,"navigation":4800,"ogImage":4802,"path":4804,"seo":4805,"stem":4806,"summary":4807,"tags":4808,"time":4811,"__hash__":4812},"blog\u002Fblog\u002F2026\u002F2026-08-04-the-representations-in-llm.md","大语言模型中的表征流形",{"type":8,"value":9,"toc":4786},"minimark",[10,22,25,34,38,41,44,47,388,738,841,845,848,992,1288,1291,1294,1907,2545,2548,2551,2554,2571,2574,2577,2580,2583,3179,3182,3255,3258,3261,3264,3704,3909,4115,4377,4450,4453,4456,4459,4462,4502,4569,4654,4657,4660,4663,4666,4669,4672,4683,4686,4689,4692,4695,4698,4701,4704,4777,4780,4783],[11,12,13,14,21],"p",{},"如果你关注了过去两年机制可解释性研究的进展，很可能见过这样一类图：把大语言模型某一层神经元的激活向量投影到二维或三维空间并作图，会发现「月份」「星期几」「年份」「颜色」等概念的表示并不是杂乱无章地散布在空间中，而是沿着一条优美的曲线排列——有时甚至是闭合的圆圈或环面。Chris Olah、Anthropic 的 Josh Batson 以及 Engels 等人，都在包括 ",[15,16,20],"a",{"href":17,"rel":18},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2405.14860",[19],"nofollow","并非所有语言模型特征都是一维线性的"," 在内的研究中展示了这一现象。",[11,23,24],{},"这些发现引人入胜，却也留下了一个令人不安的空白：我们看到了流形，却无法精确地说出这个流形与它所表征的「概念」之间究竟是什么关系。为什么「年份」这样一个本该是直线的东西，会在模型内部被扭曲成一条在高维空间中蜿蜒盘绕的曲线？「颜色」排列成圆形是巧合还是必然？向量之间的余弦相似度，真的能告诉我们两个概念在语义上有多接近吗？",[11,26,27,28,33],{},"三位数学家——分别来自帝国理工学院、爱丁堡大学和布里斯托大学的 Alexander Modell、Patrick Rubin-Delanchy 与 Nick Whiteley——在他们的论文 ",[15,29,32],{"href":30,"rel":31},"https:\u002F\u002Farxiv.org\u002Fpdf\u002F2505.18235",[19],"《大语言模型中表征流形的起源》"," 中，首次尝试为这些问题提供一个「最小可用」的数学理论。本文梳理他们论证的脉络，并给出我个人的一些思考。",[35,36,37],"h2",{"id":37},"线性表征假说",[11,39,40],{},"机制可解释性领域长期以来由一个核心信念所锚定，即线性表征假说（LRH）：模型将人类可解释的「特征」——例如「长着毛茸茸的耳朵」「提到了埃菲尔铁塔」「使用阿拉伯语」——编码为表征空间中一组近似正交的方向向量。给定输入的表示，则是这些方向向量的稀疏线性组合，权重由各特征是否存在及其程度决定。如今被广泛采用的可解释性工具稀疏自编码器（SAE），正是直接建立在这一假说之上：训练一个带有稀疏性惩罚的自编码器，去逼近这些「字典向量」。",[11,42,43],{},"然而，越来越多的证据表明，这种「非黑即白、一特征一方向」的模型无法解释某些现象。作者引用了大量的文献：模加任务中的数字被编码为圆；多语言模型中出现了环状结构；「日期」与「星期几」呈现出扭曲的环面几何；甚至在模拟的隐马尔可夫模型中涌现出分形几何。这些例子有一个共同点：特征不再是一条直线，而是一个完整、连续、可能非线性弯曲的流形。",[11,45,46],{},"因此，该领域提出了 LRH 的推广版本——多维线性表征假说：",[48,49,53],"span",{"className":50,"translate":52},[51],"katex-display","no",[48,54,57,146],{"className":55,"translate":52},[56],"katex",[48,58,61],{"className":59},[60],"katex-mathml",[62,63,66],"math",{"xmlns":64,"display":65},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[67,68,69,141],"semantics",{},[70,71,72,77,82,85,88,91,114,122,124,126,128,135,137,139],"mrow",{},[73,74,76],"mi",{"mathvariant":75},"normal","Ψ",[78,79,81],"mo",{"stretchy":80},"false","(",[73,83,84],{},"x",[78,86,87],{"stretchy":80},")",[78,89,90],{},"=",[92,93,94,97],"munder",{},[78,95,96],{},"∑",[70,98,99,102,105,108,110,112],{},[73,100,101],{},"f",[78,103,104],{},"∈",[73,106,107],{},"F",[78,109,81],{"stretchy":80},[73,111,84],{},[78,113,87],{"stretchy":80},[115,116,117,120],"msub",{},[73,118,119],{},"ρ",[73,121,101],{},[78,123,81],{"stretchy":80},[73,125,84],{},[78,127,87],{"stretchy":80},[115,129,130,133],{},[73,131,132],{},"v",[73,134,101],{},[78,136,81],{"stretchy":80},[73,138,84],{},[78,140,87],{"stretchy":80},[142,143,145],"annotation",{"encoding":144},"application\u002Fx-tex","\\Psi(x) = \\sum_{f \\in F(x)} \\rho_f(x) v_f(x)",[48,147,151,188],{"className":148,"ariaHidden":150},[149],"katex-html","true",[48,152,155,160,164,168,172,176,181,185],{"className":153},[154],"base",[48,156],{"className":157,"style":159},[158],"strut","height:1em;vertical-align:-0.25em;",[48,161,76],{"className":162},[163],"mord",[48,165,81],{"className":166},[167],"mopen",[48,169,84],{"className":170},[163,171],"mathnormal",[48,173,87],{"className":174},[175],"mclose",[48,177],{"className":178,"style":180},[179],"mspace","margin-right:0.2778em;",[48,182,90],{"className":183},[184],"mrel",[48,186],{"className":187,"style":180},[179],[48,189,191,195,279,283,328,331,334,337,379,382,385],{"className":190},[154],[48,192],{"className":193,"style":194},[158],"height:2.566em;vertical-align:-1.516em;",[48,196,200],{"className":197},[198,199],"mop","op-limits",[48,201,205,270],{"className":202},[203,204],"vlist-t","vlist-t2",[48,206,209,265],{"className":207},[208],"vlist-r",[48,210,214,252],{"className":211,"style":213},[212],"vlist","height:1.05em;",[48,215,217,222],{"style":216},"top:-1.809em;margin-left:0em;",[48,218],{"className":219,"style":221},[220],"pstrut","height:3.05em;",[48,223,229],{"className":224},[225,226,227,228],"sizing","reset-size6","size3","mtight",[48,230,232,236,239,243,246,249],{"className":231},[163,228],[48,233,101],{"className":234,"style":235},[163,171,228],"margin-right:0.1076em;",[48,237,104],{"className":238},[184,228],[48,240,107],{"className":241,"style":242},[163,171,228],"margin-right:0.1389em;",[48,244,81],{"className":245},[167,228],[48,247,84],{"className":248},[163,171,228],[48,250,87],{"className":251},[175,228],[48,253,255,258],{"style":254},"top:-3.05em;",[48,256],{"className":257,"style":221},[220],[48,259,260],{},[48,261,96],{"className":262},[198,263,264],"op-symbol","large-op",[48,266,269],{"className":267},[268],"vlist-s","​",[48,271,273],{"className":272},[208],[48,274,277],{"className":275,"style":276},[212],"height:1.516em;",[48,278],{},[48,280],{"className":281,"style":282},[179],"margin-right:0.1667em;",[48,284,286,289],{"className":285},[163],[48,287,119],{"className":288},[163,171],[48,290,293],{"className":291},[292],"msupsub",[48,294,296,319],{"className":295},[203,204],[48,297,299,316],{"className":298},[208],[48,300,303],{"className":301,"style":302},[212],"height:0.3361em;",[48,304,306,310],{"style":305},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[48,307],{"className":308,"style":309},[220],"height:2.7em;",[48,311,313],{"className":312},[225,226,227,228],[48,314,101],{"className":315,"style":235},[163,171,228],[48,317,269],{"className":318},[268],[48,320,322],{"className":321},[208],[48,323,326],{"className":324,"style":325},[212],"height:0.2861em;",[48,327],{},[48,329,81],{"className":330},[167],[48,332,84],{"className":333},[163,171],[48,335,87],{"className":336},[175],[48,338,340,344],{"className":339},[163],[48,341,132],{"className":342,"style":343},[163,171],"margin-right:0.0359em;",[48,345,347],{"className":346},[292],[48,348,350,371],{"className":349},[203,204],[48,351,353,368],{"className":352},[208],[48,354,356],{"className":355,"style":302},[212],[48,357,359,362],{"style":358},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[48,360],{"className":361,"style":309},[220],[48,363,365],{"className":364},[225,226,227,228],[48,366,101],{"className":367,"style":235},[163,171,228],[48,369,269],{"className":370},[268],[48,372,374],{"className":373},[208],[48,375,377],{"className":376,"style":325},[212],[48,378],{},[48,380,81],{"className":381},[167],[48,383,84],{"className":384},[163,171],[48,386,87],{"className":387},[175],[11,389,390,391,477,478,552,553,582,583,667,668,737],{},"其中 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不再是固定方向，而是可以在某个子空间 ",[48,479,481,500],{"className":480,"translate":52},[56],[48,482,484],{"className":483},[60],[62,485,486],{"xmlns":64},[67,487,488,497],{},[70,489,490],{},[115,491,492,495],{},[73,493,494],{},"V",[73,496,101],{},[142,498,499],{"encoding":144},"V_f",[48,501,503],{"className":502,"ariaHidden":150},[149],[48,504,506,510],{"className":505},[154],[48,507],{"className":508,"style":509},[158],"height:0.9694em;vertical-align:-0.2861em;",[48,511,513,517],{"className":512},[163],[48,514,494],{"className":515,"style":516},[163,171],"margin-right:0.2222em;",[48,518,520],{"className":519},[292],[48,521,523,544],{"className":522},[203,204],[48,524,526,541],{"className":525},[208],[48,527,529],{"className":528,"style":302},[212],[48,530,532,535],{"style":531},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[48,533],{"className":534,"style":309},[220],[48,536,538],{"className":537},[225,226,227,228],[48,539,101],{"className":540,"style":235},[163,171,228],[48,542,269],{"className":543},[268],[48,545,547],{"className":546},[208],[48,548,550],{"className":549,"style":325},[212],[48,551],{}," 内随输入 ",[48,554,556,569],{"className":555,"translate":52},[56],[48,557,559],{"className":558},[60],[62,560,561],{"xmlns":64},[67,562,563,567],{},[70,564,565],{},[73,566,84],{},[142,568,84],{"encoding":144},[48,570,572],{"className":571,"ariaHidden":150},[149],[48,573,575,579],{"className":574},[154],[48,576],{"className":577,"style":578},[158],"height:0.4306em;",[48,580,84],{"className":581},[163,171]," 连续变化。标准的 LRH 只是 ",[48,584,586,609],{"className":585,"translate":52},[56],[48,587,589],{"className":588},[60],[62,590,591],{"xmlns":64},[67,592,593,607],{},[70,594,595,601,603,605],{},[115,596,597,599],{},[73,598,132],{},[73,600,101],{},[78,602,81],{"stretchy":80},[73,604,84],{},[78,606,87],{"stretchy":80},[142,608,417],{"encoding":144},[48,610,612],{"className":611,"ariaHidden":150},[149],[48,613,615,618,658,661,664],{"className":614},[154],[48,616],{"className":617,"style":427},[158],[48,619,621,624],{"className":620},[163],[48,622,132],{"className":623,"style":343},[163,171],[48,625,627],{"className":626},[292],[48,628,630,650],{"className":629},[203,204],[48,631,633,647],{"className":632},[208],[48,634,636],{"className":635,"style":302},[212],[48,637,638,641],{"style":358},[48,639],{"className":640,"style":309},[220],[48,642,644],{"className":643},[225,226,227,228],[48,645,101],{"className":646,"style":235},[163,171,228],[48,648,269],{"className":649},[268],[48,651,653],{"className":652},[208],[48,654,656],{"className":655,"style":325},[212],[48,657],{},[48,659,81],{"className":660},[167],[48,662,84],{"className":663},[163,171],[48,665,87],{"className":666},[175]," 恒定且 ",[48,669,671,688],{"className":670,"translate":52},[56],[48,672,674],{"className":673},[60],[62,675,676],{"xmlns":64},[67,677,678,686],{},[70,679,680],{},[115,681,682,684],{},[73,683,494],{},[73,685,101],{},[142,687,499],{"encoding":144},[48,689,691],{"className":690,"ariaHidden":150},[149],[48,692,694,697],{"className":693},[154],[48,695],{"className":696,"style":509},[158],[48,698,700,703],{"className":699},[163],[48,701,494],{"className":702,"style":516},[163,171],[48,704,706],{"className":705},[292],[48,707,709,729],{"className":708},[203,204],[48,710,712,726],{"className":711},[208],[48,713,715],{"className":714,"style":302},[212],[48,716,717,720],{"style":531},[48,718],{"className":719,"style":309},[220],[48,721,723],{"className":722},[225,226,227,228],[48,724,101],{"className":725,"style":235},[163,171,228],[48,727,269],{"className":728},[268],[48,730,732],{"className":731},[208],[48,733,735],{"className":734,"style":325},[212],[48,736],{}," 一维的特例。",[11,739,740,741,770,771,840],{},"这篇论文想要处理的，恰恰是这个推广假说中最核心也最含糊的部分：特征 ",[48,742,744,757],{"className":743,"translate":52},[56],[48,745,747],{"className":746},[60],[62,748,749],{"xmlns":64},[67,750,751,755],{},[70,752,753],{},[73,754,101],{},[142,756,101],{"encoding":144},[48,758,760],{"className":759,"ariaHidden":150},[149],[48,761,763,767],{"className":762},[154],[48,764],{"className":765,"style":766},[158],"height:0.8889em;vertical-align:-0.1944em;",[48,768,101],{"className":769,"style":235},[163,171]," 在子空间 ",[48,772,774,791],{"className":773,"translate":52},[56],[48,775,777],{"className":776},[60],[62,778,779],{"xmlns":64},[67,780,781,789],{},[70,782,783],{},[115,784,785,787],{},[73,786,494],{},[73,788,101],{},[142,790,499],{"encoding":144},[48,792,794],{"className":793,"ariaHidden":150},[149],[48,795,797,800],{"className":796},[154],[48,798],{"className":799,"style":509},[158],[48,801,803,806],{"className":802},[163],[48,804,494],{"className":805,"style":516},[163,171],[48,807,809],{"className":808},[292],[48,810,812,832],{"className":811},[203,204],[48,813,815,829],{"className":814},[208],[48,816,818],{"className":817,"style":302},[212],[48,819,820,823],{"style":531},[48,821],{"className":822,"style":309},[220],[48,824,826],{"className":825},[225,226,227,228],[48,827,101],{"className":828,"style":235},[163,171,228],[48,830,269],{"className":831},[268],[48,833,835],{"className":834},[208],[48,836,838],{"className":837,"style":325},[212],[48,839],{}," 中以何种流形形态呈现，这个流形与「特征」本身之间又是什么关系？",[35,842,844],{"id":843},"特征即度量空间","「特征」即度量空间",[11,846,847],{},"论文中最优雅的一步，是先解决一个听起来有些哲学、实际上至关重要的问题：「特征」究竟是什么？",[11,849,850,851,991],{},"作者给出的答案出人意料地简洁：特征是一个度量空间 ",[48,852,854,887],{"className":853,"translate":52},[56],[48,855,857],{"className":856},[60],[62,858,859],{"xmlns":64},[67,860,861,884],{},[70,862,863,865,872,875,882],{},[78,864,81],{"stretchy":80},[115,866,867,870],{},[73,868,869],{},"Z",[73,871,101],{},[78,873,874],{"separator":150},",",[115,876,877,880],{},[73,878,879],{},"d",[73,881,101],{},[78,883,87],{"stretchy":80},[142,885,886],{"encoding":144},"(Z_f, d_f)",[48,888,890],{"className":889,"ariaHidden":150},[149],[48,891,893,896,899,941,945,948,988],{"className":892},[154],[48,894],{"className":895,"style":427},[158],[48,897,81],{"className":898},[167],[48,900,902,906],{"className":901},[163],[48,903,869],{"className":904,"style":905},[163,171],"margin-right:0.0715em;",[48,907,909],{"className":908},[292],[48,910,912,933],{"className":911},[203,204],[48,913,915,930],{"className":914},[208],[48,916,918],{"className":917,"style":302},[212],[48,919,921,924],{"style":920},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[48,922],{"className":923,"style":309},[220],[48,925,927],{"className":926},[225,226,227,228],[48,928,101],{"className":929,"style":235},[163,171,228],[48,931,269],{"className":932},[268],[48,934,936],{"className":935},[208],[48,937,939],{"className":938,"style":325},[212],[48,940],{},[48,942,874],{"className":943},[944],"mpunct",[48,946],{"className":947,"style":282},[179],[48,949,951,954],{"className":950},[163],[48,952,879],{"className":953},[163,171],[48,955,957],{"className":956},[292],[48,958,960,980],{"className":959},[203,204],[48,961,963,977],{"className":962},[208],[48,964,966],{"className":965,"style":302},[212],[48,967,968,971],{"style":305},[48,969],{"className":970,"style":309},[220],[48,972,974],{"className":973},[225,226,227,228],[48,975,101],{"className":976,"style":235},[163,171,228],[48,978,269],{"className":979},[268],[48,981,983],{"className":982},[208],[48,984,986],{"className":985,"style":325},[212],[48,987],{},[48,989,87],{"className":990},[175],"——一个集合连同定义在该集合上的某种「距离」概念。这个定义看似平淡，实则表现力极强：",[993,994,995,1070,1215],"ul",{},[996,997,998,999,1069],"li",{},"原子特征（是否有猫）：",[48,1000,1002,1020],{"className":1001,"translate":52},[56],[48,1003,1005],{"className":1004},[60],[62,1006,1007],{"xmlns":64},[67,1008,1009,1017],{},[70,1010,1011],{},[115,1012,1013,1015],{},[73,1014,869],{},[73,1016,101],{},[142,1018,1019],{"encoding":144},"Z_f",[48,1021,1023],{"className":1022,"ariaHidden":150},[149],[48,1024,1026,1029],{"className":1025},[154],[48,1027],{"className":1028,"style":509},[158],[48,1030,1032,1035],{"className":1031},[163],[48,1033,869],{"className":1034,"style":905},[163,171],[48,1036,1038],{"className":1037},[292],[48,1039,1041,1061],{"className":1040},[203,204],[48,1042,1044,1058],{"className":1043},[208],[48,1045,1047],{"className":1046,"style":302},[212],[48,1048,1049,1052],{"style":920},[48,1050],{"className":1051,"style":309},[220],[48,1053,1055],{"className":1054},[225,226,227,228],[48,1056,101],{"className":1057,"style":235},[163,171,228],[48,1059,269],{"className":1060},[268],[48,1062,1064],{"className":1063},[208],[48,1065,1067],{"className":1066,"style":325},[212],[48,1068],{}," 是单元素集合；",[996,1071,1072,1073,1142,1143,1214],{},"层次特征（动物的分类学树）：",[48,1074,1076,1093],{"className":1075,"translate":52},[56],[48,1077,1079],{"className":1078},[60],[62,1080,1081],{"xmlns":64},[67,1082,1083,1091],{},[70,1084,1085],{},[115,1086,1087,1089],{},[73,1088,869],{},[73,1090,101],{},[142,1092,1019],{"encoding":144},[48,1094,1096],{"className":1095,"ariaHidden":150},[149],[48,1097,1099,1102],{"className":1098},[154],[48,1100],{"className":1101,"style":509},[158],[48,1103,1105,1108],{"className":1104},[163],[48,1106,869],{"className":1107,"style":905},[163,171],[48,1109,1111],{"className":1110},[292],[48,1112,1114,1134],{"className":1113},[203,204],[48,1115,1117,1131],{"className":1116},[208],[48,1118,1120],{"className":1119,"style":302},[212],[48,1121,1122,1125],{"style":920},[48,1123],{"className":1124,"style":309},[220],[48,1126,1128],{"className":1127},[225,226,227,228],[48,1129,101],{"className":1130,"style":235},[163,171,228],[48,1132,269],{"className":1133},[268],[48,1135,1137],{"className":1136},[208],[48,1138,1140],{"className":1139,"style":325},[212],[48,1141],{}," 是离散集合，",[48,1144,1146,1164],{"className":1145,"translate":52},[56],[48,1147,1149],{"className":1148},[60],[62,1150,1151],{"xmlns":64},[67,1152,1153,1161],{},[70,1154,1155],{},[115,1156,1157,1159],{},[73,1158,879],{},[73,1160,101],{},[142,1162,1163],{"encoding":144},"d_f",[48,1165,1167],{"className":1166,"ariaHidden":150},[149],[48,1168,1170,1174],{"className":1169},[154],[48,1171],{"className":1172,"style":1173},[158],"height:0.9805em;vertical-align:-0.2861em;",[48,1175,1177,1180],{"className":1176},[163],[48,1178,879],{"className":1179},[163,171],[48,1181,1183],{"className":1182},[292],[48,1184,1186,1206],{"className":1185},[203,204],[48,1187,1189,1203],{"className":1188},[208],[48,1190,1192],{"className":1191,"style":302},[212],[48,1193,1194,1197],{"style":305},[48,1195],{"className":1196,"style":309},[220],[48,1198,1200],{"className":1199},[225,226,227,228],[48,1201,101],{"className":1202,"style":235},[163,171,228],[48,1204,269],{"className":1205},[268],[48,1207,1209],{"className":1208},[208],[48,1210,1212],{"className":1211,"style":325},[212],[48,1213],{}," 是其上的树距离；",[996,1216,1217,1218,1287],{},"连续特征（颜色色相、一年中的第几天、日历年）：",[48,1219,1221,1238],{"className":1220,"translate":52},[56],[48,1222,1224],{"className":1223},[60],[62,1225,1226],{"xmlns":64},[67,1227,1228,1236],{},[70,1229,1230],{},[115,1231,1232,1234],{},[73,1233,869],{},[73,1235,101],{},[142,1237,1019],{"encoding":144},[48,1239,1241],{"className":1240,"ariaHidden":150},[149],[48,1242,1244,1247],{"className":1243},[154],[48,1245],{"className":1246,"style":509},[158],[48,1248,1250,1253],{"className":1249},[163],[48,1251,869],{"className":1252,"style":905},[163,171],[48,1254,1256],{"className":1255},[292],[48,1257,1259,1279],{"className":1258},[203,204],[48,1260,1262,1276],{"className":1261},[208],[48,1263,1265],{"className":1264,"style":302},[212],[48,1266,1267,1270],{"style":920},[48,1268],{"className":1269,"style":309},[220],[48,1271,1273],{"className":1272},[225,226,227,228],[48,1274,101],{"className":1275,"style":235},[163,171,228],[48,1277,269],{"className":1278},[268],[48,1280,1282],{"className":1281},[208],[48,1283,1285],{"className":1284,"style":325},[212],[48,1286],{}," 可以是区间、圆或更高维的欧几里得空间。",[11,1289,1290],{},"这一框架比学习理论中常见的「欧几里得空间」或「超球面」假设更灵活，又比解耦文献中带群结构的黎曼流形简单得多、也更易于处理。这是典型的数学家式选择：在表达力与可处理性之间取得恰当的平衡。",[11,1292,1293],{},"在「特征即度量空间」这一定义就位后，作者提出了论文的第一个核心假说：",[1295,1296,1297],"blockquote",{},[11,1298,1299,1300,1388,1389,1473,1474,1688,1689,1906],{},"假说 1（连续对应假说）。特征值 ",[48,1301,1303,1328],{"className":1302,"translate":52},[56],[48,1304,1306],{"className":1305},[60],[62,1307,1308],{"xmlns":64},[67,1309,1310,1325],{},[70,1311,1312,1319,1321,1323],{},[115,1313,1314,1317],{},[73,1315,1316],{},"z",[73,1318,101],{},[78,1320,81],{"stretchy":80},[73,1322,84],{},[78,1324,87],{"stretchy":80},[142,1326,1327],{"encoding":144},"z_f(x)",[48,1329,1331],{"className":1330,"ariaHidden":150},[149],[48,1332,1334,1337,1379,1382,1385],{"className":1333},[154],[48,1335],{"className":1336,"style":427},[158],[48,1338,1340,1344],{"className":1339},[163],[48,1341,1316],{"className":1342,"style":1343},[163,171],"margin-right:0.044em;",[48,1345,1347],{"className":1346},[292],[48,1348,1350,1371],{"className":1349},[203,204],[48,1351,1353,1368],{"className":1352},[208],[48,1354,1356],{"className":1355,"style":302},[212],[48,1357,1359,1362],{"style":1358},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[48,1360],{"className":1361,"style":309},[220],[48,1363,1365],{"className":1364},[225,226,227,228],[48,1366,101],{"className":1367,"style":235},[163,171,228],[48,1369,269],{"className":1370},[268],[48,1372,1374],{"className":1373},[208],[48,1375,1377],{"className":1376,"style":325},[212],[48,1378],{},[48,1380,81],{"className":1381},[167],[48,1383,84],{"className":1384},[163,171],[48,1386,87],{"className":1387},[175]," 与表示方向 ",[48,1390,1392,1415],{"className":1391,"translate":52},[56],[48,1393,1395],{"className":1394},[60],[62,1396,1397],{"xmlns":64},[67,1398,1399,1413],{},[70,1400,1401,1407,1409,1411],{},[115,1402,1403,1405],{},[73,1404,132],{},[73,1406,101],{},[78,1408,81],{"stretchy":80},[73,1410,84],{},[78,1412,87],{"stretchy":80},[142,1414,417],{"encoding":144},[48,1416,1418],{"className":1417,"ariaHidden":150},[149],[48,1419,1421,1424,1464,1467,1470],{"className":1420},[154],[48,1422],{"className":1423,"style":427},[158],[48,1425,1427,1430],{"className":1426},[163],[48,1428,132],{"className":1429,"style":343},[163,171],[48,1431,1433],{"className":1432},[292],[48,1434,1436,1456],{"className":1435},[203,204],[48,1437,1439,1453],{"className":1438},[208],[48,1440,1442],{"className":1441,"style":302},[212],[48,1443,1444,1447],{"style":358},[48,1445],{"className":1446,"style":309},[220],[48,1448,1450],{"className":1449},[225,226,227,228],[48,1451,101],{"className":1452,"style":235},[163,171,228],[48,1454,269],{"className":1455},[268],[48,1457,1459],{"className":1458},[208],[48,1460,1462],{"className":1461,"style":325},[212],[48,1463],{},[48,1465,81],{"className":1466},[167],[48,1468,84],{"className":1469},[163,171],[48,1471,87],{"className":1472},[175]," 之间存在连续、可逆的一一对应；即存在连续映射 ",[48,1475,1477,1526],{"className":1476,"translate":52},[56],[48,1478,1480],{"className":1479},[60],[62,1481,1482],{"xmlns":64},[67,1483,1484,1523],{},[70,1485,1486,1493,1496,1502,1505],{},[115,1487,1488,1491],{},[73,1489,1490],{},"ϕ",[73,1492,101],{},[78,1494,1495],{},":",[115,1497,1498,1500],{},[73,1499,869],{},[73,1501,101],{},[78,1503,1504],{},"→",[1506,1507,1508,1511],"msup",{},[73,1509,1510],{},"S",[70,1512,1513,1516,1519],{},[73,1514,1515],{},"D",[78,1517,1518],{},"−",[1520,1521,1522],"mn",{},"1",[142,1524,1525],{"encoding":144},"\\phi_f: Z_f \\to S^{D-1}",[48,1527,1529,1584,1639],{"className":1528,"ariaHidden":150},[149],[48,1530,1532,1535,1575,1578,1581],{"className":1531},[154],[48,1533],{"className":1534,"style":1173},[158],[48,1536,1538,1541],{"className":1537},[163],[48,1539,1490],{"className":1540},[163,171],[48,1542,1544],{"className":1543},[292],[48,1545,1547,1567],{"className":1546},[203,204],[48,1548,1550,1564],{"className":1549},[208],[48,1551,1553],{"className":1552,"style":302},[212],[48,1554,1555,1558],{"style":305},[48,1556],{"className":1557,"style":309},[220],[48,1559,1561],{"className":1560},[225,226,227,228],[48,1562,101],{"className":1563,"style":235},[163,171,228],[48,1565,269],{"className":1566},[268],[48,1568,1570],{"className":1569},[208],[48,1571,1573],{"className":1572,"style":325},[212],[48,1574],{},[48,1576],{"className":1577,"style":180},[179],[48,1579,1495],{"className":1580},[184],[48,1582],{"className":1583,"style":180},[179],[48,1585,1587,1590,1630,1633,1636],{"className":1586},[154],[48,1588],{"className":1589,"style":509},[158],[48,1591,1593,1596],{"className":1592},[163],[48,1594,869],{"className":1595,"style":905},[163,171],[48,1597,1599],{"className":1598},[292],[48,1600,1602,1622],{"className":1601},[203,204],[48,1603,1605,1619],{"className":1604},[208],[48,1606,1608],{"className":1607,"style":302},[212],[48,1609,1610,1613],{"style":920},[48,1611],{"className":1612,"style":309},[220],[48,1614,1616],{"className":1615},[225,226,227,228],[48,1617,101],{"className":1618,"style":235},[163,171,228],[48,1620,269],{"className":1621},[268],[48,1623,1625],{"className":1624},[208],[48,1626,1628],{"className":1627,"style":325},[212],[48,1629],{},[48,1631],{"className":1632,"style":180},[179],[48,1634,1504],{"className":1635},[184],[48,1637],{"className":1638,"style":180},[179],[48,1640,1642,1646],{"className":1641},[154],[48,1643],{"className":1644,"style":1645},[158],"height:0.8413em;",[48,1647,1649,1653],{"className":1648},[163],[48,1650,1510],{"className":1651,"style":1652},[163,171],"margin-right:0.0576em;",[48,1654,1656],{"className":1655},[292],[48,1657,1659],{"className":1658},[203],[48,1660,1662],{"className":1661},[208],[48,1663,1665],{"className":1664,"style":1645},[212],[48,1666,1668,1671],{"style":1667},"top:-3.063em;margin-right:0.05em;",[48,1669],{"className":1670,"style":309},[220],[48,1672,1674],{"className":1673},[225,226,227,228],[48,1675,1677,1681,1685],{"className":1676},[163,228],[48,1678,1515],{"className":1679,"style":1680},[163,171,228],"margin-right:0.0278em;",[48,1682,1518],{"className":1683},[1684,228],"mbin",[48,1686,1522],{"className":1687},[163,228],"（映到单位超球面），使得 ",[48,1690,1692,1740],{"className":1691,"translate":52},[56],[48,1693,1695],{"className":1694},[60],[62,1696,1697],{"xmlns":64},[67,1698,1699,1737],{},[70,1700,1701,1707,1709,1711,1713,1715,1721,1723,1729,1731,1733,1735],{},[115,1702,1703,1705],{},[73,1704,132],{},[73,1706,101],{},[78,1708,81],{"stretchy":80},[73,1710,84],{},[78,1712,87],{"stretchy":80},[78,1714,90],{},[115,1716,1717,1719],{},[73,1718,1490],{},[73,1720,101],{},[78,1722,81],{"stretchy":80},[115,1724,1725,1727],{},[73,1726,1316],{},[73,1728,101],{},[78,1730,81],{"stretchy":80},[73,1732,84],{},[78,1734,87],{"stretchy":80},[78,1736,87],{"stretchy":80},[142,1738,1739],{"encoding":144},"v_f(x) = \\phi_f(z_f(x))",[48,1741,1743,1807],{"className":1742,"ariaHidden":150},[149],[48,1744,1746,1749,1789,1792,1795,1798,1801,1804],{"className":1745},[154],[48,1747],{"className":1748,"style":427},[158],[48,1750,1752,1755],{"className":1751},[163],[48,1753,132],{"className":1754,"style":343},[163,171],[48,1756,1758],{"className":1757},[292],[48,1759,1761,1781],{"className":1760},[203,204],[48,1762,1764,1778],{"className":1763},[208],[48,1765,1767],{"className":1766,"style":302},[212],[48,1768,1769,1772],{"style":358},[48,1770],{"className":1771,"style":309},[220],[48,1773,1775],{"className":1774},[225,226,227,228],[48,1776,101],{"className":1777,"style":235},[163,171,228],[48,1779,269],{"className":1780},[268],[48,1782,1784],{"className":1783},[208],[48,1785,1787],{"className":1786,"style":325},[212],[48,1788],{},[48,1790,81],{"className":1791},[167],[48,1793,84],{"className":1794},[163,171],[48,1796,87],{"className":1797},[175],[48,1799],{"className":1800,"style":180},[179],[48,1802,90],{"className":1803},[184],[48,1805],{"className":1806,"style":180},[179],[48,1808,1810,1813,1853,1856,1896,1899,1902],{"className":1809},[154],[48,1811],{"className":1812,"style":427},[158],[48,1814,1816,1819],{"className":1815},[163],[48,1817,1490],{"className":1818},[163,171],[48,1820,1822],{"className":1821},[292],[48,1823,1825,1845],{"className":1824},[203,204],[48,1826,1828,1842],{"className":1827},[208],[48,1829,1831],{"className":1830,"style":302},[212],[48,1832,1833,1836],{"style":305},[48,1834],{"className":1835,"style":309},[220],[48,1837,1839],{"className":1838},[225,226,227,228],[48,1840,101],{"className":1841,"style":235},[163,171,228],[48,1843,269],{"className":1844},[268],[48,1846,1848],{"className":1847},[208],[48,1849,1851],{"className":1850,"style":325},[212],[48,1852],{},[48,1854,81],{"className":1855},[167],[48,1857,1859,1862],{"className":1858},[163],[48,1860,1316],{"className":1861,"style":1343},[163,171],[48,1863,1865],{"className":1864},[292],[48,1866,1868,1888],{"className":1867},[203,204],[48,1869,1871,1885],{"className":1870},[208],[48,1872,1874],{"className":1873,"style":302},[212],[48,1875,1876,1879],{"style":1358},[48,1877],{"className":1878,"style":309},[220],[48,1880,1882],{"className":1881},[225,226,227,228],[48,1883,101],{"className":1884,"style":235},[163,171,228],[48,1886,269],{"className":1887},[268],[48,1889,1891],{"className":1890},[208],[48,1892,1894],{"className":1893,"style":325},[212],[48,1895],{},[48,1897,81],{"className":1898},[167],[48,1900,84],{"className":1901},[163,171],[48,1903,1905],{"className":1904},[175],"))","。",[11,1908,1909,1910,1979,1980,2050,2051,2124,2125,2194,2195,2264,2265,2334,2335,2404,2405,2474,2475,2544],{},"结合 ",[48,1911,1913,1930],{"className":1912,"translate":52},[56],[48,1914,1916],{"className":1915},[60],[62,1917,1918],{"xmlns":64},[67,1919,1920,1928],{},[70,1921,1922],{},[115,1923,1924,1926],{},[73,1925,869],{},[73,1927,101],{},[142,1929,1019],{"encoding":144},[48,1931,1933],{"className":1932,"ariaHidden":150},[149],[48,1934,1936,1939],{"className":1935},[154],[48,1937],{"className":1938,"style":509},[158],[48,1940,1942,1945],{"className":1941},[163],[48,1943,869],{"className":1944,"style":905},[163,171],[48,1946,1948],{"className":1947},[292],[48,1949,1951,1971],{"className":1950},[203,204],[48,1952,1954,1968],{"className":1953},[208],[48,1955,1957],{"className":1956,"style":302},[212],[48,1958,1959,1962],{"style":920},[48,1960],{"className":1961,"style":309},[220],[48,1963,1965],{"className":1964},[225,226,227,228],[48,1966,101],{"className":1967,"style":235},[163,171,228],[48,1969,269],{"className":1970},[268],[48,1972,1974],{"className":1973},[208],[48,1975,1977],{"className":1976,"style":325},[212],[48,1978],{}," 是紧空间的这一技术前提，该假说立即导出一个干净的推论（命题 1）：",[48,1981,1983,2001],{"className":1982,"translate":52},[56],[48,1984,1986],{"className":1985},[60],[62,1987,1988],{"xmlns":64},[67,1989,1990,1998],{},[70,1991,1992],{},[115,1993,1994,1996],{},[73,1995,1490],{},[73,1997,101],{},[142,1999,2000],{"encoding":144},"\\phi_f",[48,2002,2004],{"className":2003,"ariaHidden":150},[149],[48,2005,2007,2010],{"className":2006},[154],[48,2008],{"className":2009,"style":1173},[158],[48,2011,2013,2016],{"className":2012},[163],[48,2014,1490],{"className":2015},[163,171],[48,2017,2019],{"className":2018},[292],[48,2020,2022,2042],{"className":2021},[203,204],[48,2023,2025,2039],{"className":2024},[208],[48,2026,2028],{"className":2027,"style":302},[212],[48,2029,2030,2033],{"style":305},[48,2031],{"className":2032,"style":309},[220],[48,2034,2036],{"className":2035},[225,226,227,228],[48,2037,101],{"className":2038,"style":235},[163,171,228],[48,2040,269],{"className":2041},[268],[48,2043,2045],{"className":2044},[208],[48,2046,2048],{"className":2047,"style":325},[212],[48,2049],{}," 是一个同胚。换言之，表征流形 ",[48,2052,2054,2073],{"className":2053,"translate":52},[56],[48,2055,2057],{"className":2056},[60],[62,2058,2059],{"xmlns":64},[67,2060,2061,2070],{},[70,2062,2063],{},[115,2064,2065,2068],{},[73,2066,2067],{},"M",[73,2069,101],{},[142,2071,2072],{"encoding":144},"M_f",[48,2074,2076],{"className":2075,"ariaHidden":150},[149],[48,2077,2079,2082],{"className":2078},[154],[48,2080],{"className":2081,"style":509},[158],[48,2083,2085,2089],{"className":2084},[163],[48,2086,2067],{"className":2087,"style":2088},[163,171],"margin-right:0.109em;",[48,2090,2092],{"className":2091},[292],[48,2093,2095,2116],{"className":2094},[203,204],[48,2096,2098,2113],{"className":2097},[208],[48,2099,2101],{"className":2100,"style":302},[212],[48,2102,2104,2107],{"style":2103},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[48,2105],{"className":2106,"style":309},[220],[48,2108,2110],{"className":2109},[225,226,227,228],[48,2111,101],{"className":2112,"style":235},[163,171,228],[48,2114,269],{"className":2115},[268],[48,2117,2119],{"className":2118},[208],[48,2120,2122],{"className":2121,"style":325},[212],[48,2123],{},"（即 ",[48,2126,2128,2145],{"className":2127,"translate":52},[56],[48,2129,2131],{"className":2130},[60],[62,2132,2133],{"xmlns":64},[67,2134,2135,2143],{},[70,2136,2137],{},[115,2138,2139,2141],{},[73,2140,1490],{},[73,2142,101],{},[142,2144,2000],{"encoding":144},[48,2146,2148],{"className":2147,"ariaHidden":150},[149],[48,2149,2151,2154],{"className":2150},[154],[48,2152],{"className":2153,"style":1173},[158],[48,2155,2157,2160],{"className":2156},[163],[48,2158,1490],{"className":2159},[163,171],[48,2161,2163],{"className":2162},[292],[48,2164,2166,2186],{"className":2165},[203,204],[48,2167,2169,2183],{"className":2168},[208],[48,2170,2172],{"className":2171,"style":302},[212],[48,2173,2174,2177],{"style":305},[48,2175],{"className":2176,"style":309},[220],[48,2178,2180],{"className":2179},[225,226,227,228],[48,2181,101],{"className":2182,"style":235},[163,171,228],[48,2184,269],{"className":2185},[268],[48,2187,2189],{"className":2188},[208],[48,2190,2192],{"className":2191,"style":325},[212],[48,2193],{}," 的像）与原始特征空间 ",[48,2196,2198,2215],{"className":2197,"translate":52},[56],[48,2199,2201],{"className":2200},[60],[62,2202,2203],{"xmlns":64},[67,2204,2205,2213],{},[70,2206,2207],{},[115,2208,2209,2211],{},[73,2210,869],{},[73,2212,101],{},[142,2214,1019],{"encoding":144},[48,2216,2218],{"className":2217,"ariaHidden":150},[149],[48,2219,2221,2224],{"className":2220},[154],[48,2222],{"className":2223,"style":509},[158],[48,2225,2227,2230],{"className":2226},[163],[48,2228,869],{"className":2229,"style":905},[163,171],[48,2231,2233],{"className":2232},[292],[48,2234,2236,2256],{"className":2235},[203,204],[48,2237,2239,2253],{"className":2238},[208],[48,2240,2242],{"className":2241,"style":302},[212],[48,2243,2244,2247],{"style":920},[48,2245],{"className":2246,"style":309},[220],[48,2248,2250],{"className":2249},[225,226,227,228],[48,2251,101],{"className":2252,"style":235},[163,171,228],[48,2254,269],{"className":2255},[268],[48,2257,2259],{"className":2258},[208],[48,2260,2262],{"className":2261,"style":325},[212],[48,2263],{}," 在拓扑上「形状相同」：若 ",[48,2266,2268,2285],{"className":2267,"translate":52},[56],[48,2269,2271],{"className":2270},[60],[62,2272,2273],{"xmlns":64},[67,2274,2275,2283],{},[70,2276,2277],{},[115,2278,2279,2281],{},[73,2280,869],{},[73,2282,101],{},[142,2284,1019],{"encoding":144},[48,2286,2288],{"className":2287,"ariaHidden":150},[149],[48,2289,2291,2294],{"className":2290},[154],[48,2292],{"className":2293,"style":509},[158],[48,2295,2297,2300],{"className":2296},[163],[48,2298,869],{"className":2299,"style":905},[163,171],[48,2301,2303],{"className":2302},[292],[48,2304,2306,2326],{"className":2305},[203,204],[48,2307,2309,2323],{"className":2308},[208],[48,2310,2312],{"className":2311,"style":302},[212],[48,2313,2314,2317],{"style":920},[48,2315],{"className":2316,"style":309},[220],[48,2318,2320],{"className":2319},[225,226,227,228],[48,2321,101],{"className":2322,"style":235},[163,171,228],[48,2324,269],{"className":2325},[268],[48,2327,2329],{"className":2328},[208],[48,2330,2332],{"className":2331,"style":325},[212],[48,2333],{}," 是区间，",[48,2336,2338,2355],{"className":2337,"translate":52},[56],[48,2339,2341],{"className":2340},[60],[62,2342,2343],{"xmlns":64},[67,2344,2345,2353],{},[70,2346,2347],{},[115,2348,2349,2351],{},[73,2350,2067],{},[73,2352,101],{},[142,2354,2072],{"encoding":144},[48,2356,2358],{"className":2357,"ariaHidden":150},[149],[48,2359,2361,2364],{"className":2360},[154],[48,2362],{"className":2363,"style":509},[158],[48,2365,2367,2370],{"className":2366},[163],[48,2368,2067],{"className":2369,"style":2088},[163,171],[48,2371,2373],{"className":2372},[292],[48,2374,2376,2396],{"className":2375},[203,204],[48,2377,2379,2393],{"className":2378},[208],[48,2380,2382],{"className":2381,"style":302},[212],[48,2383,2384,2387],{"style":2103},[48,2385],{"className":2386,"style":309},[220],[48,2388,2390],{"className":2389},[225,226,227,228],[48,2391,101],{"className":2392,"style":235},[163,171,228],[48,2394,269],{"className":2395},[268],[48,2397,2399],{"className":2398},[208],[48,2400,2402],{"className":2401,"style":325},[212],[48,2403],{}," 就是一条曲线；若 ",[48,2406,2408,2425],{"className":2407,"translate":52},[56],[48,2409,2411],{"className":2410},[60],[62,2412,2413],{"xmlns":64},[67,2414,2415,2423],{},[70,2416,2417],{},[115,2418,2419,2421],{},[73,2420,869],{},[73,2422,101],{},[142,2424,1019],{"encoding":144},[48,2426,2428],{"className":2427,"ariaHidden":150},[149],[48,2429,2431,2434],{"className":2430},[154],[48,2432],{"className":2433,"style":509},[158],[48,2435,2437,2440],{"className":2436},[163],[48,2438,869],{"className":2439,"style":905},[163,171],[48,2441,2443],{"className":2442},[292],[48,2444,2446,2466],{"className":2445},[203,204],[48,2447,2449,2463],{"className":2448},[208],[48,2450,2452],{"className":2451,"style":302},[212],[48,2453,2454,2457],{"style":920},[48,2455],{"className":2456,"style":309},[220],[48,2458,2460],{"className":2459},[225,226,227,228],[48,2461,101],{"className":2462,"style":235},[163,171,228],[48,2464,269],{"className":2465},[268],[48,2467,2469],{"className":2468},[208],[48,2470,2472],{"className":2471,"style":325},[212],[48,2473],{}," 是圆，",[48,2476,2478,2495],{"className":2477,"translate":52},[56],[48,2479,2481],{"className":2480},[60],[62,2482,2483],{"xmlns":64},[67,2484,2485,2493],{},[70,2486,2487],{},[115,2488,2489,2491],{},[73,2490,2067],{},[73,2492,101],{},[142,2494,2072],{"encoding":144},[48,2496,2498],{"className":2497,"ariaHidden":150},[149],[48,2499,2501,2504],{"className":2500},[154],[48,2502],{"className":2503,"style":509},[158],[48,2505,2507,2510],{"className":2506},[163],[48,2508,2067],{"className":2509,"style":2088},[163,171],[48,2511,2513],{"className":2512},[292],[48,2514,2516,2536],{"className":2515},[203,204],[48,2517,2519,2533],{"className":2518},[208],[48,2520,2522],{"className":2521,"style":302},[212],[48,2523,2524,2527],{"style":2103},[48,2525],{"className":2526,"style":309},[220],[48,2528,2530],{"className":2529},[225,226,227,228],[48,2531,101],{"className":2532,"style":235},[163,171,228],[48,2534,269],{"className":2535},[268],[48,2537,2539],{"className":2538},[208],[48,2540,2542],{"className":2541,"style":325},[212],[48,2543],{}," 就是一个圈；连通分量、孔洞、分叉点——所有这些拓扑性质都被忠实地保留下来。",[11,2546,2547],{},"这一步的意义在于，它第一次用严格的数学语言把「流形长什么样」与「概念本身的结构」联系起来，而不再停留在「看起来像个圆」这种直观描述上。",[35,2549,2550],{"id":2550},"实证验证",[11,2552,2553],{},"再优雅的理论也必须接受数据的检验。作者选取了三个有代表性的案例：",[2555,2556,2557,2565,2568],"ol",{},[996,2558,2559,2560,2564],{},"颜色：用 OpenAI 的 ",[2561,2562,2563],"code",{},"text-embedding-large-3"," 从英文颜色名称生成 3072 维嵌入，再用 PCA 降到三维用于可视化；",[996,2566,2567],{},"年份：沿用 Engels 等（2025）的做法，用 SAE 从 GPT-2-small 的第 7 层提取「20 世纪年份」特征；",[996,2569,2570],{},"日期：用「1 月 1 日」到「12 月 31 日」等提示词生成嵌入，同样使用 OpenAI 的嵌入模型。",[11,2572,2573],{},"结果令人瞩目：颜色嵌入沿一个圆环排列，色相次序（红→紫→蓝→绿→黄→橙→红）与标准色环完全吻合。年份的 token 激活在三维空间中描出一条蜿蜒的曲线，隐约唤起人类对「时间线」的直觉。作者进一步用 k-近邻图估计了流形上的序结构，并与真实年份计算秩相关，得到肯德尔相关系数 0.97、斯皮尔曼相关系数超过 0.99——近乎完美的单调对应，为同胚预测提供了有力支持。",[11,2575,2576],{},"论文里还有一个特别耐人寻味的「陷阱」细节：在 Engels 等人的原始工作中，「星期几」和「月份」的表示投影到前两个主成分上时，呈现出整齐的圆。作者指出，一旦考察第三个主成分就会发现，这个「圆」其实在第三维上持续地扭曲缠绕；被压进二维平面的整齐圆是一种视觉假象（见其图 2）。这一观察是一个告诫：低维投影作为可解释性研究不可或缺的可视化工具，同时也可能误导人。",[35,2578,2579],{"id":2579},"计算表达能力角度",[11,2581,2582],{},"至此一个自然的问题浮现出来：既然「年份」本质上是一维的量，模型为什么不在表征空间中直接把它编码成一条直线段，而要扭曲成一条穿过高维空间的曲线呢？",[11,2584,2585,2586,2614,2615,2691,2692,2820,2821,2849,2850,2911,2912,3071,3072,3116,3117,3178],{},"作者给出的答案既直观又带着数学家的典型气质：表达能力。如果目标仅仅是通过线性投影「读出」 ",[48,2587,2589,2602],{"className":2588,"translate":52},[56],[48,2590,2592],{"className":2591},[60],[62,2593,2594],{"xmlns":64},[67,2595,2596,2600],{},[70,2597,2598],{},[73,2599,1316],{},[142,2601,1316],{"encoding":144},[48,2603,2605],{"className":2604,"ariaHidden":150},[149],[48,2606,2608,2611],{"className":2607},[154],[48,2609],{"className":2610,"style":578},[158],[48,2612,1316],{"className":2613,"style":1343},[163,171]," 本身（即让恒等函数 ",[48,2616,2618,2647],{"className":2617,"translate":52},[56],[48,2619,2621],{"className":2620},[60],[62,2622,2623],{"xmlns":64},[67,2624,2625,2644],{},[70,2626,2627,2634,2636,2638,2640,2642],{},[70,2628,2629,2632],{},[73,2630,2631],{"mathvariant":75},"i",[73,2633,879],{"mathvariant":75},[78,2635,81],{"stretchy":80},[73,2637,1316],{},[78,2639,87],{"stretchy":80},[78,2641,90],{},[73,2643,1316],{},[142,2645,2646],{"encoding":144},"\\mathrm{id}(z) = z",[48,2648,2650,2682],{"className":2649,"ariaHidden":150},[149],[48,2651,2653,2656,2664,2667,2670,2673,2676,2679],{"className":2652},[154],[48,2654],{"className":2655,"style":159},[158],[48,2657,2659],{"className":2658},[163],[48,2660,2663],{"className":2661},[163,2662],"mathrm","id",[48,2665,81],{"className":2666},[167],[48,2668,1316],{"className":2669,"style":1343},[163,171],[48,2671,87],{"className":2672},[175],[48,2674],{"className":2675,"style":180},[179],[48,2677,90],{"className":2678},[184],[48,2680],{"className":2681,"style":180},[179],[48,2683,2685,2688],{"className":2684},[154],[48,2686],{"className":2687,"style":578},[158],[48,2689,1316],{"className":2690,"style":1343},[163,171]," 能通过一次线性运算算出），那么两个正交方向 ",[48,2693,2695,2722],{"className":2694,"translate":52},[56],[48,2696,2698],{"className":2697},[60],[62,2699,2700],{"xmlns":64},[67,2701,2702,2719],{},[70,2703,2704,2711,2713],{},[115,2705,2706,2708],{},[73,2707,132],{},[1520,2709,2710],{},"0",[78,2712,874],{"separator":150},[115,2714,2715,2717],{},[73,2716,132],{},[1520,2718,1522],{},[142,2720,2721],{"encoding":144},"v_0, v_1",[48,2723,2725],{"className":2724,"ariaHidden":150},[149],[48,2726,2728,2732,2774,2777,2780],{"className":2727},[154],[48,2729],{"className":2730,"style":2731},[158],"height:0.625em;vertical-align:-0.1944em;",[48,2733,2735,2738],{"className":2734},[163],[48,2736,132],{"className":2737,"style":343},[163,171],[48,2739,2741],{"className":2740},[292],[48,2742,2744,2765],{"className":2743},[203,204],[48,2745,2747,2762],{"className":2746},[208],[48,2748,2751],{"className":2749,"style":2750},[212],"height:0.3011em;",[48,2752,2753,2756],{"style":358},[48,2754],{"className":2755,"style":309},[220],[48,2757,2759],{"className":2758},[225,226,227,228],[48,2760,2710],{"className":2761},[163,228],[48,2763,269],{"className":2764},[268],[48,2766,2768],{"className":2767},[208],[48,2769,2772],{"className":2770,"style":2771},[212],"height:0.15em;",[48,2773],{},[48,2775,874],{"className":2776},[944],[48,2778],{"className":2779,"style":282},[179],[48,2781,2783,2786],{"className":2782},[163],[48,2784,132],{"className":2785,"style":343},[163,171],[48,2787,2789],{"className":2788},[292],[48,2790,2792,2812],{"className":2791},[203,204],[48,2793,2795,2809],{"className":2794},[208],[48,2796,2798],{"className":2797,"style":2750},[212],[48,2799,2800,2803],{"style":358},[48,2801],{"className":2802,"style":309},[220],[48,2804,2806],{"className":2805},[225,226,227,228],[48,2807,1522],{"className":2808},[163,228],[48,2810,269],{"className":2811},[268],[48,2813,2815],{"className":2814},[208],[48,2816,2818],{"className":2817,"style":2771},[212],[48,2819],{}," 就足够了。但如果还希望让 ",[48,2822,2824,2837],{"className":2823,"translate":52},[56],[48,2825,2827],{"className":2826},[60],[62,2828,2829],{"xmlns":64},[67,2830,2831,2835],{},[70,2832,2833],{},[73,2834,1316],{},[142,2836,1316],{"encoding":144},[48,2838,2840],{"className":2839,"ariaHidden":150},[149],[48,2841,2843,2846],{"className":2842},[154],[48,2844],{"className":2845,"style":578},[158],[48,2847,1316],{"className":2848,"style":1343},[163,171]," 的高阶多项式（如 ",[48,2851,2853,2872],{"className":2852,"translate":52},[56],[48,2854,2856],{"className":2855},[60],[62,2857,2858],{"xmlns":64},[67,2859,2860,2869],{},[70,2861,2862],{},[1506,2863,2864,2866],{},[73,2865,1316],{},[1520,2867,2868],{},"2",[142,2870,2871],{"encoding":144},"z^2",[48,2873,2875],{"className":2874,"ariaHidden":150},[149],[48,2876,2878,2882],{"className":2877},[154],[48,2879],{"className":2880,"style":2881},[158],"height:0.8141em;",[48,2883,2885,2888],{"className":2884},[163],[48,2886,1316],{"className":2887,"style":1343},[163,171],[48,2889,2891],{"className":2890},[292],[48,2892,2894],{"className":2893},[203],[48,2895,2897],{"className":2896},[208],[48,2898,2900],{"className":2899,"style":2881},[212],[48,2901,2902,2905],{"style":1667},[48,2903],{"className":2904,"style":309},[220],[48,2906,2908],{"className":2907},[225,226,227,228],[48,2909,2868],{"className":2910},[163,228],"）也能被线性读出，就需要更多的正交方向 ",[48,2913,2915,2953],{"className":2914,"translate":52},[56],[48,2916,2918],{"className":2917},[60],[62,2919,2920],{"xmlns":64},[67,2921,2922,2950],{},[70,2923,2924,2930,2932,2935,2937],{},[115,2925,2926,2928],{},[73,2927,132],{},[1520,2929,2710],{},[78,2931,874],{"separator":150},[78,2933,2934],{},"…",[78,2936,874],{"separator":150},[115,2938,2939,2941],{},[73,2940,132],{},[70,2942,2943,2945,2948],{},[73,2944,11],{},[78,2946,2947],{},"+",[1520,2949,1522],{},[142,2951,2952],{"encoding":144},"v_0, \\dots, v_{p+1}",[48,2954,2956],{"className":2955,"ariaHidden":150},[149],[48,2957,2959,2963,3003,3006,3009,3013,3016,3019,3022],{"className":2958},[154],[48,2960],{"className":2961,"style":2962},[158],"height:0.7167em;vertical-align:-0.2861em;",[48,2964,2966,2969],{"className":2965},[163],[48,2967,132],{"className":2968,"style":343},[163,171],[48,2970,2972],{"className":2971},[292],[48,2973,2975,2995],{"className":2974},[203,204],[48,2976,2978,2992],{"className":2977},[208],[48,2979,2981],{"className":2980,"style":2750},[212],[48,2982,2983,2986],{"style":358},[48,2984],{"className":2985,"style":309},[220],[48,2987,2989],{"className":2988},[225,226,227,228],[48,2990,2710],{"className":2991},[163,228],[48,2993,269],{"className":2994},[268],[48,2996,2998],{"className":2997},[208],[48,2999,3001],{"className":3000,"style":2771},[212],[48,3002],{},[48,3004,874],{"className":3005},[944],[48,3007],{"className":3008,"style":282},[179],[48,3010,2934],{"className":3011},[3012],"minner",[48,3014],{"className":3015,"style":282},[179],[48,3017,874],{"className":3018},[944],[48,3020],{"className":3021,"style":282},[179],[48,3023,3025,3028],{"className":3024},[163],[48,3026,132],{"className":3027,"style":343},[163,171],[48,3029,3031],{"className":3030},[292],[48,3032,3034,3063],{"className":3033},[203,204],[48,3035,3037,3060],{"className":3036},[208],[48,3038,3040],{"className":3039,"style":2750},[212],[48,3041,3042,3045],{"style":358},[48,3043],{"className":3044,"style":309},[220],[48,3046,3048],{"className":3047},[225,226,227,228],[48,3049,3051,3054,3057],{"className":3050},[163,228],[48,3052,11],{"className":3053},[163,171,228],[48,3055,2947],{"className":3056},[1684,228],[48,3058,1522],{"className":3059},[163,228],[48,3061,269],{"className":3062},[268],[48,3064,3066],{"className":3065},[208],[48,3067,3069],{"className":3068,"style":325},[212],[48,3070],{},"，相应地 ",[48,3073,3075,3095],{"className":3074,"translate":52},[56],[48,3076,3078],{"className":3077},[60],[62,3079,3080],{"xmlns":64},[67,3081,3082,3092],{},[70,3083,3084,3086,3088,3090],{},[73,3085,1490],{},[78,3087,81],{"stretchy":80},[73,3089,1316],{},[78,3091,87],{"stretchy":80},[142,3093,3094],{"encoding":144},"\\phi(z)",[48,3096,3098],{"className":3097,"ariaHidden":150},[149],[48,3099,3101,3104,3107,3110,3113],{"className":3100},[154],[48,3102],{"className":3103,"style":159},[158],[48,3105,1490],{"className":3106},[163,171],[48,3108,81],{"className":3109},[167],[48,3111,1316],{"className":3112,"style":1343},[163,171],[48,3114,87],{"className":3115},[175]," 描出的路径就必须在一个 ",[48,3118,3120,3142],{"className":3119,"translate":52},[56],[48,3121,3123],{"className":3122},[60],[62,3124,3125],{"xmlns":64},[67,3126,3127,3139],{},[70,3128,3129,3131,3133,3135,3137],{},[78,3130,81],{"stretchy":80},[73,3132,11],{},[78,3134,2947],{},[1520,3136,2868],{},[78,3138,87],{"stretchy":80},[142,3140,3141],{"encoding":144},"(p+2)",[48,3143,3145,3166],{"className":3144,"ariaHidden":150},[149],[48,3146,3148,3151,3154,3157,3160,3163],{"className":3147},[154],[48,3149],{"className":3150,"style":159},[158],[48,3152,81],{"className":3153},[167],[48,3155,11],{"className":3156},[163,171],[48,3158],{"className":3159,"style":516},[179],[48,3161,2947],{"className":3162},[1684],[48,3164],{"className":3165,"style":516},[179],[48,3167,3169,3172,3175],{"className":3168},[154],[48,3170],{"className":3171,"style":159},[158],[48,3173,2868],{"className":3174},[163],[48,3176,87],{"className":3177},[175]," 维子空间中弯曲。",[11,3180,3181],{},"换言之：流形在高维空间中「扭曲」的程度，在某种意义上编码了后续网络层能够通过简单线性投影直接读出多少种关于该特征的不同（非线性）函数。这为「模型为何把简单的一维概念编码成复杂的高维流形」提供了一种功能主义解释——不是因为模型「迫不得已」，而是因为这样做有利于下游计算。",[11,3183,3184,3185,3254],{},"作者随后把这一视角与叠加假说联系起来：由于表征维度远小于潜在特征的数量，模型几乎别无选择，只能让大多数特征在同一个表征空间中稀疏地、近似正交地共享。在这种约束下，",[48,3186,3188,3205],{"className":3187,"translate":52},[56],[48,3189,3191],{"className":3190},[60],[62,3192,3193],{"xmlns":64},[67,3194,3195,3203],{},[70,3196,3197],{},[115,3198,3199,3201],{},[73,3200,1490],{},[73,3202,101],{},[142,3204,2000],{"encoding":144},[48,3206,3208],{"className":3207,"ariaHidden":150},[149],[48,3209,3211,3214],{"className":3210},[154],[48,3212],{"className":3213,"style":1173},[158],[48,3215,3217,3220],{"className":3216},[163],[48,3218,1490],{"className":3219},[163,171],[48,3221,3223],{"className":3222},[292],[48,3224,3226,3246],{"className":3225},[203,204],[48,3227,3229,3243],{"className":3228},[208],[48,3230,3232],{"className":3231,"style":302},[212],[48,3233,3234,3237],{"style":305},[48,3235],{"className":3236,"style":309},[220],[48,3238,3240],{"className":3239},[225,226,227,228],[48,3241,101],{"className":3242,"style":235},[163,171,228],[48,3244,269],{"className":3245},[268],[48,3247,3249],{"className":3248},[208],[48,3250,3252],{"className":3251,"style":325},[212],[48,3253],{},"——即某个特征对应的编码方向——确实包含恒等函数的「线性可读」成分这一事实，也在一定程度上解释了为何在实际中，简单的线性探针往往出奇地擅长从叠加的表征中「钓出」某个特定特征。",[35,3256,3257],{"id":3257},"余弦相似度",[11,3259,3260],{},"如果说前面的章节是脚手架，那么这一节就是论文真正的「硬核」贡献，在我看来也是最值得记住的部分。",[11,3262,3263],{},"作者提出了假说 2（余弦相似度反映距离）：局部来看，表示之间的余弦相似度是特征值之间距离平方的某个递减函数：",[48,3265,3267],{"className":3266,"translate":52},[51],[48,3268,3270,3373],{"className":3269,"translate":52},[56],[48,3271,3273],{"className":3272},[60],[62,3274,3275],{"xmlns":64,"display":65},[67,3276,3277,3370],{},[70,3278,3279,3297,3299,3305,3307,3309,3311,3313,3319,3321,3329,3331,3333,3335,3342,3344,3350,3352,3354,3356,3362,3368],{},[70,3280,3281,3284,3287,3290,3292,3294],{},[73,3282,3283],{"mathvariant":75},"C",[73,3285,3286],{"mathvariant":75},"o",[73,3288,3289],{"mathvariant":75},"s",[73,3291,1510],{"mathvariant":75},[73,3293,2631],{"mathvariant":75},[73,3295,3296],{"mathvariant":75},"m",[78,3298,81],{"stretchy":80},[115,3300,3301,3303],{},[73,3302,1490],{},[73,3304,101],{},[78,3306,81],{"stretchy":80},[73,3308,1316],{},[78,3310,87],{"stretchy":80},[78,3312,874],{"separator":150},[115,3314,3315,3317],{},[73,3316,1490],{},[73,3318,101],{},[78,3320,81],{"stretchy":80},[1506,3322,3323,3325],{},[73,3324,1316],{},[78,3326,3328],{"mathvariant":75,"lspace":3327,"rspace":3327},"0em","′",[78,3330,87],{"stretchy":80},[78,3332,87],{"stretchy":80},[78,3334,90],{},[115,3336,3337,3340],{},[73,3338,3339],{},"g",[73,3341,101],{},[78,3343,81],{"stretchy":80},[115,3345,3346,3348],{},[73,3347,879],{},[73,3349,101],{},[78,3351,81],{"stretchy":80},[73,3353,1316],{},[78,3355,874],{"separator":150},[1506,3357,3358,3360],{},[73,3359,1316],{},[78,3361,3328],{"mathvariant":75,"lspace":3327,"rspace":3327},[1506,3363,3364,3366],{},[78,3365,87],{"stretchy":80},[1520,3367,2868],{},[78,3369,87],{"stretchy":80},[142,3371,3372],{"encoding":144},"\\mathrm{CosSim}(\\phi_f(z), \\phi_f(z')) = g_f(d_f(z, z')^2)",[48,3374,3376,3537],{"className":3375,"ariaHidden":150},[149],[48,3377,3379,3383,3390,3393,3433,3436,3439,3442,3445,3448,3488,3491,3525,3528,3531,3534],{"className":3378},[154],[48,3380],{"className":3381,"style":3382},[158],"height:1.088em;vertical-align:-0.2861em;",[48,3384,3386],{"className":3385},[163],[48,3387,3389],{"className":3388},[163,2662],"CosSim",[48,3391,81],{"className":3392},[167],[48,3394,3396,3399],{"className":3395},[163],[48,3397,1490],{"className":3398},[163,171],[48,3400,3402],{"className":3401},[292],[48,3403,3405,3425],{"className":3404},[203,204],[48,3406,3408,3422],{"className":3407},[208],[48,3409,3411],{"className":3410,"style":302},[212],[48,3412,3413,3416],{"style":305},[48,3414],{"className":3415,"style":309},[220],[48,3417,3419],{"className":3418},[225,226,227,228],[48,3420,101],{"className":3421,"style":235},[163,171,228],[48,3423,269],{"className":3424},[268],[48,3426,3428],{"className":3427},[208],[48,3429,3431],{"className":3430,"style":325},[212],[48,3432],{},[48,3434,81],{"className":3435},[167],[48,3437,1316],{"className":3438,"style":1343},[163,171],[48,3440,87],{"className":3441},[175],[48,3443,874],{"className":3444},[944],[48,3446],{"className":3447,"style":282},[179],[48,3449,3451,3454],{"className":3450},[163],[48,3452,1490],{"className":3453},[163,171],[48,3455,3457],{"className":3456},[292],[48,3458,3460,3480],{"className":3459},[203,204],[48,3461,3463,3477],{"className":3462},[208],[48,3464,3466],{"className":3465,"style":302},[212],[48,3467,3468,3471],{"style":305},[48,3469],{"className":3470,"style":309},[220],[48,3472,3474],{"className":3473},[225,226,227,228],[48,3475,101],{"className":3476,"style":235},[163,171,228],[48,3478,269],{"className":3479},[268],[48,3481,3483],{"className":3482},[208],[48,3484,3486],{"className":3485,"style":325},[212],[48,3487],{},[48,3489,81],{"className":3490},[167],[48,3492,3494,3497],{"className":3493},[163],[48,3495,1316],{"className":3496,"style":1343},[163,171],[48,3498,3500],{"className":3499},[292],[48,3501,3503],{"className":3502},[203],[48,3504,3506],{"className":3505},[208],[48,3507,3510],{"className":3508,"style":3509},[212],"height:0.8019em;",[48,3511,3513,3516],{"style":3512},"top:-3.113em;margin-right:0.05em;",[48,3514],{"className":3515,"style":309},[220],[48,3517,3519],{"className":3518},[225,226,227,228],[48,3520,3522],{"className":3521},[163,228],[48,3523,3328],{"className":3524},[163,228],[48,3526,1905],{"className":3527},[175],[48,3529],{"className":3530,"style":180},[179],[48,3532,90],{"className":3533},[184],[48,3535],{"className":3536,"style":180},[179],[48,3538,3540,3544,3584,3587,3627,3630,3633,3636,3639,3671,3701],{"className":3539},[154],[48,3541],{"className":3542,"style":3543},[158],"height:1.1502em;vertical-align:-0.2861em;",[48,3545,3547,3550],{"className":3546},[163],[48,3548,3339],{"className":3549,"style":343},[163,171],[48,3551,3553],{"className":3552},[292],[48,3554,3556,3576],{"className":3555},[203,204],[48,3557,3559,3573],{"className":3558},[208],[48,3560,3562],{"className":3561,"style":302},[212],[48,3563,3564,3567],{"style":358},[48,3565],{"className":3566,"style":309},[220],[48,3568,3570],{"className":3569},[225,226,227,228],[48,3571,101],{"className":3572,"style":235},[163,171,228],[48,3574,269],{"className":3575},[268],[48,3577,3579],{"className":3578},[208],[48,3580,3582],{"className":3581,"style":325},[212],[48,3583],{},[48,3585,81],{"className":3586},[167],[48,3588,3590,3593],{"className":3589},[163],[48,3591,879],{"className":3592},[163,171],[48,3594,3596],{"className":3595},[292],[48,3597,3599,3619],{"className":3598},[203,204],[48,3600,3602,3616],{"className":3601},[208],[48,3603,3605],{"className":3604,"style":302},[212],[48,3606,3607,3610],{"style":305},[48,3608],{"className":3609,"style":309},[220],[48,3611,3613],{"className":3612},[225,226,227,228],[48,3614,101],{"className":3615,"style":235},[163,171,228],[48,3617,269],{"className":3618},[268],[48,3620,3622],{"className":3621},[208],[48,3623,3625],{"className":3624,"style":325},[212],[48,3626],{},[48,3628,81],{"className":3629},[167],[48,3631,1316],{"className":3632,"style":1343},[163,171],[48,3634,874],{"className":3635},[944],[48,3637],{"className":3638,"style":282},[179],[48,3640,3642,3645],{"className":3641},[163],[48,3643,1316],{"className":3644,"style":1343},[163,171],[48,3646,3648],{"className":3647},[292],[48,3649,3651],{"className":3650},[203],[48,3652,3654],{"className":3653},[208],[48,3655,3657],{"className":3656,"style":3509},[212],[48,3658,3659,3662],{"style":3512},[48,3660],{"className":3661,"style":309},[220],[48,3663,3665],{"className":3664},[225,226,227,228],[48,3666,3668],{"className":3667},[163,228],[48,3669,3328],{"className":3670},[163,228],[48,3672,3674,3677],{"className":3673},[175],[48,3675,87],{"className":3676},[175],[48,3678,3680],{"className":3679},[292],[48,3681,3683],{"className":3682},[203],[48,3684,3686],{"className":3685},[208],[48,3687,3690],{"className":3688,"style":3689},[212],"h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80h400000v40h-400000z",[48,4339,269],{"className":4340},[268],[48,4342,4344],{"className":4343},[208],[48,4345,4348],{"className":4346,"style":4347},[212],"height:0.6498em;",[48,4349],{},[48,4351],{"className":4352,"style":516},[179],[48,4354,4166],{"className":4355},[1684],[48,4357],{"className":4358,"style":516},[179],[48,4360,4362,4365,4368,4371,4374],{"className":4361},[154],[48,4363],{"className":4364,"style":159},[158],[48,4366,4132],{"className":4367},[163,171],[48,4369,81],{"className":4370},[167],[48,4372,3926],{"className":4373,"style":343},[163,171],[48,4375,87],{"className":4376},[175],[11,4378,4379,4380,4449],{},"用大白话说：流形上最短路径（测地线）的长度与特征空间中对应最短路径的长度严格成正比。也就是说，即使不知道 ",[48,4381,4383,4400],{"className":4382,"translate":52},[56],[48,4384,4386],{"className":4385},[60],[62,4387,4388],{"xmlns":64},[67,4389,4390,4398],{},[70,4391,4392],{},[115,4393,4394,4396],{},[73,4395,3339],{},[73,4397,101],{},[142,4399,3727],{"encoding":144},[48,4401,4403],{"className":4402,"ariaHidden":150},[149],[48,4404,4406,4409],{"className":4405},[154],[48,4407],{"className":4408,"style":2962},[158],[48,4410,4412,4415],{"className":4411},[163],[48,4413,3339],{"className":4414,"style":343},[163,171],[48,4416,4418],{"className":4417},[292],[48,4419,4421,4441],{"className":4420},[203,204],[48,4422,4424,4438],{"className":4423},[208],[48,4425,4427],{"className":4426,"style":302},[212],[48,4428,4429,4432],{"style":358},[48,4430],{"className":4431,"style":309},[220],[48,4433,4435],{"className":4434},[225,226,227,228],[48,4436,101],{"className":4437,"style":235},[163,171,228],[48,4439,269],{"className":4440},[268],[48,4442,4444],{"className":4443},[208],[48,4445,4447],{"className":4446,"style":325},[212],[48,4448],{}," 的具体形式，只要局部余弦相似度是距离平方的某个光滑递减函数，「沿着表征流形本身行进」的测地距离，就能精确地——只差一个统一的缩放常数——恢复概念空间中的真实距离。",[11,4451,4452],{},"这一结果回应了 Olah 和 Batson 在 2024 年提出的一个悬而未决的猜想：他们曾写道，「特征流形以超出其拓扑所要求的更复杂方式嵌入，其目的或许正是为了实现某个特定的距离度量，这个想法可能相当深刻而重要。」这篇论文实际上用度量几何的严格语言，把这个直觉性的猜想结晶成一个既可证明也可证伪的定理。",[11,4454,4455],{},"证明本身采用了度量几何中「路径长度」的经典定义（多边形逼近和的上确界），核心技巧是对余弦相似度在零点附近做泰勒展开，随后用一系列仔细的三角不等式论证来界定误差项——一段扎实的解析推理（详见附录）。",[35,4457,4458],{"id":4458},"同胚容易等距困难",[11,4460,4461],{},"理论就位之后，作者回到颜色、年份、日期三个实证案例，检验假说 2 与定理 1。他们设计了两种诊断方法：",[2555,4463,4464,4499],{},[996,4465,4466,4467,4498],{},"直接方法：以散点图绘制「余弦相似度 vs. 距离平方」，观察是否出现局部递减趋势，并用查特吉相关系数 ",[48,4468,4470,4485],{"className":4469,"translate":52},[56],[48,4471,4473],{"className":4472},[60],[62,4474,4475],{"xmlns":64},[67,4476,4477,4482],{},[70,4478,4479],{},[73,4480,4481],{},"ξ",[142,4483,4484],{"encoding":144},"\\xi",[48,4486,4488],{"className":4487,"ariaHidden":150},[149],[48,4489,4491,4494],{"className":4490},[154],[48,4492],{"className":4493,"style":766},[158],[48,4495,4481],{"className":4496,"style":4497},[163,171],"margin-right:0.046em;"," 量化整体的函数依赖程度；",[996,4500,4501],{},"间接方法：检验定理 1 本身——用 k-近邻图估计流形上的测地距离，检查它们是否与特征空间中的测地距离成正比，并用皮尔逊相关系数衡量线性程度。",[11,4503,4504,4505,4568],{},"结果耐人寻味。对于「颜色」和「日期」——两者都天然具有周期结构——一个简单的圆形度量（色相角；一年中的第几天）就能为等距性提供相当有力的支持（日期的皮尔逊相关系数达 0.97）。「年份」却栽了跟头：如果假设年份之间的距离就是 ",[48,4506,4508,4532],{"className":4507,"translate":52},[56],[48,4509,4511],{"className":4510},[60],[62,4512,4513],{"xmlns":64},[67,4514,4515,4529],{},[70,4516,4517,4520,4522,4524,4527],{},[73,4518,4519],{"mathvariant":75},"∣",[73,4521,84],{},[78,4523,1518],{},[73,4525,4526],{},"y",[73,4528,4519],{"mathvariant":75},[142,4530,4531],{"encoding":144},"|x-y|",[48,4533,4535,4556],{"className":4534,"ariaHidden":150},[149],[48,4536,4538,4541,4544,4547,4550,4553],{"className":4537},[154],[48,4539],{"className":4540,"style":159},[158],[48,4542,4519],{"className":4543},[163],[48,4545,84],{"className":4546},[163,171],[48,4548],{"className":4549,"style":516},[179],[48,4551,1518],{"className":4552},[1684],[48,4554],{"className":4555,"style":516},[179],[48,4557,4559,4562,4565],{"className":4558},[154],[48,4560],{"className":4561,"style":159},[158],[48,4563,4526],{"className":4564,"style":343},[163,171],[48,4566,4519],{"className":4567},[163],"（例如 1990 与 2000 相距 10 个单位），经验数据完全不支持这一假说。图 4 显示，越接近「当下」的年份（论文以 GPT-2 的发布年份 2019 为参考点）在流形上被拉得越开——它们的距离被「放大」了。",[11,4570,4571,4572,4653],{},"于是作者做了一个非常优雅的修正：把年份的度量空间换成 ",[48,4573,4575,4606],{"className":4574,"translate":52},[56],[48,4576,4578],{"className":4577},[60],[62,4579,4580],{"xmlns":64},[67,4581,4582,4603],{},[70,4583,4584,4587,4590,4592,4595,4597,4601],{},[73,4585,4586],{},"log",[78,4588,4589],{},"⁡",[78,4591,81],{"stretchy":80},[1520,4593,4594],{},"2019",[78,4596,1518],{},[4598,4599,4600],"mtext",{},"year",[78,4602,87],{"stretchy":80},[142,4604,4605],{"encoding":144},"\\log(2019 - \\text{year})",[48,4607,4609,4637],{"className":4608,"ariaHidden":150},[149],[48,4610,4612,4615,4622,4625,4628,4631,4634],{"className":4611},[154],[48,4613],{"className":4614,"style":159},[158],[48,4616,4618,4619],{"className":4617},[198],"lo",[48,4620,3339],{"style":4621},"margin-right:0.0139em;",[48,4623,81],{"className":4624},[167],[48,4626,4594],{"className":4627},[163],[48,4629],{"className":4630,"style":516},[179],[48,4632,1518],{"className":4633},[1684],[48,4635],{"className":4636,"style":516},[179],[48,4638,4640,4643,4650],{"className":4639},[154],[48,4641],{"className":4642,"style":159},[158],[48,4644,4647],{"className":4645},[163,4646],"text",[48,4648,4600],{"className":4649},[163],[48,4651,87],{"className":4652},[175]," 上的欧几里得距离——也就是说，模型编码的或许并不是「日历年」本身，而是对数尺度上的「多久以前」。这个新的度量空间与原年份区间拓扑等价（同胚保持，秩相关仍接近 1），但在这一新度量下检验等距性时，结果从「不支持」翻转为「强烈支持」（查特吉系数 0.84，皮尔逊相关系数 0.99）。",[11,4655,4656],{},"我认为这一节是整篇论文最见洞察力的实证部分：它清晰地表明「同胚」与「等距」是两个完全不同的几何保真度层级。前者只问「形状对不对」；后者问「数值上的距离关系对不对」。一个流形可以与你设想的某个概念空间在拓扑上完全一致，在几何上（距离的具体函数形式）却截然不同。而破解「距离究竟是如何被编码的」这一过程，本身就是对模型如何「理解」时间概念的一次侦察。GPT-2 把「较近的年份」彼此拉得更远，在某种意义上暗示：越接近模型训练截止时间的年份，承载的语义纹理越稠密——新闻和事件挤得更紧——因此被分配了更大的「表征空间预算」。这是一个相当迷人的猜想，尽管论文本身并未深入探讨这一现象背后的因果解释。",[35,4658,4659],{"id":4659},"柏拉图假说",[11,4661,4662],{},"读这篇论文时，我忍不住要把它和另一篇同样是 2024 年发表、在机器学习社区引发大量讨论的立场性论文放在一起比较——Huh、Cheung、Wang 与 Isola 提出的柏拉图式表征假说（PRH）（ICML 2024）。虽然两篇论文的具体研究对象和方法几乎完全不同，但把它们对照起来读，会发现它们是在同一个大问题的两个不同尺度上工作，而且彼此互补——甚至可以视为一条因果链条上的两个相继环节。",[11,4664,4665],{},"PRH 的核心主张是：架构不同、模态不同（视觉、语言）、训练目标不同的深度网络，随着规模和任务多样性的增大，其内部表征会趋向收敛到一个共享的几何结构。作者把这个被越来越多的模型逐步逼近的假想表征空间，类比为柏拉图式的「理想实在」（理念界）——独立于任何具体实体而存在并超越它们：每个具体模型学到的表征，只是对这个「柏拉图式表征」的一个带有噪声和偏差的投影。其实证证据包括：在处理配对的图文数据时，随着模型变大，图像模型与语言模型测量数据点间距离的方式变得越来越相似；在一批架构和训练方式各异的视觉模型中，两两间的表征相似性系统地上升。他们还提出了一个解释性猜想：这种收敛的驱动力在于，模型在学习过程中被迫逼近数据生成过程所共有的真实统计结构（即「实在」本身）——任务越多样，这种收敛压力就越强。",[11,4667,4668],{},"如果说 PRH 关心的是「在不同模型之间，整个表征空间是否正收敛到一个统一的几何」，那么 Modell、Rubin-Delanchy 和 Whiteley 的这篇论文关心的则是更小、更具体的尺度：在单个模型内部，为什么对应于单个特征（颜色、年份、日期）的子流形会呈现特定的几何形状，这个形状与它所对应的「概念」之间精确的数学关系又是什么？",[11,4670,4671],{},"把两篇论文放在一起，可以读出相当连贯的一条逻辑链：",[2555,4673,4674,4677,4680],{},[996,4675,4676],{},"PRH 提出了一个宏观猜想——不同模型的表征空间正在收敛到一个反映「世界的真实统计结构」的共同几何，而这种收敛的程度可以用「模型间测量数据点距离的一致性」来量化（PRH 论文恰恰使用了核对齐、表征相似性等评估两两距离模式一致性的工具）。但 PRH 本身停留在「存在性」和「收敛趋势」层面，并没有深入一个更基础的问题：在单个模型的表征空间内部，距离本身是如何与概念空间中的「真实」距离相对应的？",[996,4678,4679],{},"本文《大语言模型中表征流形的起源》回答的恰恰是这个更基础、更微观的问题。定理 1 告诉我们：只要局部余弦相似度与特征距离之间存在某种光滑的函数关系（假说 2），表征流形上的测地距离就能精确地——只差一个比例常数——恢复概念空间中的「真实」距离。这就为 PRH 所声称的「模型以某种一致的方式学会测量数据点之间的距离」提供了一个几何机制层面的解释：模型并不是简单地「记住」距离，而是通过把特征映射到特定形状和曲率的流形上，借助该流形上的测地路径隐式地编码了距离结构。",[996,4681,4682],{},"反过来，本文观察到的颜色和日期呈现的圆形结构——以及「颜色按标准色环的色相次序排列」这一具体发现——可以看作对 PRH 的微观印证：色相本身是客观存在于物理世界中的周期结构（可见光的连续光谱，连同人类视网膜三类视锥细胞的响应曲线，共同决定了颜色空间）。如果不同的图像模型和不同的语言模型，互不协调、各自独立地都学到「色相是一个圆」，这恰恰是 PRH 所描述的「不同模型收敛到反映世界真实统计结构的共享表征」的一个具体、可分离、可检验的实例。换言之，PRH 提供了「为什么会收敛」的宏观叙事；本文提供了「收敛之后几何长什么样，这个几何又是如何精确编码语义距离」的微观刻画。",[11,4684,4685],{},"当然，两篇工作之间也存在明显的张力，值得坦诚地指出。PRH 的收敛主张在很大程度上依赖跨模型比较——用典型相关分析、核对齐等工具比较不同模型对同一组数据点两两距离模式的测量方式，这属于「表征之间的关系」的统计范畴。而《大语言模型中表征流形的起源》则从头到尾讨论的是单个模型内部、单个特征子空间内的几何结构，几乎不涉及不同模型的表征是否可比较的问题。也就是说，即便「余弦相似度与测地距离之间的定理」对某个特定模型（例如 GPT-2）成立，也不能直接推出另一个架构完全不同的模型会用同样的方式——同样的度量空间、同样的对数尺度——编码「年份」这个特征。事实上，「年份以对数尺度编码，距离随接近参考年份而不断拉伸」这一具体发现，本身就带有相当程度的模型特异性。GPT-2 的参考点天然设定为其自身的发布年份（2019），这表明这种编码方式很可能与这个特定模型训练语料的时间分布强相关，而不是所有语言模型都收敛到的、普遍的「柏拉图式」距离编码。如果在训练截止时间和语料分布都不同的模型上重复同样的实验，对数尺度和参考点会相应地偏移吗？这其实是一个非常值得后续工作直接研究的问题——而且在某种意义上，它构成了对 PRH 收敛主张在「单个特征的特定几何参数」这一更细粒度上进行经验检验的天然实验设计。",[11,4687,4688],{},"依我之见，把这两篇论文放在一起读，比单独读任何一篇都更有启发：PRH 告诉我们「所有人都在朝同一个方向收敛」，而本文则为刻画「收敛到底收敛到什么、这种收敛如何被严格度量与证伪」提供了数学语言。如果未来有人想在具体特征层面真正检验 PRH 是否成立——例如色相的圆形结构、时间的对数编码，是否会在架构和训练数据各不相同的模型中一致地重现——那么《大语言模型中表征流形的起源》中提出的同胚检验、等距检验（查特吉系数、基于 KNN 的测地距离的皮尔逊相关）就构成一套几乎现成的、标准化的工具箱。这或许正是这篇论文除了自身数学结论之外，最具有普遍可复用性的「副产品」。",[35,4690,4691],{"id":4691},"结语",[11,4693,4694],{},"读完这篇论文，我想从几个角度给出我的评价。",[11,4696,4697],{},"首先，这是一次难得的把模糊直觉严格化的努力，其主要价值在于提供了语言和工具，而非一个终极答案。机制可解释性领域目前充斥着「看图说话」式的发现——「哦，这个特征看起来像个圆」「这个看起来像树形结构」——却缺少一种统一的数学语言来组织这些观察。把特征定义为度量空间，并把「同胚」与「等距」这两个概念干净地区分开来，本身就是一件重要的概念工具：它迫使研究者把「我觉得这个流形对应那个概念」这种模糊直觉，翻译成具体、可证伪的假说（「这是不是正确的度量空间？」），并提供具体的统计检验程序（查特吉系数、基于 k-近邻图的测地距离）。这种严谨性目前在领域内相对稀缺。",[11,4699,4700],{},"其次，定理 1 是漂亮的数学，但其「证明力」在相当程度上依赖一个相当强的前提——余弦相似度与距离之间存在光滑的函数关系。这一假说本身并非从模型训练动力学或架构设计推导而来，而是作为一个「合理的猜测」被摆上桌面，再加以经验验证。换言之，这篇论文读起来更像「如果这个假说成立，会有哪些漂亮的推论？」，而非「为什么表征空间必然呈现这种结构」（正是在这个意义上，标题中「起源」一词多少有些一厢情愿——论文并没有从优化目标或梯度下降动力学推导出流形为何「涌现」，它在很大程度上是在「流形已经存在」的前提之下，转而刻画它们应当呈现什么样子）。不过这也并非完全是批评：机制可解释性作为一个整体仍处于「观察现象、构建描述性理论」的阶段，真正从第一性原理推导表征几何的工作少之又少。这篇论文至少把「描述性理论」这一步做得比大多数同类工作更严谨。",[11,4702,4703],{},"第三，年份的对数尺度发现是全篇最令人印象深刻、也最引人深思的实证结果。它暗示了一个值得警惕的方法论陷阱：同胚检验（拓扑对不对？）的门槛非常低，很容易得到「看起来对」的东西，但这远不足以说明我们真正理解了模型的编码方式。如果作者止步于「年份沿一条曲线排列，顺序与真实时间顺序一致，秩相关 0.97，说明模型学会了年份的次序」，这个结论将相当空洞——几乎任何单调映射都能做到这一点。真正携带信息量的是等距检验揭示的距离非均匀拉伸，以及它背后的具体假说：以对数尺度编码「距参考点的时间」。这提醒我们，未来类似的可解释性研究不应止步于「形状看起来像不像？」，而应当像本文一样继续追问「距离度量对不对？」——因为只有如此，才能挖掘出模型真正编码的隐含假设。",[11,4705,4706,4707,4776],{},"第四，从应用的角度看，这篇论文对表征引导（steering）研究具有直接的方法论启示，但距离真正可操作的工具还有一段距离。论文在结尾指出，如果能够学习到 ",[48,4708,4710,4727],{"className":4709,"translate":52},[56],[48,4711,4713],{"className":4712},[60],[62,4714,4715],{"xmlns":64},[67,4716,4717,4725],{},[70,4718,4719],{},[115,4720,4721,4723],{},[73,4722,1490],{},[73,4724,101],{},[142,4726,2000],{"encoding":144},[48,4728,4730],{"className":4729,"ariaHidden":150},[149],[48,4731,4733,4736],{"className":4732},[154],[48,4734],{"className":4735,"style":1173},[158],[48,4737,4739,4742],{"className":4738},[163],[48,4740,1490],{"className":4741},[163,171],[48,4743,4745],{"className":4744},[292],[48,4746,4748,4768],{"className":4747},[203,204],[48,4749,4751,4765],{"className":4750},[208],[48,4752,4754],{"className":4753,"style":302},[212],[48,4755,4756,4759],{"style":305},[48,4757],{"className":4758,"style":309},[220],[48,4760,4762],{"className":4761},[225,226,227,228],[48,4763,101],{"className":4764,"style":235},[163,171,228],[48,4766,269],{"className":4767},[268],[48,4769,4771],{"className":4770},[208],[48,4772,4774],{"className":4773,"style":325},[212],[48,4775],{},"（从特征到流形的映射），原则上就可以进行更精细、尊重概念内在几何结构的表征编辑——例如要让「日期」特征向前平移半年，应该沿流形的测地线行进，而不是在欧几里得空间中简单加一个向量。这是一个有前景的方向，也与作者呼吁的「流形感知的 SAE」相呼应。但目前这仍停留在概念层面：如何稳健地估计一个高维、带噪声的流形，仍然是一个尚未解决的统计问题——作者在局限部分也坦诚地承认了这一点。",[11,4778,4779],{},"第五，一个可能被低估的贡献，是论文对文本嵌入服务的启示。许多从事 RAG、语义检索和推荐系统的从业者，不加思索地把余弦相似度当作「语义接近度」的度量，很少停下来追问这个数值相似度背后究竟潜藏着什么样的几何结构。这篇论文是一个提醒：余弦相似度确实能忠实地反映「沿着概念空间内在几何路径的距离」——但这是一个局部性质，依赖于假说 2 成立，而且不同特征的几何形状截然不同（圆、线段、对数线段）。把它当作跨特征的、普遍的「语义距离」度量，可能会掩盖大量重要的非线性结构（如颜色例子所示，「大距离下的余弦相似度明显偏离等距关系」）。对于调优基于嵌入的检索系统和异常检测管道的工程师来说，这是一个值得牢记的告诫。",[11,4781,4782],{},"这篇论文并不试图解释所有表征几何现象，也没有交付一个可以投入生产部署的工具。它只是做了一份「数学家的活」：对于一个被广泛观察却缺乏精确定义的现象，搭建一个最小但严格的理论框架，并诚实地把它拿到数据面前，检验它的边界在哪里。在一个日益依赖「炼金术式」经验观察的领域，这种回归第一性原理、愿意清晰陈述假设、完整写出证明、并坦诚承认自身局限的工作，本身就很值得被更多人看到。",[11,4784,4785],{},"如果要我说这篇论文给我留下的最深刻印象，大概就是「年份以对数时间距离编码」这一具体发现。它通过一个极小的例子表明，在向量空间看似平淡的几何背后，或许真的存在着某种接近人类认知直觉的东西：越近的事件越清晰、越可精细区分；遥远的事件则在记忆中被压缩。这或许是机制可解释性研究最迷人的地方：不是证明模型是一块会魔法的黑箱，而是把它内部真正在「思考」的东西，一点一点翻译成我们能够读懂的语言。",{"title":4787,"searchDepth":4788,"depth":4788,"links":4789},"",2,[4790,4791,4792,4793,4794,4795,4796,4797],{"id":37,"depth":4788,"text":37},{"id":843,"depth":4788,"text":844},{"id":2550,"depth":4788,"text":2550},{"id":2579,"depth":4788,"text":2579},{"id":3257,"depth":4788,"text":3257},{"id":4458,"depth":4788,"text":4458},{"id":4659,"depth":4788,"text":4659},{"id":4691,"depth":4788,"text":4691},"如果你关注了过去两年机制可解释性研究的进展，很可能见过这样一类图：把大语言模型某一层神经元的激活向量投影到二维或三维空间并作图，会发现「月份」「星期几」「年份」「颜色」等概念的表示并不是杂乱无章地散布在空间中，而是沿着一条优美的曲线排列——有时甚至是闭合的圆圈或环面。Chris Olah、Anthropic 的 Josh Batson 以及 Engels 等人，都在包括 并非所有语言模型特征都是一维线性的 在内的研究中展示了这一现象。",false,true,"md",null,{},"\u002Fblog\u002F2026\u002F2026-08-04-the-representations-in-llm",{"title":6,"description":4798},"blog\u002F2026\u002F2026-08-04-the-representations-in-llm","将 LLM 的表征流形形式化：以度量空间定义特征，证明余弦相似度编码了特征之间的测地距离，并在颜色、日期与年份三类数据上实证验证了同胚性与等距性。",[4809,4810],"machine-learning","ai","2026-08-04T00:00:00+08:00","7muZrPVdpjj3Hr2Jo2ta-VJnPzeZr9TL8KvVJunp_nQ",{"id":4814,"title":4815,"body":4816,"description":4820,"draft":4799,"enableComment":4800,"extension":4801,"image":4787,"important":4799,"location":4972,"meta":4973,"navigation":4800,"ogImage":4802,"path":4974,"seo":4975,"stem":4976,"summary":4977,"tags":4978,"time":4980,"__hash__":4981},"blog\u002Fblog\u002F2026\u002F2026-08-13-xie-zuo-yu-yan-de-zhuan-bian.md","写作语言的转变",{"type":8,"value":4817,"toc":4970},[4818,4821,4824,4827,4952,4955,4958,4961,4964,4967],[11,4819,4820],{},"长期以来，我对自己的博客定位从来不是分享，而更多是给自己看的「笔记」。很多情况是：记录一个刚刚搞懂的东西 → 写下来 → 以后自己查 → 写完就结束。我的博客基本都很短，基本三分钟以内就能读完。大约在 2023 年开始，同时为了学习英语，锻炼技术写作能力，我在疯狂背单词的同时，也可以保持用英语进行技术写作。现在坚持了快两年。所以长期以来，利用英语写作，对我来说压力并不大。写作时不需要追求完整论证，只需要把自己的思路编码下来。哪怕句子稍微生硬一点，只要自己以后能看懂，就已经达成目的。",[11,4822,4823],{},"过去两年的英语写作之所以轻松，是因为它服务于「笔记」功能——短、线性、结论明确。这种写作不需要复杂的逻辑嵌套，不需要反复回看和修改结构，英语的线性特征反而成了一种约束，帮你把想法压缩成清晰的短句。",[11,4825,4826],{},"但是后来，情况发生了变化，我的博客从「给自己看的笔记」的定位，开始转移到了「表达与推演」。当写作从记录变成表达与推演，语言就从工具变成了负担。写作负荷不是恒定的，它会随着结构复杂度和语言熟练度的乘积而非线性增长。",[11,4828,4829,4830,4951],{},"文章越来越长，关于技术写作的部分，有的文章我需要插入大量的 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公式、推导和演绎过程。此时再使用非母语进行技术写作时心智负担显著增大。短文里，一句话如果不知道怎么写，一般情况可以用简单句子绕一下，问题不大。但是到了长篇技术文章，可能连续几页都在描述一个复杂的思想。这时候需要同时控制数学符号、技术概念、论证结构、前后术语一致性、句法，还有段落之间的衔接。于是工作记忆里面同时跑着很多东西。",[11,4953,4954],{},"而且，长篇博客并不是一次写完的，很可能需要花上几个星期慢慢打磨，或者经常回顾我自己的博客的思想。写作的时候，我大多数时间都在思考：「这个理论到底应该怎么解释？」。但是，结果脑子里却不断出现：「这里应该用 which 还是 that？」、「这个东西应该叫 representation 还是 formulation？」、「这个词性是否合适，有没有对应的名词形式？」之类细枝末节的语法问题。久而久之，写作起来会非常累。在短文本写作里，这种差异几乎感觉不到。但当文本从几百字增长到几千、几万字以后，语言的视觉结构、信息密度、词法形态、定位效率和工作记忆负担都会开始成为写作系统的一部分。",[11,4956,4957],{},"另外一种原因，英文是线形文字，阅读效率天然就很低：想在长文中定位到某一块位置，必须要从每一段从头开始逐行扫过，无法像汉字那样逐块扫描。很多时候我在长文写作中，需要不断地往回头看。而英语的语法线性强，从句嵌套多了以后，读者和作者都容易迷失。因为汉字同时包含视觉图像和声音两种信息，它信息密度和视觉辨识特征，使中文文本非常适合视觉扫描。而英文单词之间存在大量空格，真正有语义重量的东西被拆成了很多视觉单元。所以当文章达到几千甚至上万字以后，回来看自己几周前写的东西，会出现一种很奇怪的体验：中文是在「看结构」，英文更容易变成「读句子」。所以，就导致语义单元和视觉单元不对齐——一个概念可能要三四个单词才能表达，视觉上要扫过更长的距离才能抓住一个完整意群。",[11,4959,4960],{},"其实不止是我，很多技术写作者都会遇到一个问题：语言不是中性的容器，它会反过来塑造你思考的形状。",[11,4962,4963],{},"笔记型写作的特点，思维已经完成。只是在编码一个已经清晰的结论，供未来的自己检索。表达与推演型写作则不同，写作本身就是思维过程。你在写的过程中推演、发现、修正。文字不是思维的镜像，而是思维的工具。当写作成为思考工具时，语言就不再只是输出端的问题，而是输入端的问题。非常需要语言来帮助你组织尚未成形的想法，来试探逻辑的边界，来连接不同的概念。",[11,4965,4966],{},"从此以后，我的博客功能，从以前的博客 externalized memory（外部记忆），到现在逐渐变成了 externalized thinking（外部化思考）。",[11,4968,4969],{},"也许从英语写作到中文写作，是在为思维本身让路。让认知资源从语言操作中解放出来，还给真正的思考。",{"title":4787,"searchDepth":4788,"depth":4788,"links":4971},[],"河南郑州",{},"\u002Fblog\u002F2026\u002F2026-08-13-xie-zuo-yu-yan-de-zhuan-bian",{"title":4815,"description":4820},"blog\u002F2026\u002F2026-08-13-xie-zuo-yu-yan-de-zhuan-bian","长期以来，为了学习英语并增强熟练度，在技术写作时，我都会刻意使用英语来写作。但是后来我发现使用英语的阅读和思考心智负担非常大。从笔记到表达与推演，写作目标发生变化后，认知资源也需要实现对应的重新分配。",[4979],"thoughts","2026-08-13T00:00:00+08:00","7ebbJ5U5Sd2pnIWInH_6aFeptiKE6s-ogM9nTwBT0EQ",{"id":4983,"title":4984,"body":4985,"description":4989,"draft":4799,"enableComment":4800,"extension":4801,"image":4787,"important":4799,"location":4802,"meta":21068,"navigation":4800,"ogImage":4802,"path":21069,"seo":21070,"stem":21071,"summary":21072,"tags":21073,"time":21076,"__hash__":21077},"blog\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en.md","对角线论证：改变数学与计算机的证明方法",{"type":8,"value":4986,"toc":21057},[4987,4990,4993,4996,4999,5129,5266,5815,5976,6096,6099,7023,7026,7029,7087,7142,7145,7297,8734,8937,9186,9613,9791,9823,10007,10388,10697,10792,10940,11024,11204,11479,12615,12618,12621,12624,12932,12935,12942,13104,13107,13110,13336,13412,13416,13593,13720,13910,13913,14164,14219,15091,15267,15270,16118,16443,16446,16758,16760,16766,16856,16859,16862,16981,16984,16987,17303,17306,17338,17678,17799,17846,18255,18291,18488,19330,19505,19508,19746,19748,19754,19764,19767,19770,19775,19778,20362,20365,20606,20609,20612,20615,20978,20981,21053],[11,4988,4989],{},"1891 年，德国数学家格奥尔格·康托尔发表了一篇仅几页的论文，证明了一个在当时看来有悖直觉的结论：存在不同「大小」的无穷。他所采用的证明技巧——对角线论证——不仅解决了手头的问题，更在此后一个多世纪里，成为数理逻辑、可计算性理论乃至哲学武器库中最强大的武器之一。",[11,4991,4992],{},"从「实数比自然数多」，到图灵证明「停机问题不可判定」，再到哥德尔证明「任何足够强的形式系统都包含不可判定命题」——这些看似主题各异的结论，本质上都是同一个想法的变体。本文将带领读者从集合论最基础的出发点开始，一步步推导对角线论证的完整过程，阐明它为何拥有如此强大的力量，并把抽象的证明转化为可以通过代码执行、观察的具体过程。",[35,4994,4995],{"id":4995},"等势",[11,4997,4998],{},"在讨论无穷集之前，必须先澄清一个预备问题：如何比较两个集合的大小？",[11,5000,5001,5002,5128],{},"对于有限集，答案很简单：数元素个数即可。但自然数集 ",[48,5003,5005,5051],{"className":5004,"translate":52},[56],[48,5006,5008],{"className":5007},[60],[62,5009,5010],{"xmlns":64},[67,5011,5012,5048],{},[70,5013,5014,5018,5020,5023,5025,5027,5029,5031,5033,5035,5038,5040,5042,5045],{},[73,5015,5017],{"mathvariant":5016},"double-struck","N",[78,5019,90],{},[78,5021,5022],{"stretchy":80},"{",[1520,5024,2710],{},[78,5026,874],{"separator":150},[1520,5028,1522],{},[78,5030,874],{"separator":150},[1520,5032,2868],{},[78,5034,874],{"separator":150},[1520,5036,5037],{},"3",[78,5039,874],{"separator":150},[78,5041,2934],{},[4598,5043,5044],{}," ",[78,5046,5047],{"stretchy":80},"}",[142,5049,5050],{"encoding":144},"\\mathbb{N} = \\{0, 1, 2, 3, \\dots\\}",[48,5052,5054,5074],{"className":5053,"ariaHidden":150},[149],[48,5055,5057,5061,5065,5068,5071],{"className":5056},[154],[48,5058],{"className":5059,"style":5060},[158],"height:0.6889em;",[48,5062,5017],{"className":5063},[163,5064],"mathbb",[48,5066],{"className":5067,"style":180},[179],[48,5069,90],{"className":5070},[184],[48,5072],{"className":5073,"style":180},[179],[48,5075,5077,5080,5083,5086,5089,5092,5095,5098,5101,5104,5107,5110,5113,5116,5119,5122,5125],{"className":5076},[154],[48,5078],{"className":5079,"style":159},[158],[48,5081,5022],{"className":5082},[167],[48,5084,2710],{"className":5085},[163],[48,5087,874],{"className":5088},[944],[48,5090],{"className":5091,"style":282},[179],[48,5093,1522],{"className":5094},[163],[48,5096,874],{"className":5097},[944],[48,5099],{"className":5100,"style":282},[179],[48,5102,2868],{"className":5103},[163],[48,5105,874],{"className":5106},[944],[48,5108],{"className":5109,"style":282},[179],[48,5111,5037],{"className":5112},[163],[48,5114,874],{"className":5115},[944],[48,5117],{"className":5118,"style":282},[179],[48,5120,2934],{"className":5121},[3012],[48,5123],{"className":5124,"style":282},[179],[48,5126,5047],{"className":5127},[175]," 是无穷的，无法「数到底」；因此我们需要一种新的工具。",[11,5130,5131,5132,5160,5161,5191,5192,5265],{},"定义（双射）。给定两个集合 ",[48,5133,5135,5148],{"className":5134,"translate":52},[56],[48,5136,5138],{"className":5137},[60],[62,5139,5140],{"xmlns":64},[67,5141,5142,5146],{},[70,5143,5144],{},[73,5145,4891],{},[142,5147,4891],{"encoding":144},[48,5149,5151],{"className":5150,"ariaHidden":150},[149],[48,5152,5154,5157],{"className":5153},[154],[48,5155],{"className":5156,"style":4878},[158],[48,5158,4891],{"className":5159},[163,171]," 与 ",[48,5162,5164,5178],{"className":5163,"translate":52},[56],[48,5165,5167],{"className":5166},[60],[62,5168,5169],{"xmlns":64},[67,5170,5171,5176],{},[70,5172,5173],{},[73,5174,5175],{},"B",[142,5177,5175],{"encoding":144},[48,5179,5181],{"className":5180,"ariaHidden":150},[149],[48,5182,5184,5187],{"className":5183},[154],[48,5185],{"className":5186,"style":4878},[158],[48,5188,5175],{"className":5189,"style":5190},[163,171],"margin-right:0.0502em;","，若存在函数 ",[48,5193,5195,5217],{"className":5194,"translate":52},[56],[48,5196,5198],{"className":5197},[60],[62,5199,5200],{"xmlns":64},[67,5201,5202,5214],{},[70,5203,5204,5206,5208,5210,5212],{},[73,5205,101],{},[78,5207,1495],{},[73,5209,4891],{},[78,5211,1504],{},[73,5213,5175],{},[142,5215,5216],{"encoding":144},"f: A \\to B",[48,5218,5220,5238,5256],{"className":5219,"ariaHidden":150},[149],[48,5221,5223,5226,5229,5232,5235],{"className":5222},[154],[48,5224],{"className":5225,"style":766},[158],[48,5227,101],{"className":5228,"style":235},[163,171],[48,5230],{"className":5231,"style":180},[179],[48,5233,1495],{"className":5234},[184],[48,5236],{"className":5237,"style":180},[179],[48,5239,5241,5244,5247,5250,5253],{"className":5240},[154],[48,5242],{"className":5243,"style":4878},[158],[48,5245,4891],{"className":5246},[163,171],[48,5248],{"className":5249,"style":180},[179],[48,5251,1504],{"className":5252},[184],[48,5254],{"className":5255,"style":180},[179],[48,5257,5259,5262],{"className":5258},[154],[48,5260],{"className":5261,"style":4878},[158],[48,5263,5175],{"className":5264,"style":5190},[163,171]," 满足：",[993,5267,5268,5637],{},[996,5269,5270,5271,5636],{},"单射：",[48,5272,5274,5330],{"className":5273,"translate":52},[56],[48,5275,5277],{"className":5276},[60],[62,5278,5279],{"xmlns":64},[67,5280,5281,5327],{},[70,5282,5283,5289,5292,5298,5301,5303,5305,5311,5313,5315,5317,5319,5325],{},[115,5284,5285,5287],{},[73,5286,15],{},[1520,5288,1522],{},[78,5290,5291],{"mathvariant":75},"≠",[115,5293,5294,5296],{},[73,5295,15],{},[1520,5297,2868],{},[78,5299,5300],{},"⇒",[73,5302,101],{},[78,5304,81],{"stretchy":80},[115,5306,5307,5309],{},[73,5308,15],{},[1520,5310,1522],{},[78,5312,87],{"stretchy":80},[78,5314,5291],{"mathvariant":75},[73,5316,101],{},[78,5318,81],{"stretchy":80},[115,5320,5321,5323],{},[73,5322,15],{},[1520,5324,2868],{},[78,5326,87],{"stretchy":80},[142,5328,5329],{"encoding":144},"a_1 \\neq a_2 \\Rightarrow f(a_1) \\neq f(a_2)",[48,5331,5333,5428,5484,5581],{"className":5332,"ariaHidden":150},[149],[48,5334,5336,5339,5379,5382,5425],{"className":5335},[154],[48,5337],{"className":5338,"style":766},[158],[48,5340,5342,5345],{"className":5341},[163],[48,5343,15],{"className":5344},[163,171],[48,5346,5348],{"className":5347},[292],[48,5349,5351,5371],{"className":5350},[203,204],[48,5352,5354,5368],{"className":5353},[208],[48,5355,5357],{"className":5356,"style":2750},[212],[48,5358,5359,5362],{"style":305},[48,5360],{"className":5361,"style":309},[220],[48,5363,5365],{"className":5364},[225,226,227,228],[48,5366,1522],{"className":5367},[163,228],[48,5369,269],{"className":5370},[268],[48,5372,5374],{"className":5373},[208],[48,5375,5377],{"className":5376,"style":2771},[212],[48,5378],{},[48,5380],{"className":5381,"style":180},[179],[48,5383,5385,5418,5422],{"className":5384},[184],[48,5386,5388],{"className":5387},[184],[48,5389,5392],{"className":5390},[163,5391],"vbox",[48,5393,5396],{"className":5394},[5395],"thinbox",[48,5397,5400,5403,5414],{"className":5398},[5399],"rlap",[48,5401],{"className":5402,"style":766},[158],[48,5404,5407],{"className":5405},[5406],"inner",[48,5408,5410],{"className":5409},[163],[48,5411,5413],{"className":5412},[184],"",[48,5415],{"className":5416},[5417],"fix",[48,5419],{"className":5420},[179,5421],"nobreak",[48,5423,90],{"className":5424},[184],[48,5426],{"className":5427,"style":180},[179],[48,5429,5431,5435,5475,5478,5481],{"className":5430},[154],[48,5432],{"className":5433,"style":5434},[158],"height:0.5806em;vertical-align:-0.15em;",[48,5436,5438,5441],{"className":5437},[163],[48,5439,15],{"className":5440},[163,171],[48,5442,5444],{"className":5443},[292],[48,5445,5447,5467],{"className":5446},[203,204],[48,5448,5450,5464],{"className":5449},[208],[48,5451,5453],{"className":5452,"style":2750},[212],[48,5454,5455,5458],{"style":305},[48,5456],{"className":5457,"style":309},[220],[48,5459,5461],{"className":5460},[225,226,227,228],[48,5462,2868],{"className":5463},[163,228],[48,5465,269],{"className":5466},[268],[48,5468,5470],{"className":5469},[208],[48,5471,5473],{"className":5472,"style":2771},[212],[48,5474],{},[48,5476],{"className":5477,"style":180},[179],[48,5479,5300],{"className":5480},[184],[48,5482],{"className":5483,"style":180},[179],[48,5485,5487,5490,5493,5496,5536,5539,5542,5578],{"className":5486},[154],[48,5488],{"className":5489,"style":159},[158],[48,5491,101],{"className":5492,"style":235},[163,171],[48,5494,81],{"className":5495},[167],[48,5497,5499,5502],{"className":5498},[163],[48,5500,15],{"className":5501},[163,171],[48,5503,5505],{"className":5504},[292],[48,5506,5508,5528],{"className":5507},[203,204],[48,5509,5511,5525],{"className":5510},[208],[48,5512,5514],{"className":5513,"style":2750},[212],[48,5515,5516,5519],{"style":305},[48,5517],{"className":5518,"style":309},[220],[48,5520,5522],{"className":5521},[225,226,227,228],[48,5523,1522],{"className":5524},[163,228],[48,5526,269],{"className":5527},[268],[48,5529,5531],{"className":5530},[208],[48,5532,5534],{"className":5533,"style":2771},[212],[48,5535],{},[48,5537,87],{"className":5538},[175],[48,5540],{"className":5541,"style":180},[179],[48,5543,5545,5572,5575],{"className":5544},[184],[48,5546,5548],{"className":5547},[184],[48,5549,5551],{"className":5550},[163,5391],[48,5552,5554],{"className":5553},[5395],[48,5555,5557,5560,5569],{"className":5556},[5399],[48,5558],{"className":5559,"style":766},[158],[48,5561,5563],{"className":5562},[5406],[48,5564,5566],{"className":5565},[163],[48,5567,5413],{"className":5568},[184],[48,5570],{"className":5571},[5417],[48,5573],{"className":5574},[179,5421],[48,5576,90],{"className":5577},[184],[48,5579],{"className":5580,"style":180},[179],[48,5582,5584,5587,5590,5593,5633],{"className":5583},[154],[48,5585],{"className":5586,"style":159},[158],[48,5588,101],{"className":5589,"style":235},[163,171],[48,5591,81],{"className":5592},[167],[48,5594,5596,5599],{"className":5595},[163],[48,5597,15],{"className":5598},[163,171],[48,5600,5602],{"className":5601},[292],[48,5603,5605,5625],{"className":5604},[203,204],[48,5606,5608,5622],{"className":5607},[208],[48,5609,5611],{"className":5610,"style":2750},[212],[48,5612,5613,5616],{"style":305},[48,5614],{"className":5615,"style":309},[220],[48,5617,5619],{"className":5618},[225,226,227,228],[48,5620,2868],{"className":5621},[163,228],[48,5623,269],{"className":5624},[268],[48,5626,5628],{"className":5627},[208],[48,5629,5631],{"className":5630,"style":2771},[212],[48,5632],{},[48,5634,87],{"className":5635},[175],"（不同的输入产生不同的输出）",[996,5638,5639,5640,5693,5694,5746,5747,5814],{},"满射：对每个 ",[48,5641,5643,5662],{"className":5642,"translate":52},[56],[48,5644,5646],{"className":5645},[60],[62,5647,5648],{"xmlns":64},[67,5649,5650,5659],{},[70,5651,5652,5655,5657],{},[73,5653,5654],{},"b",[78,5656,104],{},[73,5658,5175],{},[142,5660,5661],{"encoding":144},"b \\in B",[48,5663,5665,5684],{"className":5664,"ariaHidden":150},[149],[48,5666,5668,5672,5675,5678,5681],{"className":5667},[154],[48,5669],{"className":5670,"style":5671},[158],"height:0.7335em;vertical-align:-0.0391em;",[48,5673,5654],{"className":5674},[163,171],[48,5676],{"className":5677,"style":180},[179],[48,5679,104],{"className":5680},[184],[48,5682],{"className":5683,"style":180},[179],[48,5685,5687,5690],{"className":5686},[154],[48,5688],{"className":5689,"style":4878},[158],[48,5691,5175],{"className":5692,"style":5190},[163,171],"，都存在 ",[48,5695,5697,5715],{"className":5696,"translate":52},[56],[48,5698,5700],{"className":5699},[60],[62,5701,5702],{"xmlns":64},[67,5703,5704,5712],{},[70,5705,5706,5708,5710],{},[73,5707,15],{},[78,5709,104],{},[73,5711,4891],{},[142,5713,5714],{"encoding":144},"a \\in A",[48,5716,5718,5737],{"className":5717,"ariaHidden":150},[149],[48,5719,5721,5725,5728,5731,5734],{"className":5720},[154],[48,5722],{"className":5723,"style":5724},[158],"height:0.5782em;vertical-align:-0.0391em;",[48,5726,15],{"className":5727},[163,171],[48,5729],{"className":5730,"style":180},[179],[48,5732,104],{"className":5733},[184],[48,5735],{"className":5736,"style":180},[179],[48,5738,5740,5743],{"className":5739},[154],[48,5741],{"className":5742,"style":4878},[158],[48,5744,4891],{"className":5745},[163,171]," 使得 ",[48,5748,5750,5774],{"className":5749,"translate":52},[56],[48,5751,5753],{"className":5752},[60],[62,5754,5755],{"xmlns":64},[67,5756,5757,5771],{},[70,5758,5759,5761,5763,5765,5767,5769],{},[73,5760,101],{},[78,5762,81],{"stretchy":80},[73,5764,15],{},[78,5766,87],{"stretchy":80},[78,5768,90],{},[73,5770,5654],{},[142,5772,5773],{"encoding":144},"f(a) = b",[48,5775,5777,5804],{"className":5776,"ariaHidden":150},[149],[48,5778,5780,5783,5786,5789,5792,5795,5798,5801],{"className":5779},[154],[48,5781],{"className":5782,"style":159},[158],[48,5784,101],{"className":5785,"style":235},[163,171],[48,5787,81],{"className":5788},[167],[48,5790,15],{"className":5791},[163,171],[48,5793,87],{"className":5794},[175],[48,5796],{"className":5797,"style":180},[179],[48,5799,90],{"className":5800},[184],[48,5802],{"className":5803,"style":180},[179],[48,5805,5807,5811],{"className":5806},[154],[48,5808],{"className":5809,"style":5810},[158],"height:0.6944em;",[48,5812,5654],{"className":5813},[163,171],"（每个输出都被覆盖）",[11,5816,5817,5818,5846,5847,5160,5875,5903,5904,5975],{},"则称 ",[48,5819,5821,5834],{"className":5820,"translate":52},[56],[48,5822,5824],{"className":5823},[60],[62,5825,5826],{"xmlns":64},[67,5827,5828,5832],{},[70,5829,5830],{},[73,5831,101],{},[142,5833,101],{"encoding":144},[48,5835,5837],{"className":5836,"ariaHidden":150},[149],[48,5838,5840,5843],{"className":5839},[154],[48,5841],{"className":5842,"style":766},[158],[48,5844,101],{"className":5845,"style":235},[163,171]," 为双射，并称 ",[48,5848,5850,5863],{"className":5849,"translate":52},[56],[48,5851,5853],{"className":5852},[60],[62,5854,5855],{"xmlns":64},[67,5856,5857,5861],{},[70,5858,5859],{},[73,5860,4891],{},[142,5862,4891],{"encoding":144},[48,5864,5866],{"className":5865,"ariaHidden":150},[149],[48,5867,5869,5872],{"className":5868},[154],[48,5870],{"className":5871,"style":4878},[158],[48,5873,4891],{"className":5874},[163,171],[48,5876,5878,5891],{"className":5877,"translate":52},[56],[48,5879,5881],{"className":5880},[60],[62,5882,5883],{"xmlns":64},[67,5884,5885,5889],{},[70,5886,5887],{},[73,5888,5175],{},[142,5890,5175],{"encoding":144},[48,5892,5894],{"className":5893,"ariaHidden":150},[149],[48,5895,5897,5900],{"className":5896},[154],[48,5898],{"className":5899,"style":4878},[158],[48,5901,5175],{"className":5902,"style":5190},[163,171]," 等势（记为 ",[48,5905,5907,5933],{"className":5906,"translate":52},[56],[48,5908,5910],{"className":5909},[60],[62,5911,5912],{"xmlns":64},[67,5913,5914,5930],{},[70,5915,5916,5918,5920,5922,5924,5926,5928],{},[73,5917,4519],{"mathvariant":75},[73,5919,4891],{},[73,5921,4519],{"mathvariant":75},[78,5923,90],{},[73,5925,4519],{"mathvariant":75},[73,5927,5175],{},[73,5929,4519],{"mathvariant":75},[142,5931,5932],{"encoding":144},"|A| = |B|",[48,5934,5936,5960],{"className":5935,"ariaHidden":150},[149],[48,5937,5939,5942,5945,5948,5951,5954,5957],{"className":5938},[154],[48,5940],{"className":5941,"style":159},[158],[48,5943,4519],{"className":5944},[163],[48,5946,4891],{"className":5947},[163,171],[48,5949,4519],{"className":5950},[163],[48,5952],{"className":5953,"style":180},[179],[48,5955,90],{"className":5956},[184],[48,5958],{"className":5959,"style":180},[179],[48,5961,5963,5966,5969,5972],{"className":5962},[154],[48,5964],{"className":5965,"style":159},[158],[48,5967,4519],{"className":5968},[163],[48,5970,5175],{"className":5971,"style":5190},[163,171],[48,5973,4519],{"className":5974},[163],"）——即两者「大小相等」。",[11,5977,5978,5979,6007,6008,6037,6038,6066,6067,6095],{},"定义（可数集）。若集合 ",[48,5980,5982,5995],{"className":5981,"translate":52},[56],[48,5983,5985],{"className":5984},[60],[62,5986,5987],{"xmlns":64},[67,5988,5989,5993],{},[70,5990,5991],{},[73,5992,4891],{},[142,5994,4891],{"encoding":144},[48,5996,5998],{"className":5997,"ariaHidden":150},[149],[48,5999,6001,6004],{"className":6000},[154],[48,6002],{"className":6003,"style":4878},[158],[48,6005,4891],{"className":6006},[163,171]," 能与自然数集 ",[48,6009,6011,6025],{"className":6010,"translate":52},[56],[48,6012,6014],{"className":6013},[60],[62,6015,6016],{"xmlns":64},[67,6017,6018,6022],{},[70,6019,6020],{},[73,6021,5017],{"mathvariant":5016},[142,6023,6024],{"encoding":144},"\\mathbb{N}",[48,6026,6028],{"className":6027,"ariaHidden":150},[149],[48,6029,6031,6034],{"className":6030},[154],[48,6032],{"className":6033,"style":5060},[158],[48,6035,5017],{"className":6036},[163,5064]," 建立双射（或 ",[48,6039,6041,6054],{"className":6040,"translate":52},[56],[48,6042,6044],{"className":6043},[60],[62,6045,6046],{"xmlns":64},[67,6047,6048,6052],{},[70,6049,6050],{},[73,6051,4891],{},[142,6053,4891],{"encoding":144},[48,6055,6057],{"className":6056,"ariaHidden":150},[149],[48,6058,6060,6063],{"className":6059},[154],[48,6061],{"className":6062,"style":4878},[158],[48,6064,4891],{"className":6065},[163,171]," 为有限集），则称 ",[48,6068,6070,6083],{"className":6069,"translate":52},[56],[48,6071,6073],{"className":6072},[60],[62,6074,6075],{"xmlns":64},[67,6076,6077,6081],{},[70,6078,6079],{},[73,6080,4891],{},[142,6082,4891],{"encoding":144},[48,6084,6086],{"className":6085,"ariaHidden":150},[149],[48,6087,6089,6092],{"className":6088},[154],[48,6090],{"className":6091,"style":4878},[158],[48,6093,4891],{"className":6094},[163,171]," 为可数集。",[11,6097,6098],{},"下面几个例子会揭示，「可数」这个概念远比直觉所认为的更加宽松：",[993,6100,6101,6275,6947],{},[996,6102,6103,6104,6202,6203,1906],{},"偶数集 ",[48,6105,6107,6145],{"className":6106,"translate":52},[56],[48,6108,6110],{"className":6109},[60],[62,6111,6112],{"xmlns":64},[67,6113,6114,6142],{},[70,6115,6116,6118,6120,6122,6124,6126,6129,6131,6134,6136,6138,6140],{},[78,6117,5022],{"stretchy":80},[1520,6119,2710],{},[78,6121,874],{"separator":150},[1520,6123,2868],{},[78,6125,874],{"separator":150},[1520,6127,6128],{},"4",[78,6130,874],{"separator":150},[1520,6132,6133],{},"6",[78,6135,874],{"separator":150},[78,6137,2934],{},[4598,6139,5044],{},[78,6141,5047],{"stretchy":80},[142,6143,6144],{"encoding":144},"\\{0, 2, 4, 6, \\dots\\}",[48,6146,6148],{"className":6147,"ariaHidden":150},[149],[48,6149,6151,6154,6157,6160,6163,6166,6169,6172,6175,6178,6181,6184,6187,6190,6193,6196,6199],{"className":6150},[154],[48,6152],{"className":6153,"style":159},[158],[48,6155,5022],{"className":6156},[167],[48,6158,2710],{"className":6159},[163],[48,6161,874],{"className":6162},[944],[48,6164],{"className":6165,"style":282},[179],[48,6167,2868],{"className":6168},[163],[48,6170,874],{"className":6171},[944],[48,6173],{"className":6174,"style":282},[179],[48,6176,6128],{"className":6177},[163],[48,6179,874],{"className":6180},[944],[48,6182],{"className":6183,"style":282},[179],[48,6185,6133],{"className":6186},[163],[48,6188,874],{"className":6189},[944],[48,6191],{"className":6192,"style":282},[179],[48,6194,2934],{"className":6195},[3012],[48,6197],{"className":6198,"style":282},[179],[48,6200,5047],{"className":6201},[175]," 可数：双射为 ",[48,6204,6206,6233],{"className":6205,"translate":52},[56],[48,6207,6209],{"className":6208},[60],[62,6210,6211],{"xmlns":64},[67,6212,6213,6230],{},[70,6214,6215,6217,6219,6222,6224,6226,6228],{},[73,6216,101],{},[78,6218,81],{"stretchy":80},[73,6220,6221],{},"n",[78,6223,87],{"stretchy":80},[78,6225,90],{},[1520,6227,2868],{},[73,6229,6221],{},[142,6231,6232],{"encoding":144},"f(n) = 2n",[48,6234,6236,6263],{"className":6235,"ariaHidden":150},[149],[48,6237,6239,6242,6245,6248,6251,6254,6257,6260],{"className":6238},[154],[48,6240],{"className":6241,"style":159},[158],[48,6243,101],{"className":6244,"style":235},[163,171],[48,6246,81],{"className":6247},[167],[48,6249,6221],{"className":6250},[163,171],[48,6252,87],{"className":6253},[175],[48,6255],{"className":6256,"style":180},[179],[48,6258,90],{"className":6259},[184],[48,6261],{"className":6262,"style":180},[179],[48,6264,6266,6269,6272],{"className":6265},[154],[48,6267],{"className":6268,"style":3904},[158],[48,6270,2868],{"className":6271},[163],[48,6273,6221],{"className":6274},[163,171],[996,6276,6277,6278,6435,6436,6746,6747,6842,6843,1906],{},"整数集 ",[48,6279,6281,6333],{"className":6280,"translate":52},[56],[48,6282,6284],{"className":6283},[60],[62,6285,6286],{"xmlns":64},[67,6287,6288,6330],{},[70,6289,6290,6292,6294,6296,6298,6300,6302,6304,6306,6308,6310,6312,6314,6316,6318,6320,6322,6324,6326,6328],{},[73,6291,869],{"mathvariant":5016},[78,6293,90],{},[78,6295,5022],{"stretchy":80},[78,6297,2934],{},[78,6299,874],{"separator":150},[78,6301,1518],{},[1520,6303,2868],{},[78,6305,874],{"separator":150},[78,6307,1518],{},[1520,6309,1522],{},[78,6311,874],{"separator":150},[1520,6313,2710],{},[78,6315,874],{"separator":150},[1520,6317,1522],{},[78,6319,874],{"separator":150},[1520,6321,2868],{},[78,6323,874],{"separator":150},[78,6325,2934],{},[4598,6327,5044],{},[78,6329,5047],{"stretchy":80},[142,6331,6332],{"encoding":144},"\\mathbb{Z} = \\{\\dots, -2, -1, 0, 1, 2, \\dots\\}",[48,6334,6336,6354],{"className":6335,"ariaHidden":150},[149],[48,6337,6339,6342,6345,6348,6351],{"className":6338},[154],[48,6340],{"className":6341,"style":5060},[158],[48,6343,869],{"className":6344},[163,5064],[48,6346],{"className":6347,"style":180},[179],[48,6349,90],{"className":6350},[184],[48,6352],{"className":6353,"style":180},[179],[48,6355,6357,6360,6363,6366,6369,6372,6375,6378,6381,6384,6387,6390,6393,6396,6399,6402,6405,6408,6411,6414,6417,6420,6423,6426,6429,6432],{"className":6356},[154],[48,6358],{"className":6359,"style":159},[158],[48,6361,5022],{"className":6362},[167],[48,6364,2934],{"className":6365},[3012],[48,6367],{"className":6368,"style":282},[179],[48,6370,874],{"className":6371},[944],[48,6373],{"className":6374,"style":282},[179],[48,6376,1518],{"className":6377},[163],[48,6379,2868],{"className":6380},[163],[48,6382,874],{"className":6383},[944],[48,6385],{"className":6386,"style":282},[179],[48,6388,1518],{"className":6389},[163],[48,6391,1522],{"className":6392},[163],[48,6394,874],{"className":6395},[944],[48,6397],{"className":6398,"style":282},[179],[48,6400,2710],{"className":6401},[163],[48,6403,874],{"className":6404},[944],[48,6406],{"className":6407,"style":282},[179],[48,6409,1522],{"className":6410},[163],[48,6412,874],{"className":6413},[944],[48,6415],{"className":6416,"style":282},[179],[48,6418,2868],{"className":6419},[163],[48,6421,874],{"className":6422},[944],[48,6424],{"className":6425,"style":282},[179],[48,6427,2934],{"className":6428},[3012],[48,6430],{"className":6431,"style":282},[179],[48,6433,5047],{"className":6434},[175]," 可数：可以用 ",[48,6437,6439,6535],{"className":6438,"translate":52},[56],[48,6440,6442],{"className":6441},[60],[62,6443,6444],{"xmlns":64},[67,6445,6446,6532],{},[70,6447,6448,6450,6452,6454,6456,6458],{},[73,6449,101],{},[78,6451,81],{"stretchy":80},[73,6453,6221],{},[78,6455,87],{"stretchy":80},[78,6457,90],{},[70,6459,6460,6462],{},[78,6461,5022],{"fence":150},[6463,6464,6468,6497],"mtable",{"rowspacing":6465,"columnalign":6466,"columnspacing":6467},"0.36em","left left","1em",[6469,6470,6471,6486],"mtr",{},[6472,6473,6474],"mtd",{},[6475,6476,6477],"mstyle",{"scriptlevel":2710,"displaystyle":80},[70,6478,6479,6481,6484],{},[73,6480,6221],{},[73,6482,6483],{"mathvariant":75},"\u002F",[1520,6485,2868],{},[6472,6487,6488],{},[6475,6489,6490],{"scriptlevel":2710,"displaystyle":80},[70,6491,6492,6494],{},[73,6493,6221],{},[4598,6495,6496],{}," 为偶数",[6469,6498,6499,6521],{},[6472,6500,6501],{},[6475,6502,6503],{"scriptlevel":2710,"displaystyle":80},[70,6504,6505,6507,6509,6511,6513,6515,6517,6519],{},[78,6506,1518],{},[78,6508,81],{"stretchy":80},[73,6510,6221],{},[78,6512,2947],{},[1520,6514,1522],{},[78,6516,87],{"stretchy":80},[73,6518,6483],{"mathvariant":75},[1520,6520,2868],{},[6472,6522,6523],{},[6475,6524,6525],{"scriptlevel":2710,"displaystyle":80},[70,6526,6527,6529],{},[73,6528,6221],{},[4598,6530,6531],{}," 为奇数",[142,6533,6534],{"encoding":144},"f(n) = \\begin{cases} n\u002F2 & n \\text{ 为偶数} \\\\ -(n+1)\u002F2 & n \\text{ 为奇数} \\end{cases}",[48,6536,6538,6565],{"className":6537,"ariaHidden":150},[149],[48,6539,6541,6544,6547,6550,6553,6556,6559,6562],{"className":6540},[154],[48,6542],{"className":6543,"style":159},[158],[48,6545,101],{"className":6546,"style":235},[163,171],[48,6548,81],{"className":6549},[167],[48,6551,6221],{"className":6552},[163,171],[48,6554,87],{"className":6555},[175],[48,6557],{"className":6558,"style":180},[179],[48,6560,90],{"className":6561},[184],[48,6563],{"className":6564,"style":180},[179],[48,6566,6568,6572],{"className":6567},[154],[48,6569],{"className":6570,"style":6571},[158],"height:3em;vertical-align:-1.25em;",[48,6573,6575,6585,6742],{"className":6574},[3012],[48,6576,6580],{"className":6577,"style":6579},[167,6578],"delimcenter","top:0em;",[48,6581,5022],{"className":6582},[6583,6584],"delimsizing","size4",[48,6586,6588],{"className":6587},[163],[48,6589,6591,6670,6675],{"className":6590},[6463],[48,6592,6595],{"className":6593},[6594],"col-align-l",[48,6596,6598,6661],{"className":6597},[203,204],[48,6599,6601,6658],{"className":6600},[208],[48,6602,6605,6622],{"className":6603,"style":6604},[212],"height:1.69em;",[48,6606,6608,6612],{"style":6607},"top:-3.69em;",[48,6609],{"className":6610,"style":6611},[220],"height:3.008em;",[48,6613,6615,6618],{"className":6614},[163],[48,6616,6221],{"className":6617},[163,171],[48,6619,6621],{"className":6620},[163],"\u002F2",[48,6623,6625,6628],{"style":6624},"top:-2.25em;",[48,6626],{"className":6627,"style":6611},[220],[48,6629,6631,6634,6637,6640,6643,6646,6649,6652,6655],{"className":6630},[163],[48,6632,1518],{"className":6633},[163],[48,6635,81],{"className":6636},[167],[48,6638,6221],{"className":6639},[163,171],[48,6641],{"className":6642,"style":516},[179],[48,6644,2947],{"className":6645},[1684],[48,6647],{"className":6648,"style":516},[179],[48,6650,1522],{"className":6651},[163],[48,6653,87],{"className":6654},[175],[48,6656,6621],{"className":6657},[163],[48,6659,269],{"className":6660},[268],[48,6662,6664],{"className":6663},[208],[48,6665,6668],{"className":6666,"style":6667},[212],"height:1.19em;",[48,6669],{},[48,6671],{"className":6672,"style":6674},[6673],"arraycolsep","width:1em;",[48,6676,6678],{"className":6677},[6594],[48,6679,6681,6734],{"className":6680},[203,204],[48,6682,6684,6731],{"className":6683},[208],[48,6685,6687,6710],{"className":6686,"style":6604},[212],[48,6688,6689,6692],{"style":6607},[48,6690],{"className":6691,"style":6611},[220],[48,6693,6695,6698],{"className":6694},[163],[48,6696,6221],{"className":6697},[163,171],[48,6699,6701,6705],{"className":6700},[163,4646],[48,6702,6704],{"className":6703},[163]," ",[48,6706,6709],{"className":6707},[163,6708],"cjk_fallback","为偶数",[48,6711,6712,6715],{"style":6624},[48,6713],{"className":6714,"style":6611},[220],[48,6716,6718,6721],{"className":6717},[163],[48,6719,6221],{"className":6720},[163,171],[48,6722,6724,6727],{"className":6723},[163,4646],[48,6725,6704],{"className":6726},[163],[48,6728,6730],{"className":6729},[163,6708],"为奇数",[48,6732,269],{"className":6733},[268],[48,6735,6737],{"className":6736},[208],[48,6738,6740],{"className":6739,"style":6667},[212],[48,6741],{},[48,6743],{"className":6744},[175,6745],"nulldelimiter"," 把 ",[48,6748,6750,6784],{"className":6749,"translate":52},[56],[48,6751,6753],{"className":6752},[60],[62,6754,6755],{"xmlns":64},[67,6756,6757,6781],{},[70,6758,6759,6761,6763,6765,6767,6769,6771,6773,6775,6777,6779],{},[1520,6760,2710],{},[78,6762,874],{"separator":150},[1520,6764,1522],{},[78,6766,874],{"separator":150},[1520,6768,2868],{},[78,6770,874],{"separator":150},[1520,6772,5037],{},[78,6774,874],{"separator":150},[1520,6776,6128],{},[78,6778,874],{"separator":150},[78,6780,2934],{},[142,6782,6783],{"encoding":144},"0,1,2,3,4,\\dots",[48,6785,6787],{"className":6786,"ariaHidden":150},[149],[48,6788,6790,6794,6797,6800,6803,6806,6809,6812,6815,6818,6821,6824,6827,6830,6833,6836,6839],{"className":6789},[154],[48,6791],{"className":6792,"style":6793},[158],"height:0.8389em;vertical-align:-0.1944em;",[48,6795,2710],{"className":6796},[163],[48,6798,874],{"className":6799},[944],[48,6801],{"className":6802,"style":282},[179],[48,6804,1522],{"className":6805},[163],[48,6807,874],{"className":6808},[944],[48,6810],{"className":6811,"style":282},[179],[48,6813,2868],{"className":6814},[163],[48,6816,874],{"className":6817},[944],[48,6819],{"className":6820,"style":282},[179],[48,6822,5037],{"className":6823},[163],[48,6825,874],{"className":6826},[944],[48,6828],{"className":6829,"style":282},[179],[48,6831,6128],{"className":6832},[163],[48,6834,874],{"className":6835},[944],[48,6837],{"className":6838,"style":282},[179],[48,6840,2934],{"className":6841},[3012]," 映射到 ",[48,6844,6846,6884],{"className":6845,"translate":52},[56],[48,6847,6849],{"className":6848},[60],[62,6850,6851],{"xmlns":64},[67,6852,6853,6881],{},[70,6854,6855,6857,6859,6861,6863,6865,6867,6869,6871,6873,6875,6877,6879],{},[1520,6856,2710],{},[78,6858,874],{"separator":150},[78,6860,1518],{},[1520,6862,1522],{},[78,6864,874],{"separator":150},[1520,6866,1522],{},[78,6868,874],{"separator":150},[78,6870,1518],{},[1520,6872,2868],{},[78,6874,874],{"separator":150},[1520,6876,2868],{},[78,6878,874],{"separator":150},[78,6880,2934],{},[142,6882,6883],{"encoding":144},"0,-1,1,-2,2,\\dots",[48,6885,6887],{"className":6886,"ariaHidden":150},[149],[48,6888,6890,6893,6896,6899,6902,6905,6908,6911,6914,6917,6920,6923,6926,6929,6932,6935,6938,6941,6944],{"className":6889},[154],[48,6891],{"className":6892,"style":6793},[158],[48,6894,2710],{"className":6895},[163],[48,6897,874],{"className":6898},[944],[48,6900],{"className":6901,"style":282},[179],[48,6903,1518],{"className":6904},[163],[48,6906,1522],{"className":6907},[163],[48,6909,874],{"className":6910},[944],[48,6912],{"className":6913,"style":282},[179],[48,6915,1522],{"className":6916},[163],[48,6918,874],{"className":6919},[944],[48,6921],{"className":6922,"style":282},[179],[48,6924,1518],{"className":6925},[163],[48,6927,2868],{"className":6928},[163],[48,6930,874],{"className":6931},[944],[48,6933],{"className":6934,"style":282},[179],[48,6936,2868],{"className":6937},[163],[48,6939,874],{"className":6940},[944],[48,6942],{"className":6943,"style":282},[179],[48,6945,2934],{"className":6946},[3012],[996,6948,6949,6950,6981,6982,7022],{},"有理数集 ",[48,6951,6953,6968],{"className":6952,"translate":52},[56],[48,6954,6956],{"className":6955},[60],[62,6957,6958],{"xmlns":64},[67,6959,6960,6965],{},[70,6961,6962],{},[73,6963,6964],{"mathvariant":5016},"Q",[142,6966,6967],{"encoding":144},"\\mathbb{Q}",[48,6969,6971],{"className":6970,"ariaHidden":150},[149],[48,6972,6974,6978],{"className":6973},[154],[48,6975],{"className":6976,"style":6977},[158],"height:0.8556em;vertical-align:-0.1667em;",[48,6979,6964],{"className":6980},[163,5064]," 同样可数（可以用「对角线枚举法」把所有分数 ",[48,6983,6985,7004],{"className":6984,"translate":52},[56],[48,6986,6988],{"className":6987},[60],[62,6989,6990],{"xmlns":64},[67,6991,6992,7001],{},[70,6993,6994,6996,6998],{},[73,6995,11],{},[73,6997,6483],{"mathvariant":75},[73,6999,7000],{},"q",[142,7002,7003],{"encoding":144},"p\u002Fq",[48,7005,7007],{"className":7006,"ariaHidden":150},[149],[48,7008,7010,7013,7016,7019],{"className":7009},[154],[48,7011],{"className":7012,"style":159},[158],[48,7014,11],{"className":7015},[163,171],[48,7017,6483],{"className":7018},[163],[48,7020,7000],{"className":7021,"style":343},[163,171]," 排成一个序列——这是康托尔的另一个经典成果，与本文讨论的论证共享「对角线」之名，但用法不同；请读者注意不要把两者混淆）。",[11,7024,7025],{},"看上去，似乎所有无穷集都能被装进自然数的框架之内——直到我们遇到实数。",[35,7027,7028],{"id":7028},"实数的不可数",[1295,7030,7031],{},[11,7032,7033,7034,7086],{},"定理（康托尔，1891）。区间 ",[48,7035,7037,7059],{"className":7036,"translate":52},[56],[48,7038,7040],{"className":7039},[60],[62,7041,7042],{"xmlns":64},[67,7043,7044,7056],{},[70,7045,7046,7048,7050,7052,7054],{},[78,7047,81],{"stretchy":80},[1520,7049,2710],{},[78,7051,874],{"separator":150},[1520,7053,1522],{},[78,7055,87],{"stretchy":80},[142,7057,7058],{"encoding":144},"(0,1)",[48,7060,7062],{"className":7061,"ariaHidden":150},[149],[48,7063,7065,7068,7071,7074,7077,7080,7083],{"className":7064},[154],[48,7066],{"className":7067,"style":159},[158],[48,7069,81],{"className":7070},[167],[48,7072,2710],{"className":7073},[163],[48,7075,874],{"className":7076},[944],[48,7078],{"className":7079,"style":282},[179],[48,7081,1522],{"className":7082},[163],[48,7084,87],{"className":7085},[175]," 中的实数是不可数的。",[11,7088,7089,7090,7141],{},"这意味着：不存在任何方法能把 ",[48,7091,7093,7114],{"className":7092,"translate":52},[56],[48,7094,7096],{"className":7095},[60],[62,7097,7098],{"xmlns":64},[67,7099,7100,7112],{},[70,7101,7102,7104,7106,7108,7110],{},[78,7103,81],{"stretchy":80},[1520,7105,2710],{},[78,7107,874],{"separator":150},[1520,7109,1522],{},[78,7111,87],{"stretchy":80},[142,7113,7058],{"encoding":144},[48,7115,7117],{"className":7116,"ariaHidden":150},[149],[48,7118,7120,7123,7126,7129,7132,7135,7138],{"className":7119},[154],[48,7121],{"className":7122,"style":159},[158],[48,7124,81],{"className":7125},[167],[48,7127,2710],{"className":7128},[163],[48,7130,874],{"className":7131},[944],[48,7133],{"className":7134,"style":282},[179],[48,7136,1522],{"className":7137},[163],[48,7139,87],{"className":7140},[175]," 中的实数与自然数一一对应。换言之，实数的无穷「大于」自然数的无穷。",[11,7143,7144],{},"我们采用反证法。",[11,7146,7147,7148,7199,7200,7296],{},"假设：",[48,7149,7151,7172],{"className":7150,"translate":52},[56],[48,7152,7154],{"className":7153},[60],[62,7155,7156],{"xmlns":64},[67,7157,7158,7170],{},[70,7159,7160,7162,7164,7166,7168],{},[78,7161,81],{"stretchy":80},[1520,7163,2710],{},[78,7165,874],{"separator":150},[1520,7167,1522],{},[78,7169,87],{"stretchy":80},[142,7171,7058],{"encoding":144},[48,7173,7175],{"className":7174,"ariaHidden":150},[149],[48,7176,7178,7181,7184,7187,7190,7193,7196],{"className":7177},[154],[48,7179],{"className":7180,"style":159},[158],[48,7182,81],{"className":7183},[167],[48,7185,2710],{"className":7186},[163],[48,7188,874],{"className":7189},[944],[48,7191],{"className":7192,"style":282},[179],[48,7194,1522],{"className":7195},[163],[48,7197,87],{"className":7198},[175]," 中的实数可数。即存在双射 ",[48,7201,7203,7233],{"className":7202,"translate":52},[56],[48,7204,7206],{"className":7205},[60],[62,7207,7208],{"xmlns":64},[67,7209,7210,7230],{},[70,7211,7212,7214,7216,7218,7220,7222,7224,7226,7228],{},[73,7213,101],{},[78,7215,1495],{},[73,7217,5017],{"mathvariant":5016},[78,7219,1504],{},[78,7221,81],{"stretchy":80},[1520,7223,2710],{},[78,7225,874],{"separator":150},[1520,7227,1522],{},[78,7229,87],{"stretchy":80},[142,7231,7232],{"encoding":144},"f: \\mathbb{N} \\to (0,1)",[48,7234,7236,7254,7272],{"className":7235,"ariaHidden":150},[149],[48,7237,7239,7242,7245,7248,7251],{"className":7238},[154],[48,7240],{"className":7241,"style":766},[158],[48,7243,101],{"className":7244,"style":235},[163,171],[48,7246],{"className":7247,"style":180},[179],[48,7249,1495],{"className":7250},[184],[48,7252],{"className":7253,"style":180},[179],[48,7255,7257,7260,7263,7266,7269],{"className":7256},[154],[48,7258],{"className":7259,"style":5060},[158],[48,7261,5017],{"className":7262},[163,5064],[48,7264],{"className":7265,"style":180},[179],[48,7267,1504],{"className":7268},[184],[48,7270],{"className":7271,"style":180},[179],[48,7273,7275,7278,7281,7284,7287,7290,7293],{"className":7274},[154],[48,7276],{"className":7277,"style":159},[158],[48,7279,81],{"className":7280},[167],[48,7282,2710],{"className":7283},[163],[48,7285,874],{"className":7286},[944],[48,7288],{"className":7289,"style":282},[179],[48,7291,1522],{"className":7292},[163],[48,7294,87],{"className":7295},[175],"，使得我们可以把这些实数全部排成一个无穷列表：",[48,7298,7300],{"className":7299,"translate":52},[51],[48,7301,7303,7589],{"className":7302,"translate":52},[56],[48,7304,7306],{"className":7305},[60],[62,7307,7308],{"xmlns":64,"display":65},[67,7309,7310,7586],{},[6463,7311,7314,7375,7435,7495,7555],{"rowspacing":7312,"columnalign":7313,"columnspacing":3327},"0.25em","right left",[6469,7315,7316,7326],{},[6472,7317,7318],{},[6475,7319,7320],{"scriptlevel":2710,"displaystyle":150},[115,7321,7322,7324],{},[73,7323,84],{},[1520,7325,1522],{},[6472,7327,7328],{},[6475,7329,7330],{"scriptlevel":2710,"displaystyle":150},[70,7331,7332,7334,7336,7339,7346,7348,7355,7357,7364,7366,7373],{},[70,7333],{},[78,7335,90],{},[1520,7337,7338],{},"0.",[115,7340,7341,7343],{},[73,7342,879],{},[1520,7344,7345],{},"11",[4598,7347,5044],{},[115,7349,7350,7352],{},[73,7351,879],{},[1520,7353,7354],{},"12",[4598,7356,5044],{},[115,7358,7359,7361],{},[73,7360,879],{},[1520,7362,7363],{},"13",[4598,7365,5044],{},[115,7367,7368,7370],{},[73,7369,879],{},[1520,7371,7372],{},"14",[78,7374,2934],{},[6469,7376,7377,7387],{},[6472,7378,7379],{},[6475,7380,7381],{"scriptlevel":2710,"displaystyle":150},[115,7382,7383,7385],{},[73,7384,84],{},[1520,7386,2868],{},[6472,7388,7389],{},[6475,7390,7391],{"scriptlevel":2710,"displaystyle":150},[70,7392,7393,7395,7397,7399,7406,7408,7415,7417,7424,7426,7433],{},[70,7394],{},[78,7396,90],{},[1520,7398,7338],{},[115,7400,7401,7403],{},[73,7402,879],{},[1520,7404,7405],{},"21",[4598,7407,5044],{},[115,7409,7410,7412],{},[73,7411,879],{},[1520,7413,7414],{},"22",[4598,7416,5044],{},[115,7418,7419,7421],{},[73,7420,879],{},[1520,7422,7423],{},"23",[4598,7425,5044],{},[115,7427,7428,7430],{},[73,7429,879],{},[1520,7431,7432],{},"24",[78,7434,2934],{},[6469,7436,7437,7447],{},[6472,7438,7439],{},[6475,7440,7441],{"scriptlevel":2710,"displaystyle":150},[115,7442,7443,7445],{},[73,7444,84],{},[1520,7446,5037],{},[6472,7448,7449],{},[6475,7450,7451],{"scriptlevel":2710,"displaystyle":150},[70,7452,7453,7455,7457,7459,7466,7468,7475,7477,7484,7486,7493],{},[70,7454],{},[78,7456,90],{},[1520,7458,7338],{},[115,7460,7461,7463],{},[73,7462,879],{},[1520,7464,7465],{},"31",[4598,7467,5044],{},[115,7469,7470,7472],{},[73,7471,879],{},[1520,7473,7474],{},"32",[4598,7476,5044],{},[115,7478,7479,7481],{},[73,7480,879],{},[1520,7482,7483],{},"33",[4598,7485,5044],{},[115,7487,7488,7490],{},[73,7489,879],{},[1520,7491,7492],{},"34",[78,7494,2934],{},[6469,7496,7497,7507],{},[6472,7498,7499],{},[6475,7500,7501],{"scriptlevel":2710,"displaystyle":150},[115,7502,7503,7505],{},[73,7504,84],{},[1520,7506,6128],{},[6472,7508,7509],{},[6475,7510,7511],{"scriptlevel":2710,"displaystyle":150},[70,7512,7513,7515,7517,7519,7526,7528,7535,7537,7544,7546,7553],{},[70,7514],{},[78,7516,90],{},[1520,7518,7338],{},[115,7520,7521,7523],{},[73,7522,879],{},[1520,7524,7525],{},"41",[4598,7527,5044],{},[115,7529,7530,7532],{},[73,7531,879],{},[1520,7533,7534],{},"42",[4598,7536,5044],{},[115,7538,7539,7541],{},[73,7540,879],{},[1520,7542,7543],{},"43",[4598,7545,5044],{},[115,7547,7548,7550],{},[73,7549,879],{},[1520,7551,7552],{},"44",[78,7554,2934],{},[6469,7556,7557,7563],{},[6472,7558,7559],{},[6475,7560,7561],{"scriptlevel":2710,"displaystyle":150},[70,7562],{},[6472,7564,7565],{},[6475,7566,7567],{"scriptlevel":2710,"displaystyle":150},[70,7568,7569,7571,7574],{},[70,7570],{},[4598,7572,7573],{},"  ",[70,7575,7576,7579],{},[73,7577,7578],{"mathvariant":75},"⋮",[7580,7581,7582],"mpadded",{"height":3327,"voffset":3327},[179,7583],{"mathbackground":7584,"width":3327,"height":7585},"black","1.5em",[142,7587,7588],{"encoding":144},"\\begin{aligned}\nx_1 &= 0.d_{11}\\, d_{12}\\, d_{13}\\, d_{14} \\dots \\\\\nx_2 &= 0.d_{21}\\, d_{22}\\, d_{23}\\, d_{24} \\dots \\\\\nx_3 &= 0.d_{31}\\, d_{32}\\, d_{33}\\, d_{34} \\dots \\\\\nx_4 &= 0.d_{41}\\, d_{42}\\, d_{43}\\, d_{44} \\dots \\\\\n&\\ \\ 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表示第 ",[48,8819,8821,8834],{"className":8820,"translate":52},[56],[48,8822,8824],{"className":8823},[60],[62,8825,8826],{"xmlns":64},[67,8827,8828,8832],{},[70,8829,8830],{},[73,8831,2631],{},[142,8833,2631],{"encoding":144},[48,8835,8837],{"className":8836,"ariaHidden":150},[149],[48,8838,8840,8844],{"className":8839},[154],[48,8841],{"className":8842,"style":8843},[158],"height:0.6595em;",[48,8845,2631],{"className":8846},[163,171]," 个数的第 ",[48,8849,8851,8864],{"className":8850,"translate":52},[56],[48,8852,8854],{"className":8853},[60],[62,8855,8856],{"xmlns":64},[67,8857,8858,8862],{},[70,8859,8860],{},[73,8861,8758],{},[142,8863,8758],{"encoding":144},[48,8865,8867],{"className":8866,"ariaHidden":150},[149],[48,8868,8870,8874],{"className":8869},[154],[48,8871],{"className":8872,"style":8873},[158],"height:0.854em;vertical-align:-0.1944em;",[48,8875,8758],{"className":8876,"style":8804},[163,171]," 位小数（取值 ",[48,8879,8881,8894],{"className":8880,"translate":52},[56],[48,8882,8884],{"className":8883},[60],[62,8885,8886],{"xmlns":64},[67,8887,8888,8892],{},[70,8889,8890],{},[1520,8891,2710],{},[142,8893,2710],{"encoding":144},[48,8895,8897],{"className":8896,"ariaHidden":150},[149],[48,8898,8900,8903],{"className":8899},[154],[48,8901],{"className":8902,"style":3904},[158],[48,8904,2710],{"className":8905},[163]," 到 ",[48,8908,8910,8924],{"className":8909,"translate":52},[56],[48,8911,8913],{"className":8912},[60],[62,8914,8915],{"xmlns":64},[67,8916,8917,8922],{},[70,8918,8919],{},[1520,8920,8921],{},"9",[142,8923,8921],{"encoding":144},[48,8925,8927],{"className":8926,"ariaHidden":150},[149],[48,8928,8930,8933],{"className":8929},[154],[48,8931],{"className":8932,"style":3904},[158],[48,8934,8921],{"className":8935},[163],"）。",[11,8938,8939,8940,9185],{},"构造对角数。现在我们沿着「对角线」走——即取出第 1 个数的第 1 位、第 2 个数的第 2 位、第 3 个数的第 3 位，依此类推——并根据以下规则，用这些数字构造一个新数 ",[48,8941,8943,8988],{"className":8942,"translate":52},[56],[48,8944,8946],{"className":8945},[60],[62,8947,8948],{"xmlns":64},[67,8949,8950,8985],{},[70,8951,8952,8954,8956,8958,8965,8971,8977,8983],{},[73,8953,4526],{},[78,8955,90],{},[1520,8957,7338],{},[115,8959,8960,8963],{},[73,8961,8962],{},"e",[1520,8964,1522],{},[115,8966,8967,8969],{},[73,8968,8962],{},[1520,8970,2868],{},[115,8972,8973,8975],{},[73,8974,8962],{},[1520,8976,5037],{},[115,8978,8979,8981],{},[73,8980,8962],{},[1520,8982,6128],{},[78,8984,2934],{},[142,8986,8987],{"encoding":144},"y = 0.e_1 e_2 e_3 e_4 \\dots",[48,8989,8991,9009],{"className":8990,"ariaHidden":150},[149],[48,8992,8994,8997,9000,9003,9006],{"className":8993},[154],[48,8995],{"className":8996,"style":2731},[158],[48,8998,4526],{"className":8999,"style":343},[163,171],[48,9001],{"className":9002,"style":180},[179],[48,9004,90],{"className":9005},[184],[48,9007],{"className":9008,"style":180},[179],[48,9010,9012,9016,9019,9059,9099,9139,9179,9182],{"className":9011},[154],[48,9013],{"className":9014,"style":9015},[158],"height:0.7944em;vertical-align:-0.15em;",[48,9017,7338],{"className":9018},[163],[48,9020,9022,9025],{"className":9021},[163],[48,9023,8962],{"className":9024},[163,171],[48,9026,9028],{"className":9027},[292],[48,9029,9031,9051],{"className":9030},[203,204],[48,9032,9034,9048],{"className":9033},[208],[48,9035,9037],{"className":9036,"style":2750},[212],[48,9038,9039,9042],{"style":305},[48,9040],{"className":9041,"style":309},[220],[48,9043,9045],{"className":9044},[225,226,227,228],[48,9046,1522],{"className":9047},[163,228],[48,9049,269],{"className":9050},[268],[48,9052,9054],{"className":9053},[208],[48,9055,9057],{"className":9056,"style":2771},[212],[48,9058],{},[48,9060,9062,9065],{"className":9061},[163],[48,9063,8962],{"className":9064},[163,171],[48,9066,9068],{"className":9067},[292],[48,9069,9071,9091],{"className":9070},[203,204],[48,9072,9074,9088],{"className":9073},[208],[48,9075,9077],{"className":9076,"style":2750},[212],[48,9078,9079,9082],{"style":305},[48,9080],{"className":9081,"style":309},[220],[48,9083,9085],{"className":9084},[225,226,227,228],[48,9086,2868],{"className":9087},[163,228],[48,9089,269],{"className":9090},[268],[48,9092,9094],{"className":9093},[208],[48,9095,9097],{"className":9096,"style":2771},[212],[48,9098],{},[48,9100,9102,9105],{"className":9101},[163],[48,9103,8962],{"className":9104},[163,171],[48,9106,9108],{"className":9107},[292],[48,9109,9111,9131],{"className":9110},[203,204],[48,9112,9114,9128],{"className":9113},[208],[48,9115,9117],{"className":9116,"style":2750},[212],[48,9118,9119,9122],{"style":305},[48,9120],{"className":9121,"style":309},[220],[48,9123,9125],{"className":9124},[225,226,227,228],[48,9126,5037],{"className":9127},[163,228],[48,9129,269],{"className":9130},[268],[48,9132,9134],{"className":9133},[208],[48,9135,9137],{"className":9136,"style":2771},[212],[48,9138],{},[48,9140,9142,9145],{"className":9141},[163],[48,9143,8962],{"className":9144},[163,171],[48,9146,9148],{"className":9147},[292],[48,9149,9151,9171],{"className":9150},[203,204],[48,9152,9154,9168],{"className":9153},[208],[48,9155,9157],{"className":9156,"style":2750},[212],[48,9158,9159,9162],{"style":305},[48,9160],{"className":9161,"style":309},[220],[48,9163,9165],{"className":9164},[225,226,227,228],[48,9166,6128],{"className":9167},[163,228],[48,9169,269],{"className":9170},[268],[48,9172,9174],{"className":9173},[208],[48,9175,9177],{"className":9176,"style":2771},[212],[48,9178],{},[48,9180],{"className":9181,"style":282},[179],[48,9183,2934],{"className":9184},[3012],"：",[48,9187,9189],{"className":9188,"translate":52},[51],[48,9190,9192,9280],{"className":9191,"translate":52},[56],[48,9193,9195],{"className":9194},[60],[62,9196,9197],{"xmlns":64,"display":65},[67,9198,9199,9277],{},[70,9200,9201,9207,9209],{},[115,9202,9203,9205],{},[73,9204,8962],{},[73,9206,6221],{},[78,9208,90],{},[70,9210,9211,9213],{},[78,9212,5022],{"fence":150},[6463,9214,9215,9247],{"rowspacing":6465,"columnalign":6466,"columnspacing":6467},[6469,9216,9217,9224],{},[6472,9218,9219],{},[6475,9220,9221],{"scriptlevel":2710,"displaystyle":80},[1520,9222,9223],{},"5",[6472,9225,9226],{},[6475,9227,9228],{"scriptlevel":2710,"displaystyle":80},[70,9229,9230,9233,9243,9245],{},[4598,9231,9232],{},"若 ",[115,9234,9235,9237],{},[73,9236,879],{},[70,9238,9239,9241],{},[73,9240,6221],{},[73,9242,6221],{},[78,9244,5291],{"mathvariant":75},[1520,9246,9223],{},[6469,9248,9249,9255],{},[6472,9250,9251],{},[6475,9252,9253],{"scriptlevel":2710,"displaystyle":80},[1520,9254,6133],{},[6472,9256,9257],{},[6475,9258,9259],{"scriptlevel":2710,"displaystyle":80},[70,9260,9261,9263,9273,9275],{},[4598,9262,9232],{},[115,9264,9265,9267],{},[73,9266,879],{},[70,9268,9269,9271],{},[73,9270,6221],{},[73,9272,6221],{},[78,9274,90],{},[1520,9276,9223],{},[142,9278,9279],{"encoding":144},"e_n = \\begin{cases} 5 & \\text{若 } d_{nn} \\neq 5 \\\\ 6 & \\text{若 } d_{nn} = 5 \\end{cases}",[48,9281,9283,9339],{"className":9282,"ariaHidden":150},[149],[48,9284,9286,9289,9330,9333,9336],{"className":9285},[154],[48,9287],{"className":9288,"style":5434},[158],[48,9290,9292,9295],{"className":9291},[163],[48,9293,8962],{"className":9294},[163,171],[48,9296,9298],{"className":9297},[292],[48,9299,9301,9322],{"className":9300},[203,204],[48,9302,9304,9319],{"className":9303},[208],[48,9305,9308],{"className":9306,"style":9307},[212],"height:0.1514em;",[48,9309,9310,9313],{"style":305},[48,9311],{"className":9312,"style":309},[220],[48,9314,9316],{"className":9315},[225,226,227,228],[48,9317,6221],{"className":9318},[163,171,228],[48,9320,269],{"className":9321},[268],[48,9323,9325],{"className":9324},[208],[48,9326,9328],{"className":9327,"style":2771},[212],[48,9329],{},[48,9331],{"className":9332,"style":180},[179],[48,9334,90],{"className":9335},[184],[48,9337],{"className":9338,"style":180},[179],[48,9340,9342,9345],{"className":9341},[154],[48,9343],{"className":9344,"style":6571},[158],[48,9346,9348,9354,9610],{"className":9347},[3012],[48,9349,9351],{"className":9350,"style":6579},[167,6578],[48,9352,5022],{"className":9353},[6583,6584],[48,9355,9357],{"className":9356},[163],[48,9358,9360,9405,9408],{"className":9359},[6463],[48,9361,9363],{"className":9362},[6594],[48,9364,9366,9397],{"className":9365},[203,204],[48,9367,9369,9394],{"className":9368},[208],[48,9370,9372,9383],{"className":9371,"style":6604},[212],[48,9373,9374,9377],{"style":6607},[48,9375],{"className":9376,"style":6611},[220],[48,9378,9380],{"className":9379},[163],[48,9381,9223],{"className":9382},[163],[48,9384,9385,9388],{"style":6624},[48,9386],{"className":9387,"style":6611},[220],[48,9389,9391],{"className":9390},[163],[48,9392,6133],{"className":9393},[163],[48,9395,269],{"className":9396},[268],[48,9398,9400],{"className":9399},[208],[48,9401,9403],{"className":9402,"style":6667},[212],[48,9404],{},[48,9406],{"className":9407,"style":6674},[6673],[48,9409,9411],{"className":9410},[6594],[48,9412,9414,9602],{"className":9413},[203,204],[48,9415,9417,9599],{"className":9416},[208],[48,9418,9420,9527],{"className":9419,"style":6604},[212],[48,9421,9422,9425],{"style":6607},[48,9423],{"className":9424,"style":6611},[220],[48,9426,9428,9438,9482,9485,9521,9524],{"className":9427},[163],[48,9429,9431,9435],{"className":9430},[163,4646],[48,9432,9434],{"className":9433},[163,6708],"若",[48,9436,6704],{"className":9437},[163],[48,9439,9441,9444],{"className":9440},[163],[48,9442,879],{"className":9443},[163,171],[48,9445,9447],{"className":9446},[292],[48,9448,9450,9474],{"className":9449},[203,204],[48,9451,9453,9471],{"className":9452},[208],[48,9454,9456],{"className":9455,"style":9307},[212],[48,9457,9458,9461],{"style":305},[48,9459],{"className":9460,"style":309},[220],[48,9462,9464],{"className":9463},[225,226,227,228],[48,9465,9467],{"className":9466},[163,228],[48,9468,9470],{"className":9469},[163,171,228],"nn",[48,9472,269],{"className":9473},[268],[48,9475,9477],{"className":9476},[208],[48,9478,9480],{"className":9479,"style":2771},[212],[48,9481],{},[48,9483],{"className":9484,"style":180},[179],[48,9486,9488,9515,9518],{"className":9487},[184],[48,9489,9491],{"className":9490},[184],[48,9492,9494],{"className":9493},[163,5391],[48,9495,9497],{"className":9496},[5395],[48,9498,9500,9503,9512],{"className":9499},[5399],[48,9501],{"className":9502,"style":766},[158],[48,9504,9506],{"className":9505},[5406],[48,9507,9509],{"className":9508},[163],[48,9510,5413],{"className":9511},[184],[48,9513],{"className":9514},[5417],[48,9516],{"className":9517},[179,5421],[48,9519,90],{"className":9520},[184],[48,9522],{"className":9523,"style":180},[179],[48,9525,9223],{"className":9526},[163],[48,9528,9529,9532],{"style":6624},[48,9530],{"className":9531,"style":6611},[220],[48,9533,9535,9544,9587,9590,9593,9596],{"className":9534},[163],[48,9536,9538,9541],{"className":9537},[163,4646],[48,9539,9434],{"className":9540},[163,6708],[48,9542,6704],{"className":9543},[163],[48,9545,9547,9550],{"className":9546},[163],[48,9548,879],{"className":9549},[163,171],[48,9551,9553],{"className":9552},[292],[48,9554,9556,9579],{"className":9555},[203,204],[48,9557,9559,9576],{"className":9558},[208],[48,9560,9562],{"className":9561,"style":9307},[212],[48,9563,9564,9567],{"style":305},[48,9565],{"className":9566,"style":309},[220],[48,9568,9570],{"className":9569},[225,226,227,228],[48,9571,9573],{"className":9572},[163,228],[48,9574,9470],{"className":9575},[163,171,228],[48,9577,269],{"className":9578},[268],[48,9580,9582],{"className":9581},[208],[48,9583,9585],{"className":9584,"style":2771},[212],[48,9586],{},[48,9588],{"className":9589,"style":180},[179],[48,9591,90],{"className":9592},[184],[48,9594],{"className":9595,"style":180},[179],[48,9597,9223],{"className":9598},[163],[48,9600,269],{"className":9601},[268],[48,9603,9605],{"className":9604},[208],[48,9606,9608],{"className":9607,"style":6667},[212],[48,96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",[48,9617,9619,9649],{"className":9618,"translate":52},[56],[48,9620,9622],{"className":9621},[60],[62,9623,9624],{"xmlns":64},[67,9625,9626,9646],{},[70,9627,9628,9634,9636],{},[115,9629,9630,9632],{},[73,9631,8962],{},[73,9633,6221],{},[78,9635,5291],{"mathvariant":75},[115,9637,9638,9640],{},[73,9639,879],{},[70,9641,9642,9644],{},[73,9643,6221],{},[73,9645,6221],{},[142,9647,9648],{"encoding":144},"e_n \\neq d_{nn}",[48,9650,9652,9740],{"className":9651,"ariaHidden":150},[149],[48,9653,9655,9658,9698,9701,9737],{"className":9654},[154],[48,9656],{"className":9657,"style":766},[158],[48,9659,9661,9664],{"className":9660},[163],[48,9662,8962],{"className":9663},[163,171],[48,9665,9667],{"className":9666},[292],[48,9668,9670,9690],{"className":9669},[203,204],[48,9671,9673,9687],{"className":9672},[208],[48,9674,9676],{"className":9675,"style":9307},[212],[48,9677,9678,9681],{"style":305},[48,9679],{"className":9680,"style":309},[220],[48,9682,9684],{"className":9683},[225,226,227,228],[48,9685,6221],{"className":9686},[163,171,228],[48,9688,269],{"className":9689},[268],[48,9691,9693],{"className":9692},[208],[48,9694,9696],{"className":9695,"style":2771},[212],[48,9697],{},[48,9699],{"className":9700,"style":180},[179],[48,9702,9704,9731,9734],{"className":9703},[184],[48,9705,9707],{"className":9706},[184],[48,9708,9710],{"className":9709},[163,5391],[48,9711,9713],{"className":9712},[5395],[48,9714,9716,9719,9728],{"className":9715},[5399],[48,9717],{"className":9718,"style":766},[158],[48,9720,9722],{"className":9721},[5406],[48,9723,9725],{"className":9724},[163],[48,9726,5413],{"className":9727},[184],[48,9729],{"className":9730},[5417],[48,9732],{"className":9733},[179,5421],[48,9735,90],{"className":9736},[184],[48,9738],{"className":9739,"style":180},[179],[48,9741,9743,9747],{"className":9742},[154],[48,9744],{"className":9745,"style":9746},[158],"height:0.8444em;vertical-align:-0.15em;",[48,9748,9750,9753],{"className":9749},[163],[48,9751,879],{"className":9752},[163,171],[48,9754,9756],{"className":9755},[292],[48,9757,9759,9782],{"className":9758},[203,204],[48,9760,9762,9779],{"className":9761},[208],[48,9763,9765],{"className":9764,"style":9307},[212],[48,9766,9767,9770],{"style":305},[48,9768],{"className":9769,"style":309},[220],[48,9771,9773],{"className":9772},[225,226,227,228],[48,9774,9776],{"className":9775},[163,228],[48,9777,9470],{"className":9778},[163,171,228],[48,9780,269],{"className":9781},[268],[48,9783,9785],{"className":9784},[208],[48,9786,9788],{"className":9787,"style":2771},[212],[48,9789],{},"；具体选哪两个数字无关紧要，只要每位数字刻意「避开」相应的对角线数字，同时避免出现全 0 或全 9 的小数表示——那会在十进制记法中引入歧义。）",[11,9792,9793,9794,9822],{},"现在关键问题出现了：",[48,9795,9797,9810],{"className":9796,"translate":52},[56],[48,9798,9800],{"className":9799},[60],[62,9801,9802],{"xmlns":64},[67,9803,9804,9808],{},[70,9805,9806],{},[73,9807,4526],{},[142,9809,4526],{"encoding":144},[48,9811,9813],{"className":9812,"ariaHidden":150},[149],[48,9814,9816,9819],{"className":9815},[154],[48,9817],{"className":9818,"style":2731},[158],[48,9820,4526],{"className":9821,"style":343},[163,171]," 会出现在原来的列表里吗？",[11,9824,9825,9826,9854,9855,9883,9884,9978,9979,8936],{},"假设 ",[48,9827,9829,9842],{"className":9828,"translate":52},[56],[48,9830,9832],{"className":9831},[60],[62,9833,9834],{"xmlns":64},[67,9835,9836,9840],{},[70,9837,9838],{},[73,9839,4526],{},[142,9841,4526],{"encoding":144},[48,9843,9845],{"className":9844,"ariaHidden":150},[149],[48,9846,9848,9851],{"className":9847},[154],[48,9849],{"className":9850,"style":2731},[158],[48,9852,4526],{"className":9853,"style":343},[163,171]," 出现在列表中，那么 ",[48,9856,9858,9871],{"className":9857,"translate":52},[56],[48,9859,9861],{"className":9860},[60],[62,9862,9863],{"xmlns":64},[67,9864,9865,9869],{},[70,9866,9867],{},[73,9868,4526],{},[142,9870,4526],{"encoding":144},[48,9872,9874],{"className":9873,"ariaHidden":150},[149],[48,9875,9877,9880],{"className":9876},[154],[48,9878],{"className":9879,"style":2731},[158],[48,9881,4526],{"className":9882,"style":343},[163,171]," 必定等于被枚举的某个数，不妨设 ",[48,9885,9887,9910],{"className":9886,"translate":52},[56],[48,9888,9890],{"className":9889},[60],[62,9891,9892],{"xmlns":64},[67,9893,9894,9907],{},[70,9895,9896,9898,9900],{},[73,9897,4526],{},[78,9899,90],{},[115,9901,9902,9904],{},[73,9903,84],{},[73,9905,9906],{},"k",[142,9908,9909],{"encoding":144},"y = x_k",[48,9911,9913,9931],{"className":9912,"ariaHidden":150},[149],[48,9914,9916,9919,9922,9925,9928],{"className":9915},[154],[48,9917],{"className":9918,"style":2731},[158],[48,9920,4526],{"className":9921,"style":343},[163,171],[48,9923],{"className":9924,"style":180},[179],[48,9926,90],{"className":9927},[184],[48,9929],{"className":9930,"style":180},[179],[48,9932,9934,9937],{"className":9933},[154],[48,9935],{"className":9936,"style":5434},[158],[48,9938,9940,9943],{"className":9939},[163],[48,9941,84],{"className":9942},[163,171],[48,9944,9946],{"className":9945},[292],[48,9947,9949,9970],{"className":9948},[203,204],[48,9950,9952,9967],{"className":9951},[208],[48,9953,9955],{"className":9954,"style":302},[212],[48,9956,9957,9960],{"style":305},[48,9958],{"className":9959,"style":309},[220],[48,9961,9963],{"className":9962},[225,226,227,228],[48,9964,9906],{"className":9965,"style":9966},[163,171,228],"margin-right:0.0315em;",[48,9968,269],{"className":9969},[268],[48,9971,9973],{"className":9972},[208],[48,9974,9976],{"className":9975,"style":2771},[212],[48,9977],{},"（对某个确定的指标 ",[48,9980,9982,9995],{"className":9981,"translate":52},[56],[48,9983,9985],{"className":9984},[60],[62,9986,9987],{"xmlns":64},[67,9988,9989,9993],{},[70,9990,9991],{},[73,9992,9906],{},[142,9994,9906],{"encoding":144},[48,9996,9998],{"className":9997,"ariaHidden":150},[149],[48,9999,10001,10004],{"className":10000},[154],[48,10002],{"className":10003,"style":5810},[158],[48,10005,9906],{"className":10006,"style":9966},[163,171],[11,10008,10009,10010,10038,10039,10067,10068,10138,10139,10208,10209,2124,10289,10038,10359,10387],{},"但根据我们的构造规则，",[48,10011,10013,10026],{"className":10012,"translate":52},[56],[48,10014,10016],{"className":10015},[60],[62,10017,10018],{"xmlns":64},[67,10019,10020,10024],{},[70,10021,10022],{},[73,10023,4526],{},[142,10025,4526],{"encoding":144},[48,10027,10029],{"className":10028,"ariaHidden":150},[149],[48,10030,10032,10035],{"className":10031},[154],[48,10033],{"className":10034,"style":2731},[158],[48,10036,4526],{"className":10037,"style":343},[163,171]," 的第 ",[48,10040,10042,10055],{"className":10041,"translate":52},[56],[48,10043,10045],{"className":10044},[60],[62,10046,10047],{"xmlns":64},[67,10048,10049,10053],{},[70,10050,10051],{},[73,10052,9906],{},[142,10054,9906],{"encoding":144},[48,10056,10058],{"className":10057,"ariaHidden":150},[149],[48,10059,10061,10064],{"className":10060},[154],[48,10062],{"className":10063,"style":5810},[158],[48,10065,9906],{"className":10066,"style":9966},[163,171]," 位小数是 ",[48,10069,10071,10089],{"className":10070,"translate":52},[56],[48,10072,10074],{"className":10073},[60],[62,10075,10076],{"xmlns":64},[67,10077,10078,10086],{},[70,10079,10080],{},[115,10081,10082,10084],{},[73,10083,8962],{},[73,10085,9906],{},[142,10087,10088],{"encoding":144},"e_k",[48,10090,10092],{"className":10091,"ariaHidden":150},[149],[48,10093,10095,10098],{"className":10094},[154],[48,10096],{"className":10097,"style":5434},[158],[48,10099,10101,10104],{"className":10100},[163],[48,10102,8962],{"className":10103},[163,171],[48,10105,10107],{"className":10106},[292],[48,10108,10110,10130],{"className":10109},[203,204],[48,10111,10113,10127],{"className":10112},[208],[48,10114,10116],{"className":10115,"style":302},[212],[48,10117,10118,10121],{"style":305},[48,10119],{"className":10120,"style":309},[220],[48,10122,10124],{"className":10123},[225,226,227,228],[48,10125,9906],{"className":10126,"style":9966},[163,171,228],[48,10128,269],{"className":10129},[268],[48,10131,10133],{"className":10132},[208],[48,10134,10136],{"className":10135,"style":2771},[212],[48,10137],{},"，而 ",[48,10140,10142,10159],{"className":10141,"translate":52},[56],[48,10143,10145],{"className":10144},[60],[62,10146,10147],{"xmlns":64},[67,10148,10149,10157],{},[70,10150,10151],{},[115,10152,10153,10155],{},[73,10154,8962],{},[73,10156,9906],{},[142,10158,10088],{"encoding":144},[48,10160,10162],{"className":10161,"ariaHidden":150},[149],[48,10163,10165,10168],{"className":10164},[154],[48,10166],{"className":10167,"style":5434},[158],[48,10169,10171,10174],{"className":10170},[163],[48,10172,8962],{"className":10173},[163,171],[48,10175,10177],{"className":10176},[292],[48,10178,10180,10200],{"className":10179},[203,204],[48,10181,10183,10197],{"className":10182},[208],[48,10184,10186],{"className":10185,"style":302},[212],[48,10187,10188,10191],{"style":305},[48,10189],{"className":10190,"style":309},[220],[48,10192,10194],{"className":10193},[225,226,227,228],[48,10195,9906],{"className":10196,"style":9966},[163,171,228],[48,10198,269],{"className":10199},[268],[48,10201,10203],{"className":10202},[208],[48,10204,10206],{"className":10205,"style":2771},[212],[48,10207],{}," 被刻意构造得与 ",[48,10210,10212,10234],{"className":10211,"translate":52},[56],[48,10213,10215],{"className":10214},[60],[62,10216,10217],{"xmlns":64},[67,10218,10219,10231],{},[70,10220,10221],{},[115,10222,10223,10225],{},[73,10224,879],{},[70,10226,10227,10229],{},[73,10228,9906],{},[73,10230,9906],{},[142,10232,10233],{"encoding":144},"d_{kk}",[48,10235,10237],{"className":10236,"ariaHidden":150},[149],[48,10238,10240,10243],{"className":10239},[154],[48,10241],{"className":10242,"style":9746},[158],[48,10244,10246,10249],{"className":10245},[163],[48,10247,879],{"className":10248},[163,171],[48,10250,10252],{"className":10251},[292],[48,10253,10255,10281],{"className":10254},[203,204],[48,10256,10258,10278],{"className":10257},[208],[48,10259,10261],{"className":10260,"style":302},[212],[48,10262,10263,10266],{"style":305},[48,10264],{"className":10265,"style":309},[220],[48,10267,10269],{"className":10268},[225,226,227,228],[48,10270,10272,10275],{"className":10271},[163,228],[48,10273,9906],{"className":10274,"style":9966},[163,171,228],[48,10276,9906],{"className":10277,"style":9966},[163,171,228],[48,10279,269],{"className":10280},[268],[48,10282,10284],{"className":10283},[208],[48,10285,10287],{"className":10286,"style":2771},[212],[48,10288],{},[48,10290,10292,10310],{"className":10291,"translate":52},[56],[48,10293,10295],{"className":10294},[60],[62,10296,10297],{"xmlns":64},[67,10298,10299,10307],{},[70,10300,10301],{},[115,10302,10303,10305],{},[73,10304,84],{},[73,10306,9906],{},[142,10308,10309],{"encoding":144},"x_k",[48,10311,10313],{"className":10312,"ariaHidden":150},[149],[48,10314,10316,10319],{"className":10315},[154],[48,10317],{"className":10318,"style":5434},[158],[48,10320,10322,10325],{"className":10321},[163],[48,10323,84],{"className":10324},[163,171],[48,10326,10328],{"className":10327},[292],[48,10329,10331,10351],{"className":10330},[203,204],[48,10332,10334,10348],{"className":10333},[208],[48,10335,10337],{"className":10336,"style":302},[212],[48,10338,10339,10342],{"style":305},[48,10340],{"className":10341,"style":309},[220],[48,10343,10345],{"className":10344},[225,226,227,228],[48,10346,9906],{"className":10347,"style":9966},[163,171,228],[48,10349,269],{"className":10350},[268],[48,10352,10354],{"className":10353},[208],[48,10355,10357],{"className":10356,"style":2771},[212],[48,10358],{},[48,10360,10362,10375],{"className":10361,"translate":52},[56],[48,10363,10365],{"className":10364},[60],[62,10366,10367],{"xmlns":64},[67,10368,10369,10373],{},[70,10370,10371],{},[73,10372,9906],{},[142,10374,9906],{"encoding":144},[48,10376,10378],{"className":10377,"ariaHidden":150},[149],[48,10379,10381,10384],{"className":10380},[154],[48,10382],{"className":10383,"style":5810},[158],[48,10385,9906],{"className":10386,"style":9966},[163,171]," 位小数）不同：",[48,10389,10391],{"className":10390,"translate":52},[51],[48,10392,10394,10442],{"className":10393,"translate":52},[56],[48,10395,10397],{"className":10396},[60],[62,10398,10399],{"xmlns":64,"display":65},[67,10400,10401,10439],{},[70,10402,10403,10409,10411,10421,10424,10427,10429,10431,10433],{},[115,10404,10405,10407],{},[73,10406,8962],{},[73,10408,9906],{},[78,10410,5291],{"mathvariant":75},[115,10412,10413,10415],{},[73,10414,879],{},[70,10416,10417,10419],{},[73,10418,9906],{},[73,10420,9906],{},[4598,10422,10423],{},"  ",[78,10425,10426],{},"⟹",[4598,10428,10423],{},[73,10430,4526],{},[78,10432,5291],{"mathvariant":75},[115,10434,10435,10437],{},[73,10436,84],{},[73,10438,9906],{},[142,10440,10441],{"encoding":144},"e_k \\neq d_{kk} \\implies y \\neq x_k",[48,10443,10445,10533,10600,10651],{"className":10444,"ariaHidden":150},[149],[48,10446,10448,10451,10491,10494,10530],{"className":10447},[154],[48,10449],{"className":10450,"style":766},[158],[48,10452,10454,10457],{"className":10453},[163],[48,10455,8962],{"className":10456},[163,171],[48,10458,10460],{"className":10459},[292],[48,10461,10463,10483],{"className":10462},[203,204],[48,10464,10466,10480],{"className":10465},[208],[48,10467,10469],{"className":10468,"style":302},[212],[48,10470,10471,10474],{"style":305},[48,10472],{"className":10473,"style":309},[220],[48,10475,10477],{"className":10476},[225,226,227,228],[48,10478,9906],{"className":10479,"style":9966},[163,171,228],[48,10481,269],{"className":10482},[268],[48,10484,10486],{"className":10485},[208],[48,10487,10489],{"className":10488,"style":2771},[212],[48,10490],{},[48,10492],{"className":10493,"style":180},[179],[48,10495,10497,10524,10527],{"className":10496},[184],[48,10498,10500],{"className":10499},[184],[48,10501,10503],{"className":10502},[163,5391],[48,10504,10506],{"className":10505},[5395],[48,10507,10509,10512,10521],{"className":10508},[5399],[48,10510],{"className":10511,"style":766},[158],[48,10513,10515],{"className":10514},[5406],[48,10516,10518],{"className":10517},[163],[48,10519,5413],{"className":10520},[184],[48,10522],{"className":10523},[5417],[48,10525],{"className":10526},[179,5421],[48,10528,90],{"className":10529},[184],[48,10531],{"className":10532,"style":180},[179],[48,10534,10536,10539,10585,10588,10591,10594,10597],{"className":10535},[154],[48,10537],{"className":10538,"style":9746},[158],[48,10540,10542,10545],{"className":10541},[163],[48,10543,879],{"className":10544},[163,171],[48,10546,10548],{"className":10547},[292],[48,10549,10551,10577],{"className":10550},[203,204],[48,10552,10554,10574],{"className":10553},[208],[48,10555,10557],{"className":10556,"style":302},[212],[48,10558,10559,10562],{"style":305},[48,10560],{"className":10561,"style":309},[220],[48,10563,10565],{"className":10564},[225,226,227,228],[48,10566,10568,10571],{"className":10567},[163,228],[48,10569,9906],{"className":10570,"style":9966},[163,171,228],[48,10572,9906],{"className":10573,"style":9966},[163,171,228],[48,10575,269],{"className":10576},[268],[48,10578,10580],{"className":10579},[208],[48,10581,10583],{"className":10582,"style":2771},[212],[48,10584],{},[48,10586],{"className":10587,"style":180},[179],[48,10589],{"className":10590,"style":180},[179],[48,10592,10426],{"className":10593},[184],[48,10595],{"className":10596,"style":180},[179],[48,10598],{"className":10599,"style":180},[179],[48,10601,10603,10606,10609,10612,10648],{"className":10602},[154],[48,10604],{"className":10605,"style":766},[158],[48,10607,4526],{"className":10608,"style":343},[163,171],[48,10610],{"className":10611,"style":180},[179],[48,10613,10615,10642,10645],{"className":10614},[184],[48,10616,10618],{"className":10617},[184],[48,10619,10621],{"className":10620},[163,5391],[48,10622,10624],{"className":10623},[5395],[48,10625,10627,10630,10639],{"className":10626},[5399],[48,10628],{"className":10629,"style":766},[158],[48,10631,10633],{"className":10632},[5406],[48,10634,10636],{"className":10635},[163],[48,10637,5413],{"className":10638},[184],[48,10640],{"className":10641},[5417],[48,10643],{"className":10644},[179,5421],[48,10646,90],{"className":10647},[184],[48,10649],{"className":10650,"style":180},[179],[48,10652,10654,10657],{"className":10653},[154],[48,10655],{"className":10656,"style":5434},[158],[48,10658,10660,10663],{"className":10659},[163],[48,10661,84],{"className":10662},[163,171],[48,10664,10666],{"className":10665},[292],[48,10667,10669,10689],{"className":10668},[203,204],[48,10670,10672,10686],{"className":10671},[208],[48,10673,10675],{"className":10674,"style":302},[212],[48,10676,10677,10680],{"style":305},[48,10678],{"className":10679,"style":309},[220],[48,10681,10683],{"className":10682},[225,226,227,228],[48,10684,9906],{"className":10685,"style":9966},[163,171,228],[48,10687,269],{"className":10688},[268],[48,10690,10692],{"className":10691},[208],[48,10693,10695],{"className":10694,"style":2771},[212],[48,10696],{},[11,10698,10699,10700,10791],{},"这与 ",[48,10701,10703,10724],{"className":10702,"translate":52},[56],[48,10704,10706],{"className":10705},[60],[62,10707,10708],{"xmlns":64},[67,10709,10710,10722],{},[70,10711,10712,10714,10716],{},[73,10713,4526],{},[78,10715,90],{},[115,10717,10718,10720],{},[73,10719,84],{},[73,10721,9906],{},[142,10723,9909],{"encoding":144},[48,10725,10727,10745],{"className":10726,"ariaHidden":150},[149],[48,10728,10730,10733,10736,10739,10742],{"className":10729},[154],[48,10731],{"className":10732,"style":2731},[158],[48,10734,4526],{"className":10735,"style":343},[163,171],[48,10737],{"className":10738,"style":180},[179],[48,10740,90],{"className":10741},[184],[48,10743],{"className":10744,"style":180},[179],[48,10746,10748,10751],{"className":10747},[154],[48,10749],{"className":10750,"style":5434},[158],[48,10752,10754,10757],{"className":10753},[163],[48,10755,84],{"className":10756},[163,171],[48,10758,10760],{"className":10759},[292],[48,10761,10763,10783],{"className":10762},[203,204],[48,10764,10766,10780],{"className":10765},[208],[48,10767,10769],{"className":10768,"style":302},[212],[48,10770,10771,10774],{"style":305},[48,10772],{"className":10773,"style":309},[220],[48,10775,10777],{"className":10776},[225,226,227,228],[48,10778,9906],{"className":10779,"style":9966},[163,171,228],[48,10781,269],{"className":10782},[268],[48,10784,10786],{"className":10785},[208],[48,10787,10789],{"className":10788,"style":2771},[212],[48,10790],{}," 的假设矛盾！",[11,10793,10794,10795,10823,10824,10852,10853,10881,10882,10910,10911,10939],{},"而且，这一矛盾对每一个 ",[48,10796,10798,10811],{"className":10797,"translate":52},[56],[48,10799,10801],{"className":10800},[60],[62,10802,10803],{"xmlns":64},[67,10804,10805,10809],{},[70,10806,10807],{},[73,10808,9906],{},[142,10810,9906],{"encoding":144},[48,10812,10814],{"className":10813,"ariaHidden":150},[149],[48,10815,10817,10820],{"className":10816},[154],[48,10818],{"className":10819,"style":5810},[158],[48,10821,9906],{"className":10822,"style":9966},[163,171]," 都成立——",[48,10825,10827,10840],{"className":10826,"translate":52},[56],[48,10828,10830],{"className":10829},[60],[62,10831,10832],{"xmlns":64},[67,10833,10834,10838],{},[70,10835,10836],{},[73,10837,4526],{},[142,10839,4526],{"encoding":144},[48,10841,10843],{"className":10842,"ariaHidden":150},[149],[48,10844,10846,10849],{"className":10845},[154],[48,10847],{"className":10848,"style":2731},[158],[48,10850,4526],{"className":10851,"style":343},[163,171]," 在第 1 位与第 1 个数不同，在第 2 位与第 2 个数不同，在第 ",[48,10854,10856,10869],{"className":10855,"translate":52},[56],[48,10857,10859],{"className":10858},[60],[62,10860,10861],{"xmlns":64},[67,10862,10863,10867],{},[70,10864,10865],{},[73,10866,6221],{},[142,10868,6221],{"encoding":144},[48,10870,10872],{"className":10871,"ariaHidden":150},[149],[48,10873,10875,10878],{"className":10874},[154],[48,10876],{"className":10877,"style":578},[158],[48,10879,6221],{"className":10880},[163,171]," 位与第 ",[48,10883,10885,10898],{"className":10884,"translate":52},[56],[48,10886,10888],{"className":10887},[60],[62,10889,10890],{"xmlns":64},[67,10891,10892,10896],{},[70,10893,10894],{},[73,10895,6221],{},[142,10897,6221],{"encoding":144},[48,10899,10901],{"className":10900,"ariaHidden":150},[149],[48,10902,10904,10907],{"className":10903},[154],[48,10905],{"className":10906,"style":578},[158],[48,10908,6221],{"className":10909},[163,171]," 个数不同……因此 ",[48,10912,10914,10927],{"className":10913,"translate":52},[56],[48,10915,10917],{"className":10916},[60],[62,10918,10919],{"xmlns":64},[67,10920,10921,10925],{},[70,10922,10923],{},[73,10924,4526],{},[142,10926,4526],{"encoding":144},[48,10928,10930],{"className":10929,"ariaHidden":150},[149],[48,10931,10933,10936],{"className":10932},[154],[48,10934],{"className":10935,"style":2731},[158],[48,10937,4526],{"className":10938,"style":343},[163,171]," 与列表中的每一个数都不同。",[11,10941,10942,10943,10971,10972,11023],{},"然而 ",[48,10944,10946,10959],{"className":10945,"translate":52},[56],[48,10947,10949],{"className":10948},[60],[62,10950,10951],{"xmlns":64},[67,10952,10953,10957],{},[70,10954,10955],{},[73,10956,4526],{},[142,10958,4526],{"encoding":144},[48,10960,10962],{"className":10961,"ariaHidden":150},[149],[48,10963,10965,10968],{"className":10964},[154],[48,10966],{"className":10967,"style":2731},[158],[48,10969,4526],{"className":10970,"style":343},[163,171]," 无疑是 ",[48,10973,10975,10996],{"className":10974,"translate":52},[56],[48,10976,10978],{"className":10977},[60],[62,10979,10980],{"xmlns":64},[67,10981,10982,10994],{},[70,10983,10984,10986,10988,10990,10992],{},[78,10985,81],{"stretchy":80},[1520,10987,2710],{},[78,10989,874],{"separator":150},[1520,10991,1522],{},[78,10993,87],{"stretchy":80},[142,10995,7058],{"encoding":144},[48,10997,10999],{"className":10998,"ariaHidden":150},[149],[48,11000,11002,11005,11008,11011,11014,11017,11020],{"className":11001},[154],[48,11003],{"className":11004,"style":159},[158],[48,11006,81],{"className":11007},[167],[48,11009,2710],{"className":11010},[163],[48,11012,874],{"className":11013},[944],[48,11015],{"className":11016,"style":282},[179],[48,11018,1522],{"className":11019},[163],[48,11021,87],{"className":11022},[175]," 中的实数（它的每一位都是合法的十进制数字）。",[11,11025,11026,11027,11055,11056,11151,11152,11203],{},"结论：无论你如何构造这份「实数大全列表」，总能构造出一个逃脱列表的实数 ",[48,11028,11030,11043],{"className":11029,"translate":52},[56],[48,11031,11033],{"className":11032},[60],[62,11034,11035],{"xmlns":64},[67,11036,11037,11041],{},[70,11038,11039],{},[73,11040,4526],{},[142,11042,4526],{"encoding":144},[48,11044,11046],{"className":11045,"ariaHidden":150},[149],[48,11047,11049,11052],{"className":11048},[154],[48,11050],{"className":11051,"style":2731},[158],[48,11053,4526],{"className":11054,"style":343},[163,171],"。这就说明最初的假设——存在双射 ",[48,11057,11059,11088],{"className":11058,"translate":52},[56],[48,11060,11062],{"className":11061},[60],[62,11063,11064],{"xmlns":64},[67,11065,11066,11086],{},[70,11067,11068,11070,11072,11074,11076,11078,11080,11082,11084],{},[73,11069,101],{},[78,11071,1495],{},[73,11073,5017],{"mathvariant":5016},[78,11075,1504],{},[78,11077,81],{"stretchy":80},[1520,11079,2710],{},[78,11081,874],{"separator":150},[1520,11083,1522],{},[78,11085,87],{"stretchy":80},[142,11087,7232],{"encoding":144},[48,11089,11091,11109,11127],{"className":11090,"ariaHidden":150},[149],[48,11092,11094,11097,11100,11103,11106],{"className":11093},[154],[48,11095],{"className":11096,"style":766},[158],[48,11098,101],{"className":11099,"style":235},[163,171],[48,11101],{"className":11102,"style":180},[179],[48,11104,1495],{"className":11105},[184],[48,11107],{"className":11108,"style":180},[179],[48,11110,11112,11115,11118,11121,11124],{"className":11111},[154],[48,11113],{"className":11114,"style":5060},[158],[48,11116,5017],{"className":11117},[163,5064],[48,11119],{"className":11120,"style":180},[179],[48,11122,1504],{"className":11123},[184],[48,11125],{"className":11126,"style":180},[179],[48,11128,11130,11133,11136,11139,11142,11145,11148],{"className":11129},[154],[48,11131],{"className":11132,"style":159},[158],[48,11134,81],{"className":11135},[167],[48,11137,2710],{"className":11138},[163],[48,11140,874],{"className":11141},[944],[48,11143],{"className":11144,"style":282},[179],[48,11146,1522],{"className":11147},[163],[48,11149,87],{"className":11150},[175],"——是假的。",[48,11153,11155,11176],{"className":11154,"translate":52},[56],[48,11156,11158],{"className":11157},[60],[62,11159,11160],{"xmlns":64},[67,11161,11162,11174],{},[70,11163,11164,11166,11168,11170,11172],{},[78,11165,81],{"stretchy":80},[1520,11167,2710],{},[78,11169,874],{"separator":150},[1520,11171,1522],{},[78,11173,87],{"stretchy":80},[142,11175,7058],{"encoding":144},[48,11177,11179],{"className":11178,"ariaHidden":150},[149],[48,11180,11182,11185,11188,11191,11194,11197,11200],{"className":11181},[154],[48,11183],{"className":11184,"style":159},[158],[48,11186,81],{"className":11187},[167],[48,11189,2710],{"className":11190},[163],[48,11192,874],{"className":11193},[944],[48,11195],{"className":11196,"style":282},[179],[48,11198,1522],{"className":11199},[163],[48,11201,87],{"className":11202},[175]," 中的实数不可数。",[11,11205,11206,11207,11407,11408,11478],{},"为什么叫「对角线」？这个名称源于证明中选取数字 ",[48,11208,11210,11248],{"className":11209,"translate":52},[56],[48,11211,11213],{"className":11212},[60],[62,11214,11215],{"xmlns":64},[67,11216,11217,11245],{},[70,11218,11219,11225,11227,11233,11235,11241,11243],{},[115,11220,11221,11223],{},[73,11222,879],{},[1520,11224,7345],{},[78,11226,874],{"separator":150},[115,11228,11229,11231],{},[73,11230,879],{},[1520,11232,7414],{},[78,11234,874],{"separator":150},[115,11236,11237,11239],{},[73,11238,879],{},[1520,11240,7483],{},[78,11242,874],{"separator":150},[78,11244,2934],{},[142,11246,11247],{"encoding":144},"d_{11}, d_{22}, d_{33}, \\dots",[48,11249,11251],{"className":11250,"ariaHidden":150},[149],[48,11252,11254,11257,11300,11303,11306,11349,11352,11355,11398,11401,11404],{"className":11253},[154],[48,11255],{"className":11256,"style":766},[158],[48,11258,11260,11263],{"className":11259},[163],[48,11261,879],{"className":11262},[163,171],[48,11264,11266],{"className":11265},[292],[48,11267,11269,11292],{"className":11268},[203,204],[48,11270,11272,11289],{"className":11271},[208],[48,11273,11275],{"className":11274,"style":2750},[212],[48,11276,11277,11280],{"style":305},[48,11278],{"className":11279,"style":309},[220],[48,11281,11283],{"className":11282},[225,226,227,228],[48,11284,11286],{"className":11285},[163,228],[48,11287,7345],{"className":11288},[163,228],[48,11290,269],{"className":11291},[268],[48,11293,11295],{"className":11294},[208],[48,11296,11298],{"className":11297,"style":2771},[212],[48,11299],{},[48,11301,874],{"className":11302},[944],[48,11304],{"className":11305,"style":282},[179],[48,11307,11309,11312],{"className":11308},[163],[48,11310,879],{"className":11311},[163,171],[48,11313,11315],{"className":11314},[292],[48,11316,11318,11341],{"className":11317},[203,204],[48,11319,11321,11338],{"className":11320},[208],[48,11322,11324],{"className":11323,"style":2750},[212],[48,11325,11326,11329],{"style":305},[48,11327],{"className":11328,"style":309},[220],[48,11330,11332],{"className":11331},[225,226,227,228],[48,11333,11335],{"className":11334},[163,228],[48,11336,7414],{"className":11337},[163,228],[48,11339,269],{"className":11340},[268],[48,11342,11344],{"className":11343},[208],[48,11345,11347],{"className":11346,"style":2771},[212],[48,11348],{},[48,11350,874],{"className":11351},[944],[48,11353],{"className":11354,"style":282},[179],[48,11356,11358,11361],{"className":11357},[163],[48,11359,879],{"className":11360},[163,171],[48,11362,11364],{"className":11363},[292],[48,11365,11367,11390],{"className":11366},[203,204],[48,11368,11370,11387],{"className":11369},[208],[48,11371,11373],{"className":11372,"style":2750},[212],[48,11374,11375,11378],{"style":305},[48,11376],{"className":11377,"style":309},[220],[48,11379,11381],{"className":11380},[225,226,227,228],[48,11382,11384],{"className":11383},[163,228],[48,11385,7483],{"className":11386},[163,228],[48,11388,269],{"className":11389},[268],[48,11391,11393],{"className":11392},[208],[48,11394,11396],{"className":11395,"style":2771},[212],[48,11397],{},[48,11399,874],{"className":11400},[944],[48,11402],{"className":11403,"style":282},[179],[48,11405,2934],{"className":11406},[3012]," 的方式——若把所有 ",[48,11409,11411,11429],{"className":11410,"translate":52},[56],[48,11412,11414],{"className":11413},[60],[62,11415,11416],{"xmlns":64},[67,11417,11418,11426],{},[70,11419,11420],{},[115,11421,11422,11424],{},[73,11423,84],{},[73,11425,2631],{},[142,11427,11428],{"encoding":144},"x_i",[48,11430,11432],{"className":11431,"ariaHidden":150},[149],[48,11433,11435,11438],{"className":11434},[154],[48,11436],{"className":11437,"style":5434},[158],[48,11439,11441,11444],{"className":11440},[163],[48,11442,84],{"className":11443},[163,171],[48,11445,11447],{"className":11446},[292],[48,11448,11450,11470],{"className":11449},[203,204],[48,11451,11453,11467],{"className":11452},[208],[48,11454,11456],{"className":11455,"style":8789},[212],[48,11457,11458,11461],{"style":305},[48,11459],{"className":11460,"style":309},[220],[48,11462,11464],{"className":11463},[225,226,227,228],[48,11465,2631],{"className":11466},[163,171,228],[48,11468,269],{"className":11469},[268],[48,11471,11473],{"className":11472},[208],[48,11474,11476],{"className":11475,"style":2771},[212],[48,11477],{}," 纵向排成无穷矩阵，这些位置恰好构成矩阵的主对角线：",[48,11480,11482],{"className":11481,"translate":52},[51],[48,11483,11485,11691],{"className":11484,"translate":52},[56],[48,11486,11488],{"className":11487},[60],[62,11489,11490],{"xmlns":64,"display":65},[67,11491,11492,11688],{},[70,11493,11494,11496,11686],{},[78,11495,81],{"fence":150},[6463,11497,11500,11549,11595,11641],{"rowspacing":11498,"columnalign":11499,"columnspacing":6467},"0.16em","center center center center",[6469,11501,11502,11522,11532,11542],{},[6472,11503,11504],{},[6475,11505,11506],{"scriptlevel":2710,"displaystyle":80},[11507,11508,11510],"menclose",{"notation":11509},"box",[6475,11511,11512],{"scriptlevel":2710,"displaystyle":80},[6475,11513,11514],{"scriptlevel":2710,"displaystyle":80},[6475,11515,11516],{"scriptlevel":2710,"displaystyle":150},[115,11517,11518,11520],{},[73,11519,879],{},[1520,11521,7345],{},[6472,11523,11524],{},[6475,11525,11526],{"scriptlevel":2710,"displaystyle":80},[115,11527,11528,11530],{},[73,11529,879],{},[1520,11531,7354],{},[6472,11533,11534],{},[6475,11535,11536],{"scriptlevel":2710,"displaystyle":80},[115,11537,11538,11540],{},[73,11539,879],{},[1520,11541,7363],{},[6472,11543,11544],{},[6475,11545,11546],{"scriptlevel":2710,"displaystyle":80},[78,11547,11548],{"lspace":3327,"rspace":3327},"⋯",[6469,11550,11551,11561,11579,11589],{},[6472,11552,11553],{},[6475,11554,11555],{"scriptlevel":2710,"displaystyle":80},[115,11556,11557,11559],{},[73,11558,879],{},[1520,11560,7405],{},[6472,11562,11563],{},[6475,11564,11565],{"scriptlevel":2710,"displaystyle":80},[11507,11566,11567],{"notation":11509},[6475,11568,11569],{"scriptlevel":2710,"displaystyle":80},[6475,11570,11571],{"scriptlevel":2710,"displaystyle":80},[6475,11572,11573],{"scriptlevel":2710,"displaystyle":150},[115,11574,11575,11577],{},[73,11576,879],{},[1520,11578,7414],{},[6472,11580,11581],{},[6475,11582,11583],{"scriptlevel":2710,"displaystyle":80},[115,11584,11585,11587],{},[73,11586,879],{},[1520,11588,7423],{},[6472,11590,11591],{},[6475,11592,11593],{"scriptlevel":2710,"displaystyle":80},[78,11594,11548],{"lspace":3327,"rspace":3327},[6469,11596,11597,11607,11617,11635],{},[6472,11598,11599],{},[6475,11600,11601],{"scriptlevel":2710,"displaystyle":80},[115,11602,11603,11605],{},[73,11604,879],{},[1520,11606,7465],{},[6472,11608,11609],{},[6475,11610,11611],{"scriptlevel":2710,"displaystyle":80},[115,11612,11613,11615],{},[73,11614,879],{},[1520,11616,7474],{},[6472,11618,11619],{},[6475,11620,11621],{"scriptlevel":2710,"displaystyle":80},[11507,11622,11623],{"notation":11509},[6475,11624,11625],{"scriptlevel":2710,"displaystyle":80},[6475,11626,11627],{"scriptlevel":2710,"displaystyle":80},[6475,11628,11629],{"scriptlevel":2710,"displaystyle":150},[115,11630,11631,11633],{},[73,11632,879],{},[1520,11634,7483],{},[6472,11636,11637],{},[6475,11638,11639],{"scriptlevel":2710,"displaystyle":80},[78,11640,11548],{"lspace":3327,"rspace":3327},[6469,11642,11643,11655,11667,11679],{},[6472,11644,11645],{},[6475,11646,11647],{"scriptlevel":2710,"displaystyle":80},[70,11648,11649,11651],{},[73,11650,7578],{"mathvariant":75},[7580,11652,11653],{"height":3327,"voffset":3327},[179,11654],{"mathbackground":7584,"width":3327,"height":7585},[6472,11656,11657],{},[6475,11658,11659],{"scriptlevel":2710,"displaystyle":80},[70,11660,11661,11663],{},[73,11662,7578],{"mathvariant":75},[7580,11664,11665],{"height":3327,"voffset":3327},[179,11666],{"mathbackground":7584,"width":3327,"height":7585},[6472,11668,11669],{},[6475,11670,11671],{"scriptlevel":2710,"displaystyle":80},[70,11672,11673,11675],{},[73,11674,7578],{"mathvariant":75},[7580,11676,11677],{"height":3327,"voffset":3327},[179,11678],{"mathbackground":7584,"width":3327,"height":7585},[6472,11680,11681],{},[6475,11682,11683],{"scriptlevel":2710,"displaystyle":80},[78,11684,11685],{"lspace":3327,"rspace":3327},"⋱",[78,11687,87],{"fence":150},[142,11689,11690],{"encoding":144},"\\begin{pmatrix}\n\\boxed{d_{11}} & d_{12} & d_{13} & \\cdots \\\\\nd_{21} & \\boxed{d_{22}} & d_{23} & \\cdots \\\\\nd_{31} & d_{32} & \\boxed{d_{33}} & \\cdots \\\\\n\\vdots & \\vdots & \\vdots & \\ddots\n\\end{pmatrix}",[48,11692,11694],{"className":11693,"ariaHidden":150},[149],[48,11695,11697,11701],{"className":11696},[154],[48,11698],{"className":11699,"style":11700},[158],"height:6.4333em;vertical-align:-2.9667em;",[48,11702,11704,11751,12577],{"className":11703},[3012],[48,11705,11707],{"className":11706},[167],[48,11708,11711],{"className":11709},[6583,11710],"mult",[48,11712,11714,11742],{"className":11713},[203,204],[48,11715,11717,11739],{"className":11716},[208],[48,11718,11721],{"className":11719,"style":11720},[212],"height:3.25em;",[48,11722,11724,11728],{"style":11723},"top:-5.25em;",[48,11725],{"className":11726,"style":11727},[220],"height:8em;",[48,11729,11731],{"style":11730},"width:0.875em;height:6em;",[4327,11732,11736],{"xmlns":4329,"width":11733,"height":11734,"viewBox":11735},"0.875em","6em","0 0 875 6000",[4335,11737],{"d":11738},"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,2484c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-2492c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z",[48,11740,269],{"className":11741},[268],[48,11743,11745],{"className":11744},[208],[48,11746,11749],{"className":11747,"style":11748},[212],"height:2.75em;",[48,11750],{},[48,11752,11754],{"className":11753},[163],[48,11755,11757,12011,12015,12018,12256,12259,12262,12500,12503,12506],{"className":11756},[6463],[48,11758,11761],{"className":11759},[11760],"col-align-c",[48,11762,11764,12002],{"className":11763},[203,204],[48,11765,11767,11999],{"className":11766},[208],[48,11768,11771,11877,11929,11981],{"className":11769,"style":11770},[212],"height:3.4667em;",[48,11772,11774,11777],{"style":11773},"top:-6.1197em;",[48,11775],{"className":11776,"style":7855},[220],[48,11778,11780],{"className":11779},[163],[48,11781,11783],{"className":11782},[163],[48,11784,11786,11868],{"className":11785},[203,204],[48,11787,11789,11865],{"className":11788},[208],[48,11790,11793,11853],{"className":11791,"style":11792},[212],"height:1.0344em;",[48,11794,11796,11800],{"style":11795},"top:-3.5244em;",[48,11797],{"className":11798,"style":11799},[220],"height:3.5244em;",[48,11801,11804],{"className":11802},[11803],"boxpad",[48,11805,11807],{"className":11806},[163],[48,11808,11810],{"className":11809},[163],[48,11811,11813,11816],{"className":11812},[163],[48,11814,879],{"className":11815},[163,171],[48,11817,11819],{"className":11818},[292],[48,11820,11822,11845],{"className":11821},[203,204],[48,11823,11825,11842],{"className":11824},[208],[48,11826,11828],{"className":11827,"style":2750},[212],[48,11829,11830,11833],{"style":305},[48,11831],{"className":11832,"style":309},[220],[48,11834,11836],{"className":11835},[225,226,227,228],[48,11837,11839],{"className":11838},[163,228],[48,11840,7345],{"className":11841},[163,228],[48,11843,269],{"className":11844},[268],[48,11846,11848],{"className":11847},[208],[48,11849,11851],{"className":11850,"style":2771},[212],[48,11852],{},[48,11854,11856,11859],{"style":11855},"top:-3.0344em;",[48,11857],{"className":11858,"style":11799},[220],[48,11860],{"className":11861,"style":11864},[11862,11863],"stretchy","fbox","height:1.5244em;border-style:solid;border-width:0.04em;",[48,11866,269],{"className":11867},[268],[48,11869,11871],{"className":11870},[208],[48,11872,11875],{"className":11873,"style":11874},[212],"height:0.49em;",[48,11876],{},[48,11878,11880,11883],{"style":11879},"top:-4.5953em;",[48,11881],{"className":11882,"style":7855},[220],[48,11884,11886],{"className":11885},[163],[48,11887,11889,11892],{"className":11888},[163],[48,11890,879],{"className":11891},[163,171],[48,11893,11895],{"className":11894},[292],[48,11896,11898,11921],{"className":11897},[203,204],[48,11899,11901,11918],{"className":11900},[208],[48,11902,11904],{"className":11903,"style":2750},[212],[48,11905,11906,11909],{"style":305},[48,11907],{"className":11908,"style":309},[220],[48,11910,11912],{"className":11911},[225,226,227,228],[48,11913,11915],{"className":11914},[163,228],[48,11916,7405],{"className":11917},[163,228],[48,11919,269],{"className":11920},[268],[48,11922,11924],{"className":11923},[208],[48,11925,11927],{"className":11926,"style":2771},[212],[48,11928],{},[48,11930,11932,11935],{"style":11931},"top:-3.0708em;",[48,11933],{"className":11934,"style":7855},[220],[48,11936,11938],{"className":11937},[163],[48,11939,11941,11944],{"className":11940},[163],[48,11942,879],{"className":11943},[163,171],[48,11945,11947],{"className":11946},[292],[48,11948,11950,11973],{"className":11949},[203,204],[48,11951,11953,11970],{"className":11952},[208],[48,11954,11956],{"className":11955,"style":2750},[212],[48,11957,11958,11961],{"style":305},[48,11959],{"className":11960,"style":309},[220],[48,11962,11964],{"className":11963},[225,226,227,228],[48,11965,11967],{"className":11966},[163,228],[48,11968,7465],{"className":11969},[163,228],[48,11971,269],{"className":11972},[268],[48,11974,11976],{"className":11975},[208],[48,11977,11979],{"className":11978,"style":2771},[212],[48,11980],{},[48,11982,11984,11987],{"style":11983},"top:-1.0808em;",[48,11985],{"className":11986,"style":7855},[220],[48,11988,11990],{"className":11989},[163],[48,11991,11993,11996],{"className":11992},[163],[48,11994,7578],{"className":11995},[163],[48,11997],{"className":11998,"style":8722},[163,8721],[48,12000,269],{"className":12001},[268],[48,12003,12005],{"className":12004},[208],[48,12006,12009],{"className":12007,"style":12008},[212],"height:2.9667em;",[48,12010],{},[48,12012],{"className":12013,"style":12014},[6673],"width:0.5em;",[48,12016],{"className":12017,"style":12014},[6673],[48,12019,12021],{"className":12020},[11760],[48,12022,12024,12248],{"className":12023},[203,204],[48,12025,12027,12245],{"className":12026},[208],[48,12028,12030,12081,12177,12228],{"className":12029,"style":11770},[212],[48,12031,12032,12035],{"style":11773},[48,12033],{"className":12034,"style":7855},[220],[48,12036,12038],{"className":12037},[163],[48,12039,12041,12044],{"className":12040},[163],[48,12042,879],{"className":12043},[163,171],[48,12045,12047],{"className":12046},[292],[48,12048,12050,12073],{"className":12049},[203,204],[48,12051,12053,12070],{"className":12052},[208],[48,12054,12056],{"className":12055,"style":2750},[212],[48,12057,12058,12061],{"style":305},[48,12059],{"className":12060,"style":309},[220],[48,12062,12064],{"className":12063},[225,226,227,228],[48,12065,12067],{"className":12066},[163,228],[48,12068,7354],{"className":12069},[163,228],[48,12071,269],{"className":12072},[268],[48,12074,12076],{"className":12075},[208],[48,12077,12079],{"className":12078,"style":2771},[212],[48,12080],{},[48,12082,12083,12086],{"style":11879},[48,12084],{"className":12085,"style":7855},[220],[48,12087,12089],{"className":12088},[163],[48,12090,12092],{"className":12091},[163],[48,12093,12095,12169],{"className":12094},[203,204],[48,12096,12098,12166],{"className":12097},[208],[48,12099,12101,12158],{"className":12100,"style":11792},[212],[48,12102,12103,12106],{"style":11795},[48,12104],{"className":12105,"style":11799},[220],[48,12107,12109],{"className":12108},[11803],[48,12110,12112],{"className":12111},[163],[48,12113,12115],{"className":12114},[163],[48,12116,12118,12121],{"className":12117},[163],[48,12119,879],{"className":12120},[163,171],[48,12122,12124],{"className":12123},[292],[48,12125,12127,12150],{"className":12126},[203,204],[48,12128,12130,12147],{"className":12129},[208],[48,12131,12133],{"className":12132,"style":2750},[212],[48,12134,12135,12138],{"style":305},[48,12136],{"className":12137,"style":309},[220],[48,12139,12141],{"className":12140},[225,226,227,228],[48,12142,12144],{"className":12143},[163,228],[48,12145,7414],{"className":12146},[163,228],[48,12148,269],{"className":12149},[268],[48,12151,12153],{"className":12152},[208],[48,12154,12156],{"className":12155,"style":2771},[212],[48,12157],{},[48,12159,12160,12163],{"style":11855},[48,12161],{"className":12162,"style":11799},[220],[48,12164],{"className":12165,"style":11864},[11862,11863],[48,12167,269],{"className":12168},[268],[48,12170,12172],{"className":12171},[208],[48,12173,12175],{"className":12174,"style":11874},[212],[48,12176],{},[48,12178,12179,12182],{"style":11931},[48,12180],{"className":12181,"style":7855},[220],[48,12183,12185],{"className":12184},[163],[48,12186,12188,12191],{"className":12187},[163],[48,12189,879],{"className":12190},[163,171],[48,12192,12194],{"className":12193},[292],[48,12195,12197,12220],{"className":12196},[203,204],[48,12198,12200,12217],{"className":12199},[208],[48,12201,12203],{"className":12202,"style":2750},[212],[48,12204,12205,12208],{"style":305},[48,12206],{"className":12207,"style":309},[220],[48,12209,12211],{"className":12210},[225,226,227,228],[48,12212,12214],{"className":12213},[163,228],[48,12215,7474],{"className":12216},[163,228],[48,12218,269],{"className":12219},[268],[48,12221,12223],{"className":12222},[208],[48,12224,12226],{"className":12225,"style":2771},[212],[48,12227],{},[48,12229,12230,12233],{"style":11983},[48,12231],{"className":12232,"style":7855},[220],[48,12234,12236],{"className":12235},[163],[48,12237,12239,12242],{"className":12238},[163],[48,12240,7578],{"className":12241},[163],[48,12243],{"className":12244,"style":8722},[163,8721],[48,12246,269],{"className":12247},[268],[48,12249,12251],{"className":12250},[208],[48,12252,12254],{"className":12253,"style":12008},[212],[48,12255],{},[48,12257],{"className":12258,"style":12014},[6673],[48,12260],{"className":12261,"style":12014},[6673],[48,12263,12265],{"className":12264},[11760],[48,12266,12268,12492],{"className":12267},[203,204],[48,12269,12271,12489],{"className":12270},[208],[48,12272,12274,12325,12376,12472],{"className":12273,"style":11770},[212],[48,12275,12276,12279],{"style":11773},[48,12277],{"className":12278,"style":7855},[220],[48,12280,12282],{"className":12281},[163],[48,12283,12285,12288],{"className":12284},[163],[48,12286,879],{"className":12287},[163,171],[48,12289,12291],{"className":12290},[292],[48,12292,12294,12317],{"className":12293},[203,204],[48,12295,12297,12314],{"className":12296},[208],[48,12298,12300],{"className":12299,"style":2750},[212],[48,12301,12302,12305],{"style":305},[48,12303],{"className":12304,"style":309},[220],[48,12306,12308],{"className":12307},[225,226,227,228],[48,12309,12311],{"className":12310},[163,228],[48,12312,7363],{"className":12313},[163,228],[48,12315,269],{"className":12316},[268],[48,12318,12320],{"className":12319},[208],[48,12321,12323],{"className":12322,"style":2771},[212],[48,12324],{},[48,12326,12327,12330],{"style":11879},[48,12328],{"className":12329,"style":7855},[220],[48,12331,12333],{"className":12332},[163],[48,12334,12336,12339],{"className":12335},[163],[48,12337,879],{"className":12338},[163,171],[48,12340,12342],{"className":12341},[292],[48,12343,12345,12368],{"className":12344},[203,204],[48,12346,12348,12365],{"className":12347},[208],[48,12349,12351],{"className":12350,"style":2750},[212],[48,12352,12353,12356],{"style":305},[48,12354],{"className":12355,"style":309},[220],[48,12357,12359],{"className":12358},[225,226,227,228],[48,12360,12362],{"className":12361},[163,228],[48,12363,7423],{"className":12364},[163,228],[48,12366,269],{"className":12367},[268],[48,12369,12371],{"className":12370},[208],[48,12372,12374],{"className":12373,"style":2771},[212],[48,12375],{},[48,12377,12378,12381],{"style":11931},[48,12379],{"className":12380,"style":7855},[220],[48,12382,12384],{"className":12383},[163],[48,12385,12387],{"className":12386},[163],[48,12388,12390,12464],{"className":12389},[203,204],[48,12391,12393,12461],{"className":12392},[208],[48,12394,12396,12453],{"className":12395,"style":11792},[212],[48,12397,12398,12401],{"style":11795},[48,12399],{"className":12400,"style":11799},[220],[48,12402,12404],{"className":12403},[11803],[48,12405,12407],{"className":12406},[163],[48,12408,12410],{"className":12409},[163],[48,12411,12413,12416],{"className":12412},[163],[48,12414,879],{"className":12415},[163,171],[48,12417,12419],{"className":12418},[292],[48,12420,12422,12445],{"className":12421},[203,204],[48,12423,12425,12442],{"className":12424},[208],[48,12426,12428],{"className":12427,"style":2750},[212],[48,12429,12430,12433],{"style":305},[48,12431],{"className":12432,"style":309},[220],[48,12434,12436],{"className":12435},[225,226,227,228],[48,12437,12439],{"className":12438},[163,228],[48,12440,7483],{"className":12441},[163,228],[48,12443,269],{"className":12444},[268],[48,12446,12448],{"className":12447},[208],[48,12449,12451],{"className":12450,"style":2771},[212],[48,12452],{},[48,12454,12455,12458],{"style":11855},[48,12456],{"className":12457,"style":11799},[220],[48,12459],{"className":12460,"style":11864},[11862,11863],[48,12462,269],{"className":12463},[268],[48,12465,12467],{"className":12466},[208],[48,12468,12470],{"className":12469,"style":11874},[212],[48,12471],{},[48,12473,12474,12477],{"style":11983},[48,12475],{"className":12476,"style":7855},[220],[48,12478,12480],{"className":12479},[163],[48,12481,12483,12486],{"className":12482},[163],[48,12484,7578],{"className":12485},[163],[48,12487],{"className":12488,"style":8722},[163,8721],[48,12490,269],{"className":12491},[268],[48,12493,12495],{"className":12494},[208],[48,12496,12498],{"className":12497,"style":12008},[212],[48,12499],{},[48,12501],{"className":12502,"style":12014},[6673],[48,12504],{"className":12505,"style":12014},[6673],[48,12507,12509],{"className":12508},[11760],[48,12510,12512,12569],{"className":12511},[203,204],[48,12513,12515,12566],{"className":12514},[208],[48,12516,12518,12530,12542,12554],{"className":12517,"style":11770},[212],[48,12519,12521,12524],{"style":12520},"top:-5.9322em;",[48,12522],{"className":12523,"style":7625},[220],[48,12525,12527],{"className":12526},[163],[48,12528,11548],{"className":12529},[3012],[48,12531,12533,12536],{"style":12532},"top:-4.4078em;",[48,12534],{"className":12535,"style":7625},[220],[48,12537,12539],{"className":12538},[163],[48,12540,11548],{"className":12541},[3012],[48,12543,12545,12548],{"style":12544},"top:-2.8833em;",[48,12546],{"className":12547,"style":7625},[220],[48,12549,12551],{"className":12550},[163],[48,12552,11548],{"className":12553},[3012],[48,12555,12557,12560],{"style":12556},"top:-0.8933em;",[48,12558],{"className":12559,"style":7625},[220],[48,12561,12563],{"className":12562},[163],[48,12564,11685],{"className":12565},[3012],[48,12567,269],{"className":12568},[268],[48,12570,12572],{"className":12571},[208],[48,12573,12575],{"className":12574,"style":12008},[212],[48,12576],{},[48,12578,12580],{"className":12579},[175],[48,12581,12583],{"className":12582},[6583,11710],[48,12584,12586,12607],{"className":12585},[203,204],[48,12587,12589,12604],{"className":12588},[208],[48,12590,12592],{"className":12591,"style":11720},[212],[48,12593,12594,12597],{"style":11723},[48,12595],{"className":12596,"style":11727},[220],[48,12598,12599],{"style":11730},[4327,12600,12601],{"xmlns":4329,"width":11733,"height":11734,"viewBox":11735},[4335,12602],{"d":12603},"M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,\n63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5\nc11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,2409\nc-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-2544c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z",[48,12605,269],{"className":12606},[268],[48,12608,12610],{"className":12609},[208],[48,12611,12613],{"className":12612,"style":11748},[212],[48,12614],{},[11,12616,12617],{},"我们「翻转」对角线上的每一位数字，构造出一个必然逃脱整个列表的新元素。",[35,12619,12620],{"id":12620},"构造逃逸者",[11,12622,12623],{},"仅凭公式可能仍有几分抽象；让我们通过代码把这一过程具体化。下面的程序取十个「实数」作为示例列表，真正执行一次对角线构造，让读者清楚地看到新数如何「避开」原来的每一个数。",[12625,12626,12630],"pre",{"className":12627,"code":12628,"language":12629,"meta":4787,"style":4787},"language-python shiki shiki-themes github-dark","def diagonal_construct(number_list, digits=10):\n    \"\"\"\n    number_list：字符串列表，每个形如 \"0.d1d2d3...\"\n    返回：通过对角线方法构造出的新数字字符串\n    \"\"\"\n    new_digits = []\n    for i, num_str in enumerate(number_list):\n        decimal_part = num_str.split('.')[1].ljust(digits, '0')\n        d = int(decimal_part[i])       # 第 i 个数的第 i 位（对角线数字）\n        new_d = (d + 5) % 10           # 规则：使新数字与原数字不同\n        new_digits.append(str(new_d))\n    return \"0.\" + \"\".join(new_digits)\n\n\nsample_list = [\n    \"0.1234567890\", \"0.9876543210\", \"0.5555555555\", \"0.1111111111\",\n    \"0.3141592653\", \"0.2718281828\", \"0.0000000001\", \"0.9999999998\",\n    \"0.4242424242\", \"0.6180339887\",\n]\n\nnew_number = diagonal_construct(sample_list)\nprint(\"对角线构造出的新数：\", new_number)\n","python",[2561,12631,12632,12657,12663,12669,12675,12680,12691,12709,12737,12755,12783,12795,12813,12819,12824,12835,12860,12883,12896,12902,12907,12918],{"__ignoreMap":4787},[48,12633,12636,12640,12644,12648,12650,12654],{"class":12634,"line":12635},"line",1,[48,12637,12639],{"class":12638},"snl16","def",[48,12641,12643],{"class":12642},"svObZ"," diagonal_construct",[48,12645,12647],{"class":12646},"s95oV","(number_list, digits",[48,12649,90],{"class":12638},[48,12651,12653],{"class":12652},"sDLfK","10",[48,12655,12656],{"class":12646},"):\n",[48,12658,12659],{"class":12634,"line":4788},[48,12660,12662],{"class":12661},"sU2Wk","    \"\"\"\n",[48,12664,12666],{"class":12634,"line":12665},3,[48,12667,12668],{"class":12661},"    number_list：字符串列表，每个形如 \"0.d1d2d3...\"\n",[48,12670,12672],{"class":12634,"line":12671},4,[48,12673,12674],{"class":12661},"    返回：通过对角线方法构造出的新数字字符串\n",[48,12676,12678],{"class":12634,"line":12677},5,[48,12679,12662],{"class":12661},[48,12681,12683,12686,12688],{"class":12634,"line":12682},6,[48,12684,12685],{"class":12646},"    new_digits ",[48,12687,90],{"class":12638},[48,12689,12690],{"class":12646}," []\n",[48,12692,12694,12697,12700,12703,12706],{"class":12634,"line":12693},7,[48,12695,12696],{"class":12638},"    for",[48,12698,12699],{"class":12646}," i, num_str ",[48,12701,12702],{"class":12638},"in",[48,12704,12705],{"class":12652}," enumerate",[48,12707,12708],{"class":12646},"(number_list):\n",[48,12710,12712,12715,12717,12720,12723,12726,12728,12731,12734],{"class":12634,"line":12711},8,[48,12713,12714],{"class":12646},"        decimal_part ",[48,12716,90],{"class":12638},[48,12718,12719],{"class":12646}," num_str.split(",[48,12721,12722],{"class":12661},"'.'",[48,12724,12725],{"class":12646},")[",[48,12727,1522],{"class":12652},[48,12729,12730],{"class":12646},"].ljust(digits, ",[48,12732,12733],{"class":12661},"'0'",[48,12735,12736],{"class":12646},")\n",[48,12738,12740,12743,12745,12748,12751],{"class":12634,"line":12739},9,[48,12741,12742],{"class":12646},"        d ",[48,12744,90],{"class":12638},[48,12746,12747],{"class":12652}," int",[48,12749,12750],{"class":12646},"(decimal_part[i])       ",[48,12752,12754],{"class":12753},"sAwPA","# 第 i 个数的第 i 位（对角线数字）\n",[48,12756,12758,12761,12763,12766,12768,12771,12774,12777,12780],{"class":12634,"line":12757},10,[48,12759,12760],{"class":12646},"        new_d ",[48,12762,90],{"class":12638},[48,12764,12765],{"class":12646}," (d ",[48,12767,2947],{"class":12638},[48,12769,12770],{"class":12652}," 5",[48,12772,12773],{"class":12646},") ",[48,12775,12776],{"class":12638},"%",[48,12778,12779],{"class":12652}," 10",[48,12781,12782],{"class":12753},"           # 规则：使新数字与原数字不同\n",[48,12784,12786,12789,12792],{"class":12634,"line":12785},11,[48,12787,12788],{"class":12646},"        new_digits.append(",[48,12790,12791],{"class":12652},"str",[48,12793,12794],{"class":12646},"(new_d))\n",[48,12796,12798,12801,12804,12807,12810],{"class":12634,"line":12797},12,[48,12799,12800],{"class":12638},"    return",[48,12802,12803],{"class":12661}," \"0.\"",[48,12805,12806],{"class":12638}," +",[48,12808,12809],{"class":12661}," \"\"",[48,12811,12812],{"class":12646},".join(new_digits)\n",[48,12814,12816],{"class":12634,"line":12815},13,[48,12817,12818],{"emptyLinePlaceholder":4800},"\n",[48,12820,12822],{"class":12634,"line":12821},14,[48,12823,12818],{"emptyLinePlaceholder":4800},[48,12825,12827,12830,12832],{"class":12634,"line":12826},15,[48,12828,12829],{"class":12646},"sample_list ",[48,12831,90],{"class":12638},[48,12833,12834],{"class":12646}," [\n",[48,12836,12838,12841,12844,12847,12849,12852,12854,12857],{"class":12634,"line":12837},16,[48,12839,12840],{"class":12661},"    \"0.1234567890\"",[48,12842,12843],{"class":12646},", ",[48,12845,12846],{"class":12661},"\"0.9876543210\"",[48,12848,12843],{"class":12646},[48,12850,12851],{"class":12661},"\"0.5555555555\"",[48,12853,12843],{"class":12646},[48,12855,12856],{"class":12661},"\"0.1111111111\"",[48,12858,12859],{"class":12646},",\n",[48,12861,12863,12866,12868,12871,12873,12876,12878,12881],{"class":12634,"line":12862},17,[48,12864,12865],{"class":12661},"    \"0.3141592653\"",[48,12867,12843],{"class":12646},[48,12869,12870],{"class":12661},"\"0.2718281828\"",[48,12872,12843],{"class":12646},[48,12874,12875],{"class":12661},"\"0.0000000001\"",[48,12877,12843],{"class":12646},[48,12879,12880],{"class":12661},"\"0.9999999998\"",[48,12882,12859],{"class":12646},[48,12884,12886,12889,12891,12894],{"class":12634,"line":12885},18,[48,12887,12888],{"class":12661},"    \"0.4242424242\"",[48,12890,12843],{"class":12646},[48,12892,12893],{"class":12661},"\"0.6180339887\"",[48,12895,12859],{"class":12646},[48,12897,12899],{"class":12634,"line":12898},19,[48,12900,12901],{"class":12646},"]\n",[48,12903,12905],{"class":12634,"line":12904},20,[48,12906,12818],{"emptyLinePlaceholder":4800},[48,12908,12910,12913,12915],{"class":12634,"line":12909},21,[48,12911,12912],{"class":12646},"new_number ",[48,12914,90],{"class":12638},[48,12916,12917],{"class":12646}," diagonal_construct(sample_list)\n",[48,12919,12921,12924,12926,12929],{"class":12634,"line":12920},22,[48,12922,12923],{"class":12652},"print",[48,12925,81],{"class":12646},[48,12927,12928],{"class":12661},"\"对角线构造出的新数：\"",[48,12930,12931],{"class":12646},", new_number)\n",[11,12933,12934],{},"输出：",[12625,12936,12940],{"className":12937,"code":12939,"language":4646},[12938],"language-text","原始列表：\n  第 1 个：0.1234567890   （第 1 位对角线数字：1）\n  第 2 个：0.9876543210   （第 2 位对角线数字：8）\n  第 3 个：0.5555555555   （第 3 位对角线数字：5）\n  第 4 个：0.1111111111   （第 4 位对角线数字：1）\n  第 5 个：0.3141592653   （第 5 位对角线数字：5）\n  第 6 个：0.2718281828   （第 6 位对角线数字：8）\n  第 7 个：0.0000000001   （第 7 位对角线数字：0）\n  第 8 个：0.9999999998   （第 8 位对角线数字：9）\n  第 9 个：0.4242424242   （第 9 位对角线数字：4）\n  第 10 个：0.6180339887  （第 10 位对角线数字：7）\n\n对角线构造出的新数：0.6306035492\n\n验证新数与列表中每个数都不同：\n  与第 1 个在第 1 位不同：原数=1，新数=6，不同=True\n  与第 2 个在第 2 位不同：原数=8，新数=3，不同=True\n  与第 3 个在第 3 位不同：原数=5，新数=0，不同=True\n  与第 4 个在第 4 位不同：原数=1，新数=6，不同=True\n  与第 5 个在第 5 位不同：原数=5，新数=0，不同=True\n  与第 6 个在第 6 位不同：原数=8，新数=3，不同=True\n  与第 7 个在第 7 位不同：原数=0，新数=5，不同=True\n  与第 8 个在第 8 位不同：原数=9，新数=4，不同=True\n  与第 9 个在第 9 位不同：原数=4，新数=9，不同=True\n  与第 10 个在第 10 位不同：原数=7，新数=2，不同=True\n",[2561,12941,12939],{"__ignoreMap":4787},[11,12943,12944,12945,12948,12949,12977,12978,13103],{},"可以清楚地看到，新构造的数 ",[2561,12946,12947],{},"0.6306035492"," 恰好在相应的位置上与列表中的每一个数都不同。当然，这里演示的只是十个数的有限情形（真正的无穷列表毕竟无法存入计算机），但这恰恰是证明核心逻辑步骤——对任意 ",[48,12950,12952,12965],{"className":12951,"translate":52},[56],[48,12953,12955],{"className":12954},[60],[62,12956,12957],{"xmlns":64},[67,12958,12959,12963],{},[70,12960,12961],{},[73,12962,9906],{},[142,12964,9906],{"encoding":144},[48,12966,12968],{"className":12967,"ariaHidden":150},[149],[48,12969,12971,12974],{"className":12970},[154],[48,12972],{"className":12973,"style":5810},[158],[48,12975,9906],{"className":12976,"style":9966},[163,171],"，",[48,12979,12981,13003],{"className":12980,"translate":52},[56],[48,12982,12984],{"className":12983},[60],[62,12985,12986],{"xmlns":64},[67,12987,12988,13000],{},[70,12989,12990,12992,12994],{},[73,12991,4526],{},[78,12993,5291],{"mathvariant":75},[115,12995,12996,12998],{},[73,12997,84],{},[73,12999,9906],{},[142,13001,13002],{"encoding":144},"y \\neq x_k",[48,13004,13006,13057],{"className":13005,"ariaHidden":150},[149],[48,13007,13009,13012,13015,13018,13054],{"className":13008},[154],[48,13010],{"className":13011,"style":766},[158],[48,13013,4526],{"className":13014,"style":343},[163,171],[48,13016],{"className":13017,"style":180},[179],[48,13019,13021,13048,13051],{"className":13020},[184],[48,13022,13024],{"className":13023},[184],[48,13025,13027],{"className":13026},[163,5391],[48,13028,13030],{"className":13029},[5395],[48,13031,13033,13036,13045],{"className":13032},[5399],[48,13034],{"className":13035,"style":766},[158],[48,13037,13039],{"className":13038},[5406],[48,13040,13042],{"className":13041},[163],[48,13043,5413],{"className":13044},[184],[48,13046],{"className":13047},[5417],[48,13049],{"className":13050},[179,5421],[48,13052,90],{"className":13053},[184],[48,13055],{"className":13056,"style":180},[179],[48,13058,13060,13063],{"className":13059},[154],[48,13061],{"className":13062,"style":5434},[158],[48,13064,13066,13069],{"className":13065},[163],[48,13067,84],{"className":13068},[163,171],[48,13070,13072],{"className":13071},[292],[48,13073,13075,13095],{"className":13074},[203,204],[48,13076,13078,13092],{"className":13077},[208],[48,13079,13081],{"className":13080,"style":302},[212],[48,13082,13083,13086],{"style":305},[48,13084],{"className":13085,"style":309},[220],[48,13087,13089],{"className":13088},[225,226,227,228],[48,13090,9906],{"className":13091,"style":9966},[163,171,228],[48,13093,269],{"className":13094},[268],[48,13096,13098],{"className":13097},[208],[48,13099,13101],{"className":13100,"style":2771},[212],[48,13102],{},"——的直观实例化。无论列表多长，这条构造规则都能保证新数逃脱每一个条目。",[35,13105,13106],{"id":13106},"康托尔定理",[11,13108,13109],{},"对角线论证不仅能证明实数不可数；它还有一个更一般、更强大的表述：康托尔定理。",[1295,13111,13112],{},[11,13113,13114,13115,13143,13144,2124,13192,13220,13221,13249,13250,1906],{},"康托尔定理。对任意集合 ",[48,13116,13118,13131],{"className":13117,"translate":52},[56],[48,13119,13121],{"className":13120},[60],[62,13122,13123],{"xmlns":64},[67,13124,13125,13129],{},[70,13126,13127],{},[73,13128,4891],{},[142,13130,4891],{"encoding":144},[48,13132,13134],{"className":13133,"ariaHidden":150},[149],[48,13135,13137,13140],{"className":13136},[154],[48,13138],{"className":13139,"style":4878},[158],[48,13141,4891],{"className":13142},[163,171],"，其幂集 ",[48,13145,13147,13169],{"className":13146,"translate":52},[56],[48,13148,13150],{"className":13149},[60],[62,13151,13152],{"xmlns":64},[67,13153,13154,13166],{},[70,13155,13156,13160,13162,13164],{},[73,13157,13159],{"mathvariant":13158},"script","P",[78,13161,81],{"stretchy":80},[73,13163,4891],{},[78,13165,87],{"stretchy":80},[142,13167,13168],{"encoding":144},"\\mathcal{P}(A)",[48,13170,13172],{"className":13171,"ariaHidden":150},[149],[48,13173,13175,13178,13183,13186,13189],{"className":13174},[154],[48,13176],{"className":13177,"style":159},[158],[48,13179,13159],{"className":13180,"style":13182},[163,13181],"mathcal","margin-right:0.0822em;",[48,13184,81],{"className":13185},[167],[48,13187,4891],{"className":13188},[163,171],[48,13190,87],{"className":13191},[175],[48,13193,13195,13208],{"className":13194,"translate":52},[56],[48,13196,13198],{"className":13197},[60],[62,13199,13200],{"xmlns":64},[67,13201,13202,13206],{},[70,13203,13204],{},[73,13205,4891],{},[142,13207,4891],{"encoding":144},[48,13209,13211],{"className":13210,"ariaHidden":150},[149],[48,13212,13214,13217],{"className":13213},[154],[48,13215],{"className":13216,"style":4878},[158],[48,13218,4891],{"className":13219},[163,171]," 的所有子集构成的集合）的势严格大于 ",[48,13222,13224,13237],{"className":13223,"translate":52},[56],[48,13225,13227],{"className":13226},[60],[62,13228,13229],{"xmlns":64},[67,13230,13231,13235],{},[70,13232,13233],{},[73,13234,4891],{},[142,13236,4891],{"encoding":144},[48,13238,13240],{"className":13239,"ariaHidden":150},[149],[48,13241,13243,13246],{"className":13242},[154],[48,13244],{"className":13245,"style":4878},[158],[48,13247,4891],{"className":13248},[163,171]," 本身，即 ",[48,13251,13253,13285],{"className":13252,"translate":52},[56],[48,13254,13256],{"className":13255},[60],[62,13257,13258],{"xmlns":64},[67,13259,13260,13282],{},[70,13261,13262,13264,13266,13268,13270,13272,13274,13276,13278,13280],{},[73,13263,4519],{"mathvariant":75},[73,13265,4891],{},[73,13267,4519],{"mathvariant":75},[78,13269,3807],{},[73,13271,4519],{"mathvariant":75},[73,13273,13159],{"mathvariant":13158},[78,13275,81],{"stretchy":80},[73,13277,4891],{},[78,13279,87],{"stretchy":80},[73,13281,4519],{"mathvariant":75},[142,13283,13284],{"encoding":144},"|A| \u003C |\\mathcal{P}(A)|",[48,13286,13288,13312],{"className":13287,"ariaHidden":150},[149],[48,13289,13291,13294,13297,13300,13303,13306,13309],{"className":13290},[154],[48,13292],{"className":13293,"style":159},[158],[48,13295,4519],{"className":13296},[163],[48,13298,4891],{"className":13299},[163,171],[48,13301,4519],{"className":13302},[163],[48,13304],{"className":13305,"style":180},[179],[48,13307,3807],{"className":13308},[184],[48,13310],{"className":13311,"style":180},[179],[48,13313,13315,13318,13321,13324,13327,13330,13333],{"className":13314},[154],[48,13316],{"className":13317,"style":159},[158],[48,13319,4519],{"className":13320},[163],[48,13322,13159],{"className":13323,"style":13182},[163,13181],[48,13325,81],{"className":13326},[167],[48,13328,4891],{"className":13329},[163,171],[48,13331,87],{"className":13332},[175],[48,13334,4519],{"className":13335},[163],[11,13337,13338,13339,13367,13368,13411],{},"需要指出，该定理对有限集和无穷集都成立，而对无穷集尤其揭示出一个惊人的事实：不存在「最大的无穷」。无论 ",[48,13340,13342,13355],{"className":13341,"translate":52},[56],[48,13343,13345],{"className":13344},[60],[62,13346,13347],{"xmlns":64},[67,13348,13349,13353],{},[70,13350,13351],{},[73,13352,4891],{},[142,13354,4891],{"encoding":144},[48,13356,13358],{"className":13357,"ariaHidden":150},[149],[48,13359,13361,13364],{"className":13360},[154],[48,13362],{"className":13363,"style":4878},[158],[48,13365,4891],{"className":13366},[163,171]," 多大，",[48,13369,13371,13390],{"className":13370,"translate":52},[56],[48,13372,13374],{"className":13373},[60],[62,13375,13376],{"xmlns":64},[67,13377,13378,13388],{},[70,13379,13380,13382,13384,13386],{},[73,13381,13159],{"mathvariant":13158},[78,13383,81],{"stretchy":80},[73,13385,4891],{},[78,13387,87],{"stretchy":80},[142,13389,13168],{"encoding":144},[48,13391,13393],{"className":13392,"ariaHidden":150},[149],[48,13394,13396,13399,13402,13405,13408],{"className":13395},[154],[48,13397],{"className":13398,"style":159},[158],[48,13400,13159],{"className":13401,"style":13182},[163,13181],[48,13403,81],{"className":13404},[167],[48,13406,4891],{"className":13407},[163,171],[48,13409,87],{"className":13410},[175]," 总更大；你可以无限地迭代幂集运算，得到一级级不断攀升的无穷层级。",[13413,13414,13415],"h3",{"id":13415},"证明",[11,13417,13418,13419,13507,13508,8936],{},"假设存在一个满射 ",[48,13420,13422,13450],{"className":13421,"translate":52},[56],[48,13423,13425],{"className":13424},[60],[62,13426,13427],{"xmlns":64},[67,13428,13429,13447],{},[70,13430,13431,13433,13435,13437,13439,13441,13443,13445],{},[73,13432,101],{},[78,13434,1495],{},[73,13436,4891],{},[78,13438,1504],{},[73,13440,13159],{"mathvariant":13158},[78,13442,81],{"stretchy":80},[73,13444,4891],{},[78,13446,87],{"stretchy":80},[142,13448,13449],{"encoding":144},"f: A \\to \\mathcal{P}(A)",[48,13451,13453,13471,13489],{"className":13452,"ariaHidden":150},[149],[48,13454,13456,13459,13462,13465,13468],{"className":13455},[154],[48,13457],{"className":13458,"style":766},[158],[48,13460,101],{"className":13461,"style":235},[163,171],[48,13463],{"className":13464,"style":180},[179],[48,13466,1495],{"className":13467},[184],[48,13469],{"className":13470,"style":180},[179],[48,13472,13474,13477,13480,13483,13486],{"className":13473},[154],[48,13475],{"className":13476,"style":4878},[158],[48,13478,4891],{"className":13479},[163,171],[48,13481],{"className":13482,"style":180},[179],[48,13484,1504],{"className":13485},[184],[48,13487],{"className":13488,"style":180},[179],[48,13490,13492,13495,13498,13501,13504],{"className":13491},[154],[48,13493],{"className":13494,"style":159},[158],[48,13496,13159],{"className":13497,"style":13182},[163,13181],[48,13499,81],{"className":13500},[167],[48,13502,4891],{"className":13503},[163,171],[48,13505,87],{"className":13506},[175],"（我们将证明这样的满射不存在，由此得到 ",[48,13509,13511,13542],{"className":13510,"translate":52},[56],[48,13512,13514],{"className":13513},[60],[62,13515,13516],{"xmlns":64},[67,13517,13518,13540],{},[70,13519,13520,13522,13524,13526,13528,13530,13532,13534,13536,13538],{},[73,13521,4519],{"mathvariant":75},[73,13523,4891],{},[73,13525,4519],{"mathvariant":75},[78,13527,3807],{},[73,13529,4519],{"mathvariant":75},[73,13531,13159],{"mathvariant":13158},[78,13533,81],{"stretchy":80},[73,13535,4891],{},[78,13537,87],{"stretchy":80},[73,13539,4519],{"mathvariant":75},[142,13541,13284],{"encoding":144},[48,13543,13545,13569],{"className":13544,"ariaHidden":150},[149],[48,13546,13548,13551,13554,13557,13560,13563,13566],{"className":13547},[154],[48,13549],{"className":13550,"style":159},[158],[48,13552,4519],{"className":13553},[163],[48,13555,4891],{"className":13556},[163,171],[48,13558,4519],{"className":13559},[163],[48,13561],{"className":13562,"style":180},[179],[48,13564,3807],{"className":13565},[184],[48,13567],{"className":13568,"style":180},[179],[48,13570,13572,13575,13578,13581,13584,13587,13590],{"className":13571},[154],[48,13573],{"className":13574,"style":159},[158],[48,13576,4519],{"className":13577},[163],[48,13579,13159],{"className":13580,"style":13182},[163,13181],[48,13582,81],{"className":13583},[167],[48,13585,4891],{"className":13586},[163,171],[48,13588,87],{"className":13589},[175],[48,13591,4519],{"className":13592},[163],[11,13594,13595,13596,12977,13646,13690,13691,13719],{},"对每个元素 ",[48,13597,13599,13616],{"className":13598,"translate":52},[56],[48,13600,13602],{"className":13601},[60],[62,13603,13604],{"xmlns":64},[67,13605,13606,13614],{},[70,13607,13608,13610,13612],{},[73,13609,15],{},[78,13611,104],{},[73,13613,4891],{},[142,13615,5714],{"encoding":144},[48,13617,13619,13637],{"className":13618,"ariaHidden":150},[149],[48,13620,13622,13625,13628,13631,13634],{"className":13621},[154],[48,13623],{"className":13624,"style":5724},[158],[48,13626,15],{"className":13627},[163,171],[48,13629],{"className":13630,"style":180},[179],[48,13632,104],{"className":13633},[184],[48,13635],{"className":13636,"style":180},[179],[48,13638,13640,13643],{"className":13639},[154],[48,13641],{"className":13642,"style":4878},[158],[48,13644,4891],{"className":13645},[163,171],[48,13647,13649,13669],{"className":13648,"translate":52},[56],[48,13650,13652],{"className":13651},[60],[62,13653,13654],{"xmlns":64},[67,13655,13656,13666],{},[70,13657,13658,13660,13662,13664],{},[73,13659,101],{},[78,13661,81],{"stretchy":80},[73,13663,15],{},[78,13665,87],{"stretchy":80},[142,13667,13668],{"encoding":144},"f(a)",[48,13670,13672],{"className":13671,"ariaHidden":150},[149],[48,13673,13675,13678,13681,13684,13687],{"className":13674},[154],[48,13676],{"className":13677,"style":159},[158],[48,13679,101],{"className":13680,"style":235},[163,171],[48,13682,81],{"className":13683},[167],[48,13685,15],{"className":13686},[163,171],[48,13688,87],{"className":13689},[175]," 都是 ",[48,13692,13694,13707],{"className":13693,"translate":52},[56],[48,13695,13697],{"className":13696},[60],[62,13698,13699],{"xmlns":64},[67,13700,13701,13705],{},[70,13702,13703],{},[73,13704,4891],{},[142,13706,4891],{"encoding":144},[48,13708,13710],{"className":13709,"ariaHidden":150},[149],[48,13711,13713,13716],{"className":13712},[154],[48,13714],{"className":13715,"style":4878},[158],[48,13717,4891],{"className":13718},[163,171]," 的一个子集。现在构造一个特殊的子集：",[48,13721,13723],{"className":13722,"translate":52},[51],[48,13724,13726,13771],{"className":13725,"translate":52},[56],[48,13727,13729],{"className":13728},[60],[62,13730,13731],{"xmlns":64,"display":65},[67,13732,13733,13768],{},[70,13734,13735,13737,13739,13741,13743,13745,13747,13749,13751,13753,13756,13758,13760,13762,13764,13766],{},[73,13736,1515],{},[78,13738,90],{},[78,13740,5022],{"stretchy":80},[4598,13742,5044],{},[73,13744,15],{},[78,13746,104],{},[73,13748,4891],{},[78,13750,1495],{},[73,13752,15],{},[78,13754,13755],{"mathvariant":75},"∉",[73,13757,101],{},[78,13759,81],{"stretchy":80},[73,13761,15],{},[78,13763,87],{"stretchy":80},[4598,13765,5044],{},[78,13767,5047],{"stretchy":80},[142,13769,13770],{"encoding":144},"D = \\{\\, a \\in A : a \\notin f(a) \\,\\}",[48,13772,13774,13792,13816,13834,13886],{"className":13773,"ariaHidden":150},[149],[48,13775,13777,13780,13783,13786,13789],{"className":13776},[154],[48,13778],{"className":13779,"style":4878},[158],[48,13781,1515],{"className":13782,"style":1680},[163,171],[48,13784],{"className":13785,"style":180},[179],[48,13787,90],{"className":13788},[184],[48,13790],{"className":13791,"style":180},[179],[48,13793,13795,13798,13801,13804,13807,13810,13813],{"className":13794},[154],[48,13796],{"className":13797,"style":159},[158],[48,13799,5022],{"className":13800},[167],[48,13802],{"className":13803,"style":282},[179],[48,13805,15],{"className":13806},[163,171],[48,13808],{"className":13809,"style":180},[179],[48,13811,104],{"className":13812},[184],[48,13814],{"className":13815,"style":180},[179],[48,13817,13819,13822,13825,13828,13831],{"className":13818},[154],[48,13820],{"className":13821,"style":4878},[158],[48,13823,4891],{"className":13824},[163,171],[48,13826],{"className":13827,"style":180},[179],[48,13829,1495],{"className":13830},[184],[48,13832],{"className":13833,"style":180},[179],[48,13835,13837,13840,13843,13846,13883],{"className":13836},[154],[48,13838],{"className":13839,"style":159},[158],[48,13841,15],{"className":13842},[163,171],[48,13844],{"className":13845,"style":180},[179],[48,13847,13849,13855],{"className":13848},[184],[48,13850,13852],{"className":13851},[163],[48,13853,104],{"className":13854},[184],[48,13856,13858],{"className":13857},[163,5391],[48,13859,13861],{"className":13860},[5395],[48,13862,13865,13868,13880],{"className":13863},[13864],"llap",[48,13866],{"className":13867,"style":159},[158],[48,13869,13871],{"className":13870},[5406],[48,13872,13874,13877],{"className":13873},[163],[48,13875,6483],{"className":13876},[163],[48,13878],{"className":13879,"style":4043},[179],[48,13881],{"className":13882},[5417],[48,13884],{"className":13885,"style":180},[179],[48,13887,13889,13892,13895,13898,13901,13904,13907],{"className":13888},[154],[48,13890],{"className":13891,"style":159},[158],[48,13893,101],{"className":13894,"style":235},[163,171],[48,13896,81],{"className":13897},[167],[48,13899,15],{"className":13900},[163,171],[48,13902,87],{"className":13903},[175],[48,13905],{"className":13906,"style":282},[179],[48,13908,5047],{"className":13909},[175],[11,13911,13912],{},"这就是「对角集」——它收集所有「不属于自己被映射到的那个子集」的元素。",[11,13914,13915,13916,13944,13945,13973,13974,14002,14003,14046,14047,5746,14098,1906],{},"由于 ",[48,13917,13919,13932],{"className":13918,"translate":52},[56],[48,13920,13922],{"className":13921},[60],[62,13923,13924],{"xmlns":64},[67,13925,13926,13930],{},[70,13927,13928],{},[73,13929,101],{},[142,13931,101],{"encoding":144},[48,13933,13935],{"className":13934,"ariaHidden":150},[149],[48,13936,13938,13941],{"className":13937},[154],[48,13939],{"className":13940,"style":766},[158],[48,13942,101],{"className":13943,"style":235},[163,171]," 是满射，",[48,13946,13948,13961],{"className":13947,"translate":52},[56],[48,13949,13951],{"className":13950},[60],[62,13952,13953],{"xmlns":64},[67,13954,13955,13959],{},[70,13956,13957],{},[73,13958,1515],{},[142,13960,1515],{"encoding":144},[48,13962,13964],{"className":13963,"ariaHidden":150},[149],[48,13965,13967,13970],{"className":13966},[154],[48,13968],{"className":13969,"style":4878},[158],[48,13971,1515],{"className":13972,"style":1680},[163,171],"（作为 ",[48,13975,13977,13990],{"className":13976,"translate":52},[56],[48,13978,13980],{"className":13979},[60],[62,13981,13982],{"xmlns":64},[67,13983,13984,13988],{},[70,13985,13986],{},[73,13987,4891],{},[142,13989,4891],{"encoding":144},[48,13991,13993],{"className":13992,"ariaHidden":150},[149],[48,13994,13996,13999],{"className":13995},[154],[48,13997],{"className":13998,"style":4878},[158],[48,14000,4891],{"className":14001},[163,171]," 的子集，因而是 ",[48,14004,14006,14025],{"className":14005,"translate":52},[56],[48,14007,14009],{"className":14008},[60],[62,14010,14011],{"xmlns":64},[67,14012,14013,14023],{},[70,14014,14015,14017,14019,14021],{},[73,14016,13159],{"mathvariant":13158},[78,14018,81],{"stretchy":80},[73,14020,4891],{},[78,14022,87],{"stretchy":80},[142,14024,13168],{"encoding":144},[48,14026,14028],{"className":14027,"ariaHidden":150},[149],[48,14029,14031,14034,14037,14040,14043],{"className":14030},[154],[48,14032],{"className":14033,"style":159},[158],[48,14035,13159],{"className":14036,"style":13182},[163,13181],[48,14038,81],{"className":14039},[167],[48,14041,4891],{"className":14042},[163,171],[48,14044,87],{"className":14045},[175]," 的元素）必有原像；即存在某个 ",[48,14048,14050,14068],{"className":14049,"translate":52},[56],[48,14051,14053],{"className":14052},[60],[62,14054,14055],{"xmlns":64},[67,14056,14057,14065],{},[70,14058,14059,14061,14063],{},[73,14060,879],{},[78,14062,104],{},[73,14064,4891],{},[142,14066,14067],{"encoding":144},"d \\in A",[48,14069,14071,14089],{"className":14070,"ariaHidden":150},[149],[48,14072,14074,14077,14080,14083,14086],{"className":14073},[154],[48,14075],{"className":14076,"style":5671},[158],[48,14078,879],{"className":14079},[163,171],[48,14081],{"className":14082,"style":180},[179],[48,14084,104],{"className":14085},[184],[48,14087],{"className":14088,"style":180},[179],[48,14090,14092,14095],{"className":14091},[154],[48,14093],{"className":14094,"style":4878},[158],[48,14096,4891],{"className":14097},[163,171],[48,14099,14101,14125],{"className":14100,"translate":52},[56],[48,14102,14104],{"className":14103},[60],[62,14105,14106],{"xmlns":64},[67,14107,14108,14122],{},[70,14109,14110,14112,14114,14116,14118,14120],{},[73,14111,101],{},[78,14113,81],{"stretchy":80},[73,14115,879],{},[78,14117,87],{"stretchy":80},[78,14119,90],{},[73,14121,1515],{},[142,14123,14124],{"encoding":144},"f(d) = D",[48,14126,14128,14155],{"className":14127,"ariaHidden":150},[149],[48,14129,14131,14134,14137,14140,14143,14146,14149,14152],{"className":14130},[154],[48,14132],{"className":14133,"style":159},[158],[48,14135,101],{"className":14136,"style":235},[163,171],[48,14138,81],{"className":14139},[167],[48,14141,879],{"className":14142},[163,171],[48,14144,87],{"className":14145},[175],[48,14147],{"className":14148,"style":180},[179],[48,14150,90],{"className":14151},[184],[48,14153],{"className":14154,"style":180},[179],[48,14156,14158,14161],{"className":14157},[154],[48,14159],{"className":14160,"style":4878},[158],[48,14162,1515],{"className":14163,"style":1680},[163,171],[11,14165,14166,14167,14218],{},"现在问：",[48,14168,14170,14188],{"className":14169,"translate":52},[56],[48,14171,14173],{"className":14172},[60],[62,14174,14175],{"xmlns":64},[67,14176,14177,14185],{},[70,14178,14179,14181,14183],{},[73,14180,879],{},[78,14182,104],{},[73,14184,1515],{},[142,14186,14187],{"encoding":144},"d \\in D",[48,14189,14191,14209],{"className":14190,"ariaHidden":150},[149],[48,14192,14194,14197,14200,14203,14206],{"className":14193},[154],[48,14195],{"className":14196,"style":5671},[158],[48,14198,879],{"className":14199},[163,171],[48,14201],{"className":14202,"style":180},[179],[48,14204,104],{"className":14205},[184],[48,14207],{"className":14208,"style":180},[179],[48,14210,14212,14215],{"className":14211},[154],[48,14213],{"className":14214,"style":4878},[158],[48,14216,1515],{"className":14217,"style":1680},[163,171]," 吗？",[993,14220,14221,14665],{},[996,14222,14223,14224,14274,14275,14303,14304,14459,14460,14525,14526,14664],{},"若 ",[48,14225,14227,14244],{"className":14226,"translate":52},[56],[48,14228,14230],{"className":14229},[60],[62,14231,14232],{"xmlns":64},[67,14233,14234,14242],{},[70,14235,14236,14238,14240],{},[73,14237,879],{},[78,14239,104],{},[73,14241,1515],{},[142,14243,14187],{"encoding":144},[48,14245,14247,14265],{"className":14246,"ariaHidden":150},[149],[48,14248,14250,14253,14256,14259,14262],{"className":14249},[154],[48,14251],{"className":14252,"style":5671},[158],[48,14254,879],{"className":14255},[163,171],[48,14257],{"className":14258,"style":180},[179],[48,14260,104],{"className":14261},[184],[48,14263],{"className":14264,"style":180},[179],[48,14266,14268,14271],{"className":14267},[154],[48,14269],{"className":14270,"style":4878},[158],[48,14272,1515],{"className":14273,"style":1680},[163,171],"：根据 ",[48,14276,14278,14291],{"className":14277,"translate":52},[56],[48,14279,14281],{"className":14280},[60],[62,14282,14283],{"xmlns":64},[67,14284,14285,14289],{},[70,14286,14287],{},[73,14288,1515],{},[142,14290,1515],{"encoding":144},[48,14292,14294],{"className":14293,"ariaHidden":150},[149],[48,14295,14297,14300],{"className":14296},[154],[48,14298],{"className":14299,"style":4878},[158],[48,14301,1515],{"className":14302,"style":1680},[163,171]," 的定义，",[48,14305,14307,14344],{"className":14306,"translate":52},[56],[48,14308,14310],{"className":14309},[60],[62,14311,14312],{"xmlns":64},[67,14313,14314,14341],{},[70,14315,14316,14318,14320,14322,14324,14327,14329,14331,14333,14335,14337,14339],{},[73,14317,879],{},[78,14319,104],{},[73,14321,1515],{},[4598,14323,10423],{},[78,14325,14326],{},"⟺",[4598,14328,10423],{},[73,14330,879],{},[78,14332,13755],{"mathvariant":75},[73,14334,101],{},[78,14336,81],{"stretchy":80},[73,14338,879],{},[78,14340,87],{"stretchy":80},[142,14342,14343],{"encoding":144},"d \\in D \\iff d \\notin f(d)",[48,14345,14347,14365,14390,14441],{"className":14346,"ariaHidden":150},[149],[48,14348,14350,14353,14356,14359,14362],{"className":14349},[154],[48,14351],{"className":14352,"style":5671},[158],[48,14354,879],{"className":14355},[163,171],[48,14357],{"className":14358,"style":180},[179],[48,14360,104],{"className":14361},[184],[48,14363],{"className":14364,"style":180},[179],[48,14366,14368,14372,14375,14378,14381,14384,14387],{"className":14367},[154],[48,14369],{"className":14370,"style":14371},[158],"height:0.7073em;vertical-align:-0.024em;",[48,14373,1515],{"className":14374,"style":1680},[163,171],[48,14376],{"className":14377,"style":180},[179],[48,14379],{"className":14380,"style":180},[179],[48,14382,14326],{"className":14383},[184],[48,14385],{"className":14386,"style":180},[179],[48,14388],{"className":14389,"style":180},[179],[48,14391,14393,14396,14399,14402,14438],{"className":14392},[154],[48,14394],{"className":14395,"style":159},[158],[48,14397,879],{"className":14398},[163,171],[48,14400],{"className":14401,"style":180},[179],[48,14403,14405,14411],{"className":14404},[184],[48,14406,14408],{"className":14407},[163],[48,14409,104],{"className":14410},[184],[48,14412,14414],{"className":14413},[163,5391],[48,14415,14417],{"className":14416},[5395],[48,14418,14420,14423,14435],{"className":14419},[13864],[48,14421],{"className":14422,"style":159},[158],[48,14424,14426],{"className":14425},[5406],[48,14427,14429,14432],{"className":14428},[163],[48,14430,6483],{"className":14431},[163],[48,14433],{"className":14434,"style":4043},[179],[48,14436],{"className":14437},[5417],[48,14439],{"className":14440,"style":180},[179],[48,14442,14444,14447,14450,14453,14456],{"className":14443},[154],[48,14445],{"className":14446,"style":159},[158],[48,14448,101],{"className":14449,"style":235},[163,171],[48,14451,81],{"className":14452},[167],[48,14454,879],{"className":14455},[163,171],[48,14457,87],{"className":14458},[175],"。由于 ",[48,14461,14463,14486],{"className":14462,"translate":52},[56],[48,14464,14466],{"className":14465},[60],[62,14467,14468],{"xmlns":64},[67,14469,14470,14484],{},[70,14471,14472,14474,14476,14478,14480,14482],{},[73,14473,101],{},[78,14475,81],{"stretchy":80},[73,14477,879],{},[78,14479,87],{"stretchy":80},[78,14481,90],{},[73,14483,1515],{},[142,14485,14124],{"encoding":144},[48,14487,14489,14516],{"className":14488,"ariaHidden":150},[149],[48,14490,14492,14495,14498,14501,14504,14507,14510,14513],{"className":14491},[154],[48,14493],{"className":14494,"style":159},[158],[48,14496,101],{"className":14497,"style":235},[163,171],[48,14499,81],{"className":14500},[167],[48,14502,879],{"className":14503},[163,171],[48,14505,87],{"className":14506},[175],[48,14508],{"className":14509,"style":180},[179],[48,14511,90],{"className":14512},[184],[48,14514],{"className":14515,"style":180},[179],[48,14517,14519,14522],{"className":14518},[154],[48,14520],{"className":14521,"style":4878},[158],[48,14523,1515],{"className":14524,"style":1680},[163,171],"，就有 ",[48,14527,14529,14559],{"className":14528,"translate":52},[56],[48,14530,14532],{"className":14531},[60],[62,14533,14534],{"xmlns":64},[67,14535,14536,14556],{},[70,14537,14538,14540,14542,14544,14546,14548,14550,14552,14554],{},[73,14539,879],{},[78,14541,104],{},[73,14543,1515],{},[4598,14545,10423],{},[78,14547,14326],{},[4598,14549,10423],{},[73,14551,879],{},[78,14553,13755],{"mathvariant":75},[73,14555,1515],{},[142,14557,14558],{"encoding":144},"d \\in D \\iff d \\notin D",[48,14560,14562,14580,14604,14655],{"className":14561,"ariaHidden":150},[149],[48,14563,14565,14568,14571,14574,14577],{"className":14564},[154],[48,14566],{"className":14567,"style":5671},[158],[48,14569,879],{"className":14570},[163,171],[48,14572],{"className":14573,"style":180},[179],[48,14575,104],{"className":14576},[184],[48,14578],{"className":14579,"style":180},[179],[48,14581,14583,14586,14589,14592,14595,14598,14601],{"className":14582},[154],[48,14584],{"className":14585,"style":14371},[158],[48,14587,1515],{"className":14588,"style":1680},[163,171],[48,14590],{"className":14591,"style":180},[179],[48,14593],{"className":14594,"style":180},[179],[48,14596,14326],{"className":14597},[184],[48,14599],{"className":14600,"style":180},[179],[48,14602],{"className":14603,"style":180},[179],[48,14605,14607,14610,14613,14616,14652],{"className":14606},[154],[48,14608],{"className":14609,"style":159},[158],[48,14611,879],{"className":14612},[163,171],[48,14614],{"className":14615,"style":180},[179],[48,14617,14619,14625],{"className":14618},[184],[48,14620,14622],{"className":14621},[163],[48,14623,104],{"className":14624},[184],[48,14626,14628],{"className":14627},[163,5391],[48,14629,14631],{"className":14630},[5395],[48,14632,14634,14637,14649],{"className":14633},[13864],[48,14635],{"className":14636,"style":159},[158],[48,14638,14640],{"className":14639},[5406],[48,14641,14643,14646],{"className":14642},[163],[48,14644,6483],{"className":14645},[163],[48,14647],{"className":14648,"style":4043},[179],[48,14650],{"className":14651},[5417],[48,14653],{"className":14654,"style":180},[179],[48,14656,14658,14661],{"className":14657},[154],[48,14659],{"className":14660,"style":4878},[158],[48,14662,1515],{"className":14663,"style":1680},[163,171],"。矛盾！",[996,14666,14223,14667,14751,14752,14951,14952,15040,15041,14664],{},[48,14668,14670,14688],{"className":14669,"translate":52},[56],[48,14671,14673],{"className":14672},[60],[62,14674,14675],{"xmlns":64},[67,14676,14677,14685],{},[70,14678,14679,14681,14683],{},[73,14680,879],{},[78,14682,13755],{"mathvariant":75},[73,14684,1515],{},[142,14686,14687],{"encoding":144},"d \\notin D",[48,14689,14691,14742],{"className":14690,"ariaHidden":150},[149],[48,14692,14694,14697,14700,14703,14739],{"className":14693},[154],[48,14695],{"className":14696,"style":159},[158],[48,14698,879],{"className":14699},[163,171],[48,14701],{"className":14702,"style":180},[179],[48,14704,14706,14712],{"className":14705},[184],[48,14707,14709],{"className":14708},[163],[48,14710,104],{"className":14711},[184],[48,14713,14715],{"className":14714},[163,5391],[48,14716,14718],{"className":14717},[5395],[48,14719,14721,14724,14736],{"className":14720},[13864],[48,14722],{"className":14723,"style":159},[158],[48,14725,14727],{"className":14726},[5406],[48,14728,14730,14733],{"className":14729},[163],[48,14731,6483],{"className":14732},[163],[48,14734],{"className":14735,"style":4043},[179],[48,14737],{"className":14738},[5417],[48,14740],{"className":14741,"style":180},[179],[48,14743,14745,14748],{"className":14744},[154],[48,14746],{"className":14747,"style":4878},[158],[48,14749,1515],{"className":14750,"style":1680},[163,171],"：同理，",[48,14753,14755,14798],{"className":14754,"translate":52},[56],[48,14756,14758],{"className":14757},[60],[62,14759,14760],{"xmlns":64},[67,14761,14762,14795],{},[70,14763,14764,14766,14768,14770,14772,14774,14776,14779,14781,14783,14785,14787,14789,14791,14793],{},[73,14765,879],{},[78,14767,13755],{"mathvariant":75},[73,14769,1515],{},[4598,14771,10423],{},[78,14773,14326],{},[4598,14775,10423],{},[73,14777,14778],{"mathvariant":75},"¬",[78,14780,81],{"stretchy":80},[73,14782,879],{},[78,14784,13755],{"mathvariant":75},[73,14786,101],{},[78,14788,81],{"stretchy":80},[73,14790,879],{},[78,14792,87],{"stretchy":80},[78,14794,87],{"stretchy":80},[142,14796,14797],{"encoding":144},"d \\notin D \\iff \\neg(d \\notin f(d))",[48,14799,14801,14852,14876,14933],{"className":14800,"ariaHidden":150},[149],[48,14802,14804,14807,14810,14813,14849],{"className":14803},[154],[48,14805],{"className":14806,"style":159},[158],[48,14808,879],{"className":14809},[163,171],[48,14811],{"className":14812,"style":180},[179],[48,14814,14816,14822],{"className":14815},[184],[48,14817,14819],{"className":14818},[163],[48,14820,104],{"className":14821},[184],[48,14823,14825],{"className":14824},[163,5391],[48,14826,14828],{"className":14827},[5395],[48,14829,14831,14834,14846],{"className":14830},[13864],[48,14832],{"className":14833,"style":159},[158],[48,14835,14837],{"className":14836},[5406],[48,14838,14840,14843],{"className":14839},[163],[48,14841,6483],{"className":14842},[163],[48,14844],{"className":14845,"style":4043},[179],[48,14847],{"className":14848},[5417],[48,14850],{"className":14851,"style":180},[179],[48,14853,14855,14858,14861,14864,14867,14870,14873],{"className":14854},[154],[48,14856],{"className":14857,"style":14371},[158],[48,14859,1515],{"className":14860,"style":1680},[163,171],[48,14862],{"className":14863,"style":180},[179],[48,14865],{"className":14866,"style":180},[179],[48,14868,14326],{"className":14869},[184],[48,14871],{"className":14872,"style":180},[179],[48,14874],{"className":14875,"style":180},[179],[48,14877,14879,14882,14885,14888,14891,14894,14930],{"className":14878},[154],[48,14880],{"className":14881,"style":159},[158],[48,14883,14778],{"className":14884},[163],[48,14886,81],{"className":14887},[167],[48,14889,879],{"className":14890},[163,171],[48,14892],{"className":14893,"style":180},[179],[48,14895,14897,14903],{"className":14896},[184],[48,14898,14900],{"className":14899},[163],[48,14901,104],{"className":14902},[184],[48,14904,14906],{"className":14905},[163,5391],[48,14907,14909],{"className":14908},[5395],[48,14910,14912,14915,14927],{"className":14911},[13864],[48,14913],{"className":14914,"style":159},[158],[48,14916,14918],{"className":14917},[5406],[48,14919,14921,14924],{"className":14920},[163],[48,14922,6483],{"className":14923},[163],[48,14925],{"className":14926,"style":4043},[179],[48,14928],{"className":14929},[5417],[48,14931],{"className":14932,"style":180},[179],[48,14934,14936,14939,14942,14945,14948],{"className":14935},[154],[48,14937],{"className":14938,"style":159},[158],[48,14940,101],{"className":14941,"style":235},[163,171],[48,14943,81],{"className":14944},[167],[48,14946,879],{"className":14947},[163,171],[48,14949,1905],{"className":14950},[175],"，即 ",[48,14953,14955,14983],{"className":14954,"translate":52},[56],[48,14956,14958],{"className":14957},[60],[62,14959,14960],{"xmlns":64},[67,14961,14962,14980],{},[70,14963,14964,14966,14968,14970,14972,14974,14976,14978],{},[73,14965,879],{},[78,14967,104],{},[73,14969,101],{},[78,14971,81],{"stretchy":80},[73,14973,879],{},[78,14975,87],{"stretchy":80},[78,14977,90],{},[73,14979,1515],{},[142,14981,14982],{"encoding":144},"d \\in f(d) = D",[48,14984,14986,15004,15031],{"className":14985,"ariaHidden":150},[149],[48,14987,14989,14992,14995,14998,15001],{"className":14988},[154],[48,14990],{"className":14991,"style":5671},[158],[48,14993,879],{"className":14994},[163,171],[48,14996],{"className":14997,"style":180},[179],[48,14999,104],{"className":15000},[184],[48,15002],{"className":15003,"style":180},[179],[48,15005,15007,15010,15013,15016,15019,15022,15025,15028],{"className":15006},[154],[48,15008],{"className":15009,"style":159},[158],[48,15011,101],{"className":15012,"style":235},[163,171],[48,15014,81],{"className":15015},[167],[48,15017,879],{"className":15018},[163,171],[48,15020,87],{"className":15021},[175],[48,15023],{"className":15024,"style":180},[179],[48,15026,90],{"className":15027},[184],[48,15029],{"className":15030,"style":180},[179],[48,15032,15034,15037],{"className":15033},[154],[48,15035],{"className":15036,"style":4878},[158],[48,15038,1515],{"className":15039,"style":1680},[163,171],"，从而 ",[48,15042,15044,15061],{"className":15043,"translate":52},[56],[48,15045,15047],{"className":15046},[60],[62,15048,15049],{"xmlns":64},[67,15050,15051,15059],{},[70,15052,15053,15055,15057],{},[73,15054,879],{},[78,15056,104],{},[73,15058,1515],{},[142,15060,14187],{"encoding":144},[48,15062,15064,15082],{"className":15063,"ariaHidden":150},[149],[48,15065,15067,15070,15073,15076,15079],{"className":15066},[154],[48,15068],{"className":15069,"style":5671},[158],[48,15071,879],{"className":15072},[163,171],[48,15074],{"className":15075,"style":180},[179],[48,15077,104],{"className":15078},[184],[48,15080],{"className":15081,"style":180},[179],[48,15083,15085,15088],{"className":15084},[154],[48,15086],{"className":15087,"style":4878},[158],[48,15089,1515],{"className":15090,"style":1680},[163,171],[11,15092,15093,15094,15181,15182,1906],{},"两种情况都导致自相矛盾，说明「存在满射 ",[48,15095,15097,15124],{"className":15096,"translate":52},[56],[48,15098,15100],{"className":15099},[60],[62,15101,15102],{"xmlns":64},[67,15103,15104,15122],{},[70,15105,15106,15108,15110,15112,15114,15116,15118,15120],{},[73,15107,101],{},[78,15109,1495],{},[73,15111,4891],{},[78,15113,1504],{},[73,15115,13159],{"mathvariant":13158},[78,15117,81],{"stretchy":80},[73,15119,4891],{},[78,15121,87],{"stretchy":80},[142,15123,13449],{"encoding":144},[48,15125,15127,15145,15163],{"className":15126,"ariaHidden":150},[149],[48,15128,15130,15133,15136,15139,15142],{"className":15129},[154],[48,15131],{"className":15132,"style":766},[158],[48,15134,101],{"className":15135,"style":235},[163,171],[48,15137],{"className":15138,"style":180},[179],[48,15140,1495],{"className":15141},[184],[48,15143],{"className":15144,"style":180},[179],[48,15146,15148,15151,15154,15157,15160],{"className":15147},[154],[48,15149],{"className":15150,"style":4878},[158],[48,15152,4891],{"className":15153},[163,171],[48,15155],{"className":15156,"style":180},[179],[48,15158,1504],{"className":15159},[184],[48,15161],{"className":15162,"style":180},[179],[48,15164,15166,15169,15172,15175,15178],{"className":15165},[154],[48,15167],{"className":15168,"style":159},[158],[48,15170,13159],{"className":15171,"style":13182},[163,13181],[48,15173,81],{"className":15174},[167],[48,15176,4891],{"className":15177},[163,171],[48,15179,87],{"className":15180},[175],"」这一假设站不住脚。因此 ",[48,15183,15185,15216],{"className":15184,"translate":52},[56],[48,15186,15188],{"className":15187},[60],[62,15189,15190],{"xmlns":64},[67,15191,15192,15214],{},[70,15193,15194,15196,15198,15200,15202,15204,15206,15208,15210,15212],{},[73,15195,4519],{"mathvariant":75},[73,15197,4891],{},[73,15199,4519],{"mathvariant":75},[78,15201,3807],{},[73,15203,4519],{"mathvariant":75},[73,15205,13159],{"mathvariant":13158},[78,15207,81],{"stretchy":80},[73,15209,4891],{},[78,15211,87],{"stretchy":80},[73,15213,4519],{"mathvariant":75},[142,15215,13284],{"encoding":144},[48,15217,15219,15243],{"className":15218,"ariaHidden":150},[149],[48,15220,15222,15225,15228,15231,15234,15237,15240],{"className":15221},[154],[48,15223],{"className":15224,"style":159},[158],[48,15226,4519],{"className":15227},[163],[48,15229,4891],{"className":15230},[163,171],[48,15232,4519],{"className":15233},[163],[48,15235],{"className":15236,"style":180},[179],[48,15238,3807],{"className":15239},[184],[48,15241],{"className":15242,"style":180},[179],[48,15244,15246,15249,15252,15255,15258,15261,15264],{"className":15245},[154],[48,15247],{"className":15248,"style":159},[158],[48,15250,4519],{"className":15251},[163],[48,15253,13159],{"className":15254,"style":13182},[163,13181],[48,15256,81],{"className":15257},[167],[48,15259,4891],{"className":15260},[163,171],[48,15262,87],{"className":15263},[175],[48,15265,4519],{"className":15266},[163],[11,15268,15269],{},"细心的读者可能已经发现，这个证明的结构与上面的不可数证明完全相同：",[15271,15272,15273,15285],"table",{},[15274,15275,15276],"thead",{},[15277,15278,15279,15283],"tr",{},[15280,15281,15282],"th",{},"实数的不可数性",[15280,15284,13106],{},[15286,15287,15288,15479,15766,16054],"tbody",{},[15277,15289,15290,15389],{},[15291,15292,15293,15294],"td",{},"假设双射 ",[48,15295,15297,15326],{"className":15296,"translate":52},[56],[48,15298,15300],{"className":15299},[60],[62,15301,15302],{"xmlns":64},[67,15303,15304,15324],{},[70,15305,15306,15308,15310,15312,15314,15316,15318,15320,15322],{},[73,15307,101],{},[78,15309,1495],{},[73,15311,5017],{"mathvariant":5016},[78,15313,1504],{},[78,15315,81],{"stretchy":80},[1520,15317,2710],{},[78,15319,874],{"separator":150},[1520,15321,1522],{},[78,15323,87],{"stretchy":80},[142,15325,7232],{"encoding":144},[48,15327,15329,15347,15365],{"className":15328,"ariaHidden":150},[149],[48,15330,15332,15335,15338,15341,15344],{"className":15331},[154],[48,15333],{"className":15334,"style":766},[158],[48,15336,101],{"className":15337,"style":235},[163,171],[48,15339],{"className":15340,"style":180},[179],[48,15342,1495],{"className":15343},[184],[48,15345],{"className":15346,"style":180},[179],[48,15348,15350,15353,15356,15359,15362],{"className":15349},[154],[48,15351],{"className":15352,"style":5060},[158],[48,15354,5017],{"className":15355},[163,5064],[48,15357],{"className":15358,"style":180},[179],[48,15360,1504],{"className":15361},[184],[48,15363],{"className":15364,"style":180},[179],[48,15366,15368,15371,15374,15377,15380,15383,15386],{"className":15367},[154],[48,15369],{"className":15370,"style":159},[158],[48,15372,81],{"className":15373},[167],[48,15375,2710],{"className":15376},[163],[48,15378,874],{"className":15379},[944],[48,15381],{"className":15382,"style":282},[179],[48,15384,1522],{"className":15385},[163],[48,15387,87],{"className":15388},[175],[15291,15390,15391,15392],{},"假设满射 ",[48,15393,15395,15422],{"className":15394,"translate":52},[56],[48,15396,15398],{"className":15397},[60],[62,15399,15400],{"xmlns":64},[67,15401,15402,15420],{},[70,15403,15404,15406,15408,15410,15412,15414,15416,15418],{},[73,15405,101],{},[78,15407,1495],{},[73,15409,4891],{},[78,15411,1504],{},[73,15413,13159],{"mathvariant":13158},[78,15415,81],{"stretchy":80},[73,15417,4891],{},[78,15419,87],{"stretchy":80},[142,15421,13449],{"encoding":144},[48,15423,15425,15443,15461],{"className":15424,"ariaHidden":150},[149],[48,15426,15428,15431,15434,15437,15440],{"className":15427},[154],[48,15429],{"className":15430,"style":766},[158],[48,15432,101],{"className":15433,"style":235},[163,171],[48,15435],{"className":15436,"style":180},[179],[48,15438,1495],{"className":15439},[184],[48,15441],{"className":15442,"style":180},[179],[48,15444,15446,15449,15452,15455,15458],{"className":15445},[154],[48,15447],{"className":15448,"style":4878},[158],[48,15450,4891],{"className":15451},[163,171],[48,15453],{"className":15454,"style":180},[179],[48,15456,1504],{"className":15457},[184],[48,15459],{"className":15460,"style":180},[179],[48,15462,15464,15467,15470,15473,15476],{"className":15463},[154],[48,15465],{"className":15466,"style":159},[158],[48,15468,13159],{"className":15469,"style":13182},[163,13181],[48,15471,81],{"className":15472},[167],[48,15474,4891],{"className":15475},[163,171],[48,15477,87],{"className":15478},[175],[15277,15480,15481,15612],{},[15291,15482,15483,15484,15512,15513,10038,15583,15611],{},"构造对角数 ",[48,15485,15487,15500],{"className":15486,"translate":52},[56],[48,15488,15490],{"className":15489},[60],[62,15491,15492],{"xmlns":64},[67,15493,15494,15498],{},[70,15495,15496],{},[73,15497,4526],{},[142,15499,4526],{"encoding":144},[48,15501,15503],{"className":15502,"ariaHidden":150},[149],[48,15504,15506,15509],{"className":15505},[154],[48,15507],{"className":15508,"style":2731},[158],[48,15510,4526],{"className":15511,"style":343},[163,171],"，每一位都与 ",[48,15514,15516,15534],{"className":15515,"translate":52},[56],[48,15517,15519],{"className":15518},[60],[62,15520,15521],{"xmlns":64},[67,15522,15523,15531],{},[70,15524,15525],{},[115,15526,15527,15529],{},[73,15528,84],{},[73,15530,6221],{},[142,15532,15533],{"encoding":144},"x_n",[48,15535,15537],{"className":15536,"ariaHidden":150},[149],[48,15538,15540,15543],{"className":15539},[154],[48,15541],{"className":15542,"style":5434},[158],[48,15544,15546,15549],{"className":15545},[163],[48,15547,84],{"className":15548},[163,171],[48,15550,15552],{"className":15551},[292],[48,15553,15555,15575],{"className":15554},[203,204],[48,15556,15558,15572],{"className":15557},[208],[48,15559,15561],{"className":15560,"style":9307},[212],[48,15562,15563,15566],{"style":305},[48,15564],{"className":15565,"style":309},[220],[48,15567,15569],{"className":15568},[225,226,227,228],[48,15570,6221],{"className":15571},[163,171,228],[48,15573,269],{"className":15574},[268],[48,15576,15578],{"className":15577},[208],[48,15579,15581],{"className":15580,"style":2771},[212],[48,15582],{},[48,15584,15586,15599],{"className":15585,"translate":52},[56],[48,15587,15589],{"className":15588},[60],[62,15590,15591],{"xmlns":64},[67,15592,15593,15597],{},[70,15594,15595],{},[73,15596,6221],{},[142,15598,6221],{"encoding":144},[48,15600,15602],{"className":15601,"ariaHidden":150},[149],[48,15603,15605,15608],{"className":15604},[154],[48,15606],{"className":15607,"style":578},[158],[48,15609,6221],{"className":15610},[163,171]," 位不同",[15291,15613,15614,15615],{},"构造对角集 ",[48,15616,15618,15654],{"className":15617,"translate":52},[56],[48,15619,15621],{"className":15620},[60],[62,15622,15623],{"xmlns":64},[67,15624,15625,15651],{},[70,15626,15627,15629,15631,15633,15635,15637,15639,15641,15643,15645,15647,15649],{},[73,15628,1515],{},[78,15630,90],{},[78,15632,5022],{"stretchy":80},[73,15634,15],{},[78,15636,1495],{},[73,15638,15],{},[78,15640,13755],{"mathvariant":75},[73,15642,101],{},[78,15644,81],{"stretchy":80},[73,15646,15],{},[78,15648,87],{"stretchy":80},[78,15650,5047],{"stretchy":80},[142,15652,15653],{"encoding":144},"D = \\{a : a \\notin f(a)\\}",[48,15655,15657,15675,15696,15747],{"className":15656,"ariaHidden":150},[149],[48,15658,15660,15663,15666,15669,15672],{"className":15659},[154],[48,15661],{"className":15662,"style":4878},[158],[48,15664,1515],{"className":15665,"style":1680},[163,171],[48,15667],{"className":15668,"style":180},[179],[48,15670,90],{"className":15671},[184],[48,15673],{"className":15674,"style":180},[179],[48,15676,15678,15681,15684,15687,15690,15693],{"className":15677},[154],[48,15679],{"className":15680,"style":159},[158],[48,15682,5022],{"className":15683},[167],[48,15685,15],{"className":15686},[163,171],[48,15688],{"className":15689,"style":180},[179],[48,15691,1495],{"className":15692},[184],[48,15694],{"className":15695,"style":180},[179],[48,15697,15699,15702,15705,15708,15744],{"className":15698},[154],[48,15700],{"className":15701,"style":159},[158],[48,15703,15],{"className":15704},[163,171],[48,15706],{"className":15707,"style":180},[179],[48,15709,15711,15717],{"className":15710},[184],[48,15712,15714],{"className":15713},[163],[48,15715,104],{"className":15716},[184],[48,15718,15720],{"className":15719},[163,5391],[48,15721,15723],{"className":15722},[5395],[48,15724,15726,15729,15741],{"className":15725},[13864],[48,15727],{"className":15728,"style":159},[158],[48,15730,15732],{"className":15731},[5406],[48,15733,15735,15738],{"className":15734},[163],[48,15736,6483],{"className":15737},[163],[48,15739],{"className":15740,"style":4043},[179],[48,15742],{"className":15743},[5417],[48,15745],{"className":15746,"style":180},[179],[48,15748,15750,15753,15756,15759,15762],{"className":15749},[154],[48,15751],{"className":15752,"style":159},[158],[48,15754,101],{"className":15755,"style":235},[163,171],[48,15757,81],{"className":15758},[167],[48,15760,15],{"className":15761},[163,171],[48,15763,15765],{"className":15764},[175],")}",[15277,15767,15768,15925],{},[15291,15769,15770,15771,15799,15800],{},"对所有 ",[48,15772,15774,15787],{"className":15773,"translate":52},[56],[48,15775,15777],{"className":15776},[60],[62,15778,15779],{"xmlns":64},[67,15780,15781,15785],{},[70,15782,15783],{},[73,15784,6221],{},[142,15786,6221],{"encoding":144},[48,15788,15790],{"className":15789,"ariaHidden":150},[149],[48,15791,15793,15796],{"className":15792},[154],[48,15794],{"className":15795,"style":578},[158],[48,15797,6221],{"className":15798},[163,171]," 有 ",[48,15801,15803,15825],{"className":15802,"translate":52},[56],[48,15804,15806],{"className":15805},[60],[62,15807,15808],{"xmlns":64},[67,15809,15810,15822],{},[70,15811,15812,15814,15816],{},[73,15813,4526],{},[78,15815,5291],{"mathvariant":75},[115,15817,15818,15820],{},[73,15819,84],{},[73,15821,6221],{},[142,15823,15824],{"encoding":144},"y \\neq x_n",[48,15826,15828,15879],{"className":15827,"ariaHidden":150},[149],[48,15829,15831,15834,15837,15840,15876],{"className":15830},[154],[48,15832],{"className":15833,"style":766},[158],[48,15835,4526],{"className":15836,"style":343},[163,171],[48,15838],{"className":15839,"style":180},[179],[48,15841,15843,15870,15873],{"className":15842},[184],[48,15844,15846],{"className":15845},[184],[48,15847,15849],{"className":15848},[163,5391],[48,15850,15852],{"className":15851},[5395],[48,15853,15855,15858,15867],{"className":15854},[5399],[48,15856],{"className":15857,"style":766},[158],[48,15859,15861],{"className":15860},[5406],[48,15862,15864],{"className":15863},[163],[48,15865,5413],{"className":15866},[184],[48,15868],{"className":15869},[5417],[48,15871],{"className":15872},[179,5421],[48,15874,90],{"className":15875},[184],[48,15877],{"className":15878,"style":180},[179],[48,15880,15882,15885],{"className":15881},[154],[48,15883],{"className":15884,"style":5434},[158],[48,15886,15888,15891],{"className":15887},[163],[48,15889,84],{"className":15890},[163,171],[48,15892,15894],{"className":15893},[292],[48,15895,15897,15917],{"className":15896},[203,204],[48,15898,15900,15914],{"className":15899},[208],[48,15901,15903],{"className":15902,"style":9307},[212],[48,15904,15905,15908],{"style":305},[48,15906],{"className":15907,"style":309},[220],[48,15909,15911],{"className":15910},[225,226,227,228],[48,15912,6221],{"className":15913},[163,171,228],[48,15915,269],{"className":15916},[268],[48,15918,15920],{"className":15919},[208],[48,15921,15923],{"className":15922,"style":2771},[212],[48,15924],{},[15291,15926,15770,15927,15799,15955],{},[48,15928,15930,15943],{"className":15929,"translate":52},[56],[48,15931,15933],{"className":15932},[60],[62,15934,15935],{"xmlns":64},[67,15936,15937,15941],{},[70,15938,15939],{},[73,15940,15],{},[142,15942,15],{"encoding":144},[48,15944,15946],{"className":15945,"ariaHidden":150},[149],[48,15947,15949,15952],{"className":15948},[154],[48,15950],{"className":15951,"style":578},[158],[48,15953,15],{"className":15954},[163,171],[48,15956,15958,15982],{"className":15957,"translate":52},[56],[48,15959,15961],{"className":15960},[60],[62,15962,15963],{"xmlns":64},[67,15964,15965,15979],{},[70,15966,15967,15969,15971,15973,15975,15977],{},[73,15968,1515],{},[78,15970,5291],{"mathvariant":75},[73,15972,101],{},[78,15974,81],{"stretchy":80},[73,15976,15],{},[78,15978,87],{"stretchy":80},[142,15980,15981],{"encoding":144},"D \\neq f(a)",[48,15983,15985,16036],{"className":15984,"ariaHidden":150},[149],[48,15986,15988,15991,15994,15997,16033],{"className":15987},[154],[48,15989],{"className":15990,"style":766},[158],[48,15992,1515],{"className":15993,"style":1680},[163,171],[48,15995],{"className":15996,"style":180},[179],[48,15998,16000,16027,16030],{"className":15999},[184],[48,16001,16003],{"className":16002},[184],[48,16004,16006],{"className":16005},[163,5391],[48,16007,16009],{"className":16008},[5395],[48,16010,16012,16015,16024],{"className":16011},[5399],[48,16013],{"className":16014,"style":766},[158],[48,16016,16018],{"className":16017},[5406],[48,16019,16021],{"className":16020},[163],[48,16022,5413],{"className":16023},[184],[48,16025],{"className":16026},[5417],[48,16028],{"className":16029},[179,5421],[48,16031,90],{"className":16032},[184],[48,16034],{"className":16035,"style":180},[179],[48,16037,16039,16042,16045,16048,16051],{"className":16038},[154],[48,16040],{"className":16041,"style":159},[158],[48,16043,101],{"className":16044,"style":235},[163,171],[48,16046,81],{"className":16047},[167],[48,16049,15],{"className":16050},[163,171],[48,16052,87],{"className":16053},[175],[15277,16055,16056,16087],{},[15291,16057,16058,16086],{},[48,16059,16061,16074],{"className":16060,"translate":52},[56],[48,16062,16064],{"className":16063},[60],[62,16065,16066],{"xmlns":64},[67,16067,16068,16072],{},[70,16069,16070],{},[73,16071,4526],{},[142,16073,4526],{"encoding":144},[48,16075,16077],{"className":16076,"ariaHidden":150},[149],[48,16078,16080,16083],{"className":16079},[154],[48,16081],{"className":16082,"style":2731},[158],[48,16084,4526],{"className":16085,"style":343},[163,171]," 无法被列表覆盖；矛盾",[15291,16088,16089,16117],{},[48,16090,16092,16105],{"className":16091,"translate":52},[56],[48,16093,16095],{"className":16094},[60],[62,16096,16097],{"xmlns":64},[67,16098,16099,16103],{},[70,16100,16101],{},[73,16102,1515],{},[142,16104,1515],{"encoding":144},[48,16106,16108],{"className":16107,"ariaHidden":150},[149],[48,16109,16111,16114],{"className":16110},[154],[48,16112],{"className":16113,"style":4878},[158],[48,16115,1515],{"className":16116,"style":1680},[163,171]," 无法被映射覆盖；矛盾",[11,16119,16120,16121,16172,16173,16260,16261,16345,16346,16390,16391,16442],{},"事实上，",[48,16122,16124,16145],{"className":16123,"translate":52},[56],[48,16125,16127],{"className":16126},[60],[62,16128,16129],{"xmlns":64},[67,16130,16131,16143],{},[70,16132,16133,16135,16137,16139,16141],{},[78,16134,81],{"stretchy":80},[1520,16136,2710],{},[78,16138,874],{"separator":150},[1520,16140,1522],{},[78,16142,87],{"stretchy":80},[142,16144,7058],{"encoding":144},[48,16146,16148],{"className":16147,"ariaHidden":150},[149],[48,16149,16151,16154,16157,16160,16163,16166,16169],{"className":16150},[154],[48,16152],{"className":16153,"style":159},[158],[48,16155,81],{"className":16156},[167],[48,16158,2710],{"className":16159},[163],[48,16161,874],{"className":16162},[944],[48,16164],{"className":16165,"style":282},[179],[48,16167,1522],{"className":16168},[163],[48,16170,87],{"className":16171},[175]," 中的实数可以与 ",[48,16174,16176,16202],{"className":16175,"translate":52},[56],[48,16177,16179],{"className":16178},[60],[62,16180,16181],{"xmlns":64},[67,16182,16183,16199],{},[70,16184,16185,16187,16189,16191,16193],{},[78,16186,5022],{"stretchy":80},[1520,16188,2710],{},[78,16190,874],{"separator":150},[1520,16192,1522],{},[1506,16194,16195,16197],{},[78,16196,5047],{"stretchy":80},[73,16198,5017],{"mathvariant":5016},[142,16200,16201],{"encoding":144},"\\{0,1\\}^{\\mathbb{N}}",[48,16203,16205],{"className":16204,"ariaHidden":150},[149],[48,16206,16208,16212,16215,16218,16221,16224,16227],{"className":16207},[154],[48,16209],{"className":16210,"style":16211},[158],"height:1.0952em;vertical-align:-0.25em;",[48,16213,5022],{"className":16214},[167],[48,16216,2710],{"className":16217},[163],[48,16219,874],{"className":16220},[944],[48,16222],{"className":16223,"style":282},[179],[48,16225,1522],{"className":16226},[163],[48,16228,16230,16233],{"className":16229},[175],[48,16231,5047],{"className":16232},[175],[48,16234,16236],{"className":16235},[292],[48,16237,16239],{"className":16238},[203],[48,16240,16242],{"className":16241},[208],[48,16243,16246],{"className":16244,"style":16245},[212],"height:0.8452em;",[48,16247,16248,16251],{"style":1667},[48,16249],{"className":16250,"style":309},[220],[48,16252,16254],{"className":16253},[225,226,227,228],[48,16255,16257],{"className":16256},[163,228],[48,16258,5017],{"className":16259},[163,5064,228],"（由 0 和 1 组成的无穷二进制序列集）近似对应，而 ",[48,16262,16264,16289],{"className":16263,"translate":52},[56],[48,16265,16267],{"className":16266},[60],[62,16268,16269],{"xmlns":64},[67,16270,16271,16287],{},[70,16272,16273,16275,16277,16279,16281],{},[78,16274,5022],{"stretchy":80},[1520,16276,2710],{},[78,16278,874],{"separator":150},[1520,16280,1522],{},[1506,16282,16283,16285],{},[78,16284,5047],{"stretchy":80},[73,16286,5017],{"mathvariant":5016},[142,16288,16201],{"encoding":144},[48,16290,16292],{"className":16291,"ariaHidden":150},[149],[48,16293,16295,16298,16301,16304,16307,16310,16313],{"className":16294},[154],[48,16296],{"className":16297,"style":16211},[158],[48,16299,5022],{"className":16300},[167],[48,16302,2710],{"className":16303},[163],[48,16305,874],{"className":16306},[944],[48,16308],{"className":16309,"style":282},[179],[48,16311,1522],{"className":16312},[163],[48,16314,16316,16319],{"className":16315},[175],[48,16317,5047],{"className":16318},[175],[48,16320,16322],{"className":16321},[292],[48,16323,16325],{"className":16324},[203],[48,16326,16328],{"className":16327},[208],[48,16329,16331],{"className":16330,"style":16245},[212],[48,16332,16333,16336],{"style":1667},[48,16334],{"className":16335,"style":309},[220],[48,16337,16339],{"className":16338},[225,226,227,228],[48,16340,16342],{"className":16341},[163,228],[48,16343,5017],{"className":16344},[163,5064,228]," 本质上就是 ",[48,16347,16349,16369],{"className":16348,"translate":52},[56],[48,16350,16352],{"className":16351},[60],[62,16353,16354],{"xmlns":64},[67,16355,16356,16366],{},[70,16357,16358,16360,16362,16364],{},[73,16359,13159],{"mathvariant":13158},[78,16361,81],{"stretchy":80},[73,16363,5017],{"mathvariant":5016},[78,16365,87],{"stretchy":80},[142,16367,16368],{"encoding":144},"\\mathcal{P}(\\mathbb{N})",[48,16370,16372],{"className":16371,"ariaHidden":150},[149],[48,16373,16375,16378,16381,16384,16387],{"className":16374},[154],[48,16376],{"className":16377,"style":159},[158],[48,16379,13159],{"className":16380,"style":13182},[163,13181],[48,16382,81],{"className":16383},[167],[48,16385,5017],{"className":16386},[163,5064],[48,16388,87],{"className":16389},[175],"（每个子集对应一个 0\u002F1 序列，标明某个位置是否属于该子集）。因此，「实数不可数」实际上是康托尔定理在 ",[48,16392,16394,16412],{"className":16393,"translate":52},[56],[48,16395,16397],{"className":16396},[60],[62,16398,16399],{"xmlns":64},[67,16400,16401,16409],{},[70,16402,16403,16405,16407],{},[73,16404,4891],{},[78,16406,90],{},[73,16408,5017],{"mathvariant":5016},[142,16410,16411],{"encoding":144},"A = \\mathbb{N}",[48,16413,16415,16433],{"className":16414,"ariaHidden":150},[149],[48,16416,16418,16421,16424,16427,16430],{"className":16417},[154],[48,16419],{"className":16420,"style":4878},[158],[48,16422,4891],{"className":16423},[163,171],[48,16425],{"className":16426,"style":180},[179],[48,16428,90],{"className":16429},[184],[48,16431],{"className":16432,"style":180},[179],[48,16434,16436,16439],{"className":16435},[154],[48,16437],{"className":16438,"style":5060},[158],[48,16440,5017],{"className":16441},[163,5064]," 这一特例下的具体体现。",[11,16444,16445],{},"虽然我们无法在计算机中表示无穷集，但可以用有限集验证证明的逻辑骨架：",[12625,16447,16449],{"className":12627,"code":16448,"language":12629,"meta":4787,"style":4787},"def cantor_diagonal_demo(A):\n    # 构造一个「任意」的映射 f: A -> P(A) 用于演示\n    f = {}\n    for i in A:\n        f[i] = frozenset(j for j in A if (i + j) % 2 == 0)\n\n    print(\"集合 A =\", set(A))\n    for i in A:\n        print(f\"  f({i}) = {set(f[i])}\")\n\n    # 对角线构造：D = { i in A : i not in f(i) }\n    D = frozenset(i for i in A if i not in f[i])\n    print(f\"对角线构造出的集合 D = {set(D)}\")\n\n    matches = [i for i in A if f[i] == D]\n    print(f\"是否存在 i 使得 f(i) = D？ -> {matches if matches else '不存在（矛盾已建立）'}\")\n\ncantor_diagonal_demo([0, 1, 2, 3])\n",[2561,16450,16451,16461,16466,16476,16488,16536,16540,16558,16568,16602,16606,16611,16644,16666,16670,16699,16732,16736],{"__ignoreMap":4787},[48,16452,16453,16455,16458],{"class":12634,"line":12635},[48,16454,12639],{"class":12638},[48,16456,16457],{"class":12642}," cantor_diagonal_demo",[48,16459,16460],{"class":12646},"(A):\n",[48,16462,16463],{"class":12634,"line":4788},[48,16464,16465],{"class":12753},"    # 构造一个「任意」的映射 f: A -> P(A) 用于演示\n",[48,16467,16468,16471,16473],{"class":12634,"line":12665},[48,16469,16470],{"class":12646},"    f ",[48,16472,90],{"class":12638},[48,16474,16475],{"class":12646}," {}\n",[48,16477,16478,16480,16483,16485],{"class":12634,"line":12671},[48,16479,12696],{"class":12638},[48,16481,16482],{"class":12646}," i ",[48,16484,12702],{"class":12638},[48,16486,16487],{"class":12646}," A:\n",[48,16489,16490,16493,16495,16498,16501,16504,16507,16509,16512,16515,16518,16520,16523,16525,16528,16531,16534],{"class":12634,"line":12677},[48,16491,16492],{"class":12646},"        f[i] ",[48,16494,90],{"class":12638},[48,16496,16497],{"class":12652}," frozenset",[48,16499,16500],{"class":12646},"(j ",[48,16502,16503],{"class":12638},"for",[48,16505,16506],{"class":12646}," j ",[48,16508,12702],{"class":12638},[48,16510,16511],{"class":12646}," A ",[48,16513,16514],{"class":12638},"if",[48,16516,16517],{"class":12646}," (i ",[48,16519,2947],{"class":12638},[48,16521,16522],{"class":12646}," j) ",[48,16524,12776],{"class":12638},[48,16526,16527],{"class":12652}," 2",[48,16529,16530],{"class":12638}," ==",[48,16532,16533],{"class":12652}," 0",[48,16535,12736],{"class":12646},[48,16537,16538],{"class":12634,"line":12682},[48,16539,12818],{"emptyLinePlaceholder":4800},[48,16541,16542,16545,16547,16550,16552,16555],{"class":12634,"line":12693},[48,16543,16544],{"class":12652},"    print",[48,16546,81],{"class":12646},[48,16548,16549],{"class":12661},"\"集合 A =\"",[48,16551,12843],{"class":12646},[48,16553,16554],{"class":12652},"set",[48,16556,16557],{"class":12646},"(A))\n",[48,16559,16560,16562,16564,16566],{"class":12634,"line":12711},[48,16561,12696],{"class":12638},[48,16563,16482],{"class":12646},[48,16565,12702],{"class":12638},[48,16567,16487],{"class":12646},[48,16569,16570,16573,16575,16577,16580,16582,16584,16586,16589,16592,16595,16597,16600],{"class":12634,"line":12739},[48,16571,16572],{"class":12652},"        print",[48,16574,81],{"class":12646},[48,16576,101],{"class":12638},[48,16578,16579],{"class":12661},"\"  f(",[48,16581,5022],{"class":12652},[48,16583,2631],{"class":12646},[48,16585,5047],{"class":12652},[48,16587,16588],{"class":12661},") = ",[48,16590,16591],{"class":12652},"{set",[48,16593,16594],{"class":12646},"(f[i])",[48,16596,5047],{"class":12652},[48,16598,16599],{"class":12661},"\"",[48,16601,12736],{"class":12646},[48,16603,16604],{"class":12634,"line":12757},[48,16605,12818],{"emptyLinePlaceholder":4800},[48,16607,16608],{"class":12634,"line":12785},[48,16609,16610],{"class":12753},"    # 对角线构造：D = { i in A : i not in f(i) }\n",[48,16612,16613,16616,16618,16620,16623,16625,16627,16629,16631,16633,16635,16638,16641],{"class":12634,"line":12797},[48,16614,16615],{"class":12646},"    D ",[48,16617,90],{"class":12638},[48,16619,16497],{"class":12652},[48,16621,16622],{"class":12646},"(i ",[48,16624,16503],{"class":12638},[48,16626,16482],{"class":12646},[48,16628,12702],{"class":12638},[48,16630,16511],{"class":12646},[48,16632,16514],{"class":12638},[48,16634,16482],{"class":12646},[48,16636,16637],{"class":12638},"not",[48,16639,16640],{"class":12638}," in",[48,16642,16643],{"class":12646}," f[i])\n",[48,16645,16646,16648,16650,16652,16655,16657,16660,16662,16664],{"class":12634,"line":12815},[48,16647,16544],{"class":12652},[48,16649,81],{"class":12646},[48,16651,101],{"class":12638},[48,16653,16654],{"class":12661},"\"对角线构造出的集合 D = ",[48,16656,16591],{"class":12652},[48,16658,16659],{"class":12646},"(D)",[48,16661,5047],{"class":12652},[48,16663,16599],{"class":12661},[48,16665,12736],{"class":12646},[48,16667,16668],{"class":12634,"line":12821},[48,16669,12818],{"emptyLinePlaceholder":4800},[48,16671,16672,16675,16677,16680,16682,16684,16686,16688,16690,16693,16696],{"class":12634,"line":12826},[48,16673,16674],{"class":12646},"    matches ",[48,16676,90],{"class":12638},[48,16678,16679],{"class":12646}," [i ",[48,16681,16503],{"class":12638},[48,16683,16482],{"class":12646},[48,16685,12702],{"class":12638},[48,16687,16511],{"class":12646},[48,16689,16514],{"class":12638},[48,16691,16692],{"class":12646}," f[i] ",[48,16694,16695],{"class":12638},"==",[48,16697,16698],{"class":12646}," D]\n",[48,16700,16701,16703,16705,16707,16710,16712,16715,16717,16720,16723,16726,16728,16730],{"class":12634,"line":12837},[48,16702,16544],{"class":12652},[48,16704,81],{"class":12646},[48,16706,101],{"class":12638},[48,16708,16709],{"class":12661},"\"是否存在 i 使得 f(i) = D？ -> ",[48,16711,5022],{"class":12652},[48,16713,16714],{"class":12646},"matches ",[48,16716,16514],{"class":12638},[48,16718,16719],{"class":12646}," matches ",[48,16721,16722],{"class":12638},"else",[48,16724,16725],{"class":12661}," '不存在（矛盾已建立）'",[48,16727,5047],{"class":12652},[48,16729,16599],{"class":12661},[48,16731,12736],{"class":12646},[48,16733,16734],{"class":12634,"line":12862},[48,16735,12818],{"emptyLinePlaceholder":4800},[48,16737,16738,16741,16743,16745,16747,16749,16751,16753,16755],{"class":12634,"line":12885},[48,16739,16740],{"class":12646},"cantor_diagonal_demo([",[48,16742,2710],{"class":12652},[48,16744,12843],{"class":12646},[48,16746,1522],{"class":12652},[48,16748,12843],{"class":12646},[48,16750,2868],{"class":12652},[48,16752,12843],{"class":12646},[48,16754,5037],{"class":12652},[48,16756,16757],{"class":12646},"])\n",[11,16759,12934],{},[12625,16761,16764],{"className":16762,"code":16763,"language":4646},[12938],"集合 A = {0, 1, 2, 3}\n假定的映射 f: A -> P(A)：\n  f(0) = {0, 2}\n  f(1) = {1, 3}\n  f(2) = {0, 2}\n  f(3) = {1, 3}\n\n对角线构造出的集合 D = { i : i ∉ f(i) } = set()\n是否存在 i 使得 f(i) = D？ -> 不存在（矛盾已建立）\n\n逐一验证 f(i) ≠ D：\n  i=0：i∉D 但 i∈f(0)，故 D ≠ f(0)\n  i=1：i∉D 但 i∈f(1)，故 D ≠ f(1)\n  i=2：i∉D 但 i∈f(2)，故 D ≠ f(2)\n  i=3：i∉D 但 i∈f(3)，故 D ≠ f(3)\n",[2561,16765,16763],{"__ignoreMap":4787},[11,16767,16768,16769,16797,16798,16826,16827,16855],{},"这里构造出的 ",[48,16770,16772,16785],{"className":16771,"translate":52},[56],[48,16773,16775],{"className":16774},[60],[62,16776,16777],{"xmlns":64},[67,16778,16779,16783],{},[70,16780,16781],{},[73,16782,1515],{},[142,16784,1515],{"encoding":144},[48,16786,16788],{"className":16787,"ariaHidden":150},[149],[48,16789,16791,16794],{"className":16790},[154],[48,16792],{"className":16793,"style":4878},[158],[48,16795,1515],{"className":16796,"style":1680},[163,171]," 恰好是空集，而这个特定的映射 ",[48,16799,16801,16814],{"className":16800,"translate":52},[56],[48,16802,16804],{"className":16803},[60],[62,16805,16806],{"xmlns":64},[67,16807,16808,16812],{},[70,16809,16810],{},[73,16811,101],{},[142,16813,101],{"encoding":144},[48,16815,16817],{"className":16816,"ariaHidden":150},[149],[48,16818,16820,16823],{"className":16819},[154],[48,16821],{"className":16822,"style":766},[158],[48,16824,101],{"className":16825,"style":235},[163,171]," 恰好没有把任何元素映射到空集——这是对角线论证「刻意构造逃逸者」这一核心思想的直接体现。当然，这只是一个具体的映射示例；真正的证明是对每一种可能的映射 ",[48,16828,16830,16843],{"className":16829,"translate":52},[56],[48,16831,16833],{"className":16832},[60],[62,16834,16835],{"xmlns":64},[67,16836,16837,16841],{},[70,16838,16839],{},[73,16840,101],{},[142,16842,101],{"encoding":144},[48,16844,16846],{"className":16845,"ariaHidden":150},[149],[48,16847,16849,16852],{"className":16848},[154],[48,16850],{"className":16851,"style":766},[158],[48,16853,101],{"className":16854,"style":235},[163,171]," 都成立的普适论证。",[35,16857,16858],{"id":16858},"停机问题",[11,16860,16861],{},"对角线论证最令人惊叹的应用之一，出现在阿兰·图灵 1936 年对停机问题不可判定性的证明中。这一结果直接奠定了整个可计算性理论领域的基础。",[11,16863,16864,16865,16893,16894,16922,16923,16951,16952,16980],{},"停机问题。给定一个程序 ",[48,16866,16868,16881],{"className":16867,"translate":52},[56],[48,16869,16871],{"className":16870},[60],[62,16872,16873],{"xmlns":64},[67,16874,16875,16879],{},[70,16876,16877],{},[73,16878,13159],{},[142,16880,13159],{"encoding":144},[48,16882,16884],{"className":16883,"ariaHidden":150},[149],[48,16885,16887,16890],{"className":16886},[154],[48,16888],{"className":16889,"style":4878},[158],[48,16891,13159],{"className":16892,"style":242},[163,171]," 和一个输入 ",[48,16895,16897,16910],{"className":16896,"translate":52},[56],[48,16898,16900],{"className":16899},[60],[62,16901,16902],{"xmlns":64},[67,16903,16904,16908],{},[70,16905,16906],{},[73,16907,84],{},[142,16909,84],{"encoding":144},[48,16911,16913],{"className":16912,"ariaHidden":150},[149],[48,16914,16916,16919],{"className":16915},[154],[48,16917],{"className":16918,"style":578},[158],[48,16920,84],{"className":16921},[163,171],"，是否存在一个通用算法，能够判断 ",[48,16924,16926,16939],{"className":16925,"translate":52},[56],[48,16927,16929],{"className":16928},[60],[62,16930,16931],{"xmlns":64},[67,16932,16933,16937],{},[70,16934,16935],{},[73,16936,13159],{},[142,16938,13159],{"encoding":144},[48,16940,16942],{"className":16941,"ariaHidden":150},[149],[48,16943,16945,16948],{"className":16944},[154],[48,16946],{"className":16947,"style":4878},[158],[48,16949,13159],{"className":16950,"style":242},[163,171]," 在输入 ",[48,16953,16955,16968],{"className":16954,"translate":52},[56],[48,16956,16958],{"className":16957},[60],[62,16959,16960],{"xmlns":64},[67,16961,16962,16966],{},[70,16963,16964],{},[73,16965,84],{},[142,16967,84],{"encoding":144},[48,16969,16971],{"className":16970,"ariaHidden":150},[149],[48,16972,16974,16977],{"className":16973},[154],[48,16975],{"className":16976,"style":578},[158],[48,16978,84],{"className":16979},[163,171]," 上运行时，究竟会停机（并返回结果）还是永不停机（陷入死循环）？",[11,16982,16983],{},"图灵证明：这样的通用算法不存在（这是对角线论证的一个变体）。",[11,16985,16986],{},"假设存在这样一个「通用停机判定器」，我们把它形式化为一个函数：",[48,16988,16990],{"className":16989,"translate":52},[51],[48,16991,16993,17082],{"className":16992,"translate":52},[56],[48,16994,16996],{"className":16995},[60],[62,16997,16998],{"xmlns":64,"display":65},[67,16999,17000,17079],{},[70,17001,17002,17005,17007,17009,17011,17013,17015,17017],{},[4598,17003,17004],{},"halts",[78,17006,81],{"stretchy":80},[73,17008,13159],{},[78,17010,874],{"separator":150},[73,17012,84],{},[78,17014,87],{"stretchy":80},[78,17016,90],{},[70,17018,17019,17021],{},[78,17020,5022],{"fence":150},[6463,17022,17023,17051],{"rowspacing":6465,"columnalign":6466,"columnspacing":6467},[6469,17024,17025,17032],{},[6472,17026,17027],{},[6475,17028,17029],{"scriptlevel":2710,"displaystyle":80},[4598,17030,17031],{},"真",[6472,17033,17034],{},[6475,17035,17036],{"scriptlevel":2710,"displaystyle":80},[70,17037,17038,17040,17042,17044,17046,17048],{},[4598,17039,9232],{},[73,17041,13159],{},[78,17043,81],{"stretchy":80},[73,17045,84],{},[78,17047,87],{"stretchy":80},[4598,17049,17050],{}," 停机",[6469,17052,17053,17060],{},[6472,17054,17055],{},[6475,17056,17057],{"scriptlevel":2710,"displaystyle":80},[4598,17058,17059],{},"假",[6472,17061,17062],{},[6475,17063,17064],{"scriptlevel":2710,"displaystyle":80},[70,17065,17066,17068,17070,17072,17074,17076],{},[4598,17067,9232],{},[73,17069,13159],{},[78,17071,81],{"stretchy":80},[73,17073,84],{},[78,17075,87],{"stretchy":80},[4598,17077,17078],{}," 不停机",[142,17080,17081],{"encoding":144},"\\text{halts}(P, x) = \\begin{cases} \\text{真} & \\text{若 } P(x) \\text{ 停机} \\\\ \\text{假} & \\text{若 } P(x) \\text{ 不停机} \\end{cases}",[48,17083,17085,17124],{"className":17084,"ariaHidden":150},[149],[48,17086,17088,17091,17097,17100,17103,17106,17109,17112,17115,17118,17121],{"className":17087},[154],[48,17089],{"className":17090,"style":159},[158],[48,17092,17094],{"className":17093},[163,4646],[48,17095,17004],{"className":17096},[163],[48,17098,81],{"className":17099},[167],[48,17101,13159],{"className":17102,"style":242},[163,171],[48,17104,874],{"className":17105},[944],[48,17107],{"className":17108,"style":282},[179],[48,17110,84],{"className":17111},[163,171],[48,17113,87],{"className":17114},[175],[48,17116],{"className":17117,"style":180},[179],[48,17119,90],{"className":17120},[184],[48,17122],{"className":17123,"style":180},[179],[48,17125,17127,17130],{"className":17126},[154],[48,17128],{"className":17129,"style":6571},[158],[48,17131,17133,17139,17300],{"className":17132},[3012],[48,17134,17136],{"className":17135,"style":6579},[167,6578],[48,17137,5022],{"className":17138},[6583,6584],[48,17140,17142],{"className":17141},[163],[48,17143,17145,17196,17199],{"className":17144},[6463],[48,17146,17148],{"className":17147},[6594],[48,17149,17151,17188],{"className":17150},[203,204],[48,17152,17154,17185],{"className":17153},[208],[48,17155,17157,17171],{"className":17156,"style":6604},[212],[48,17158,17159,17162],{"style":6607},[48,17160],{"className":17161,"style":6611},[220],[48,17163,17165],{"className":17164},[163],[48,17166,17168],{"className":17167},[163,4646],[48,17169,17031],{"className":17170},[163,6708],[48,17172,17173,17176],{"style":6624},[48,17174],{"className":17175,"style":6611},[220],[48,17177,17179],{"className":17178},[163],[48,17180,17182],{"className":17181},[163,4646],[48,17183,17059],{"className":17184},[163,6708],[48,17186,269],{"className":17187},[268],[48,17189,17191],{"className":17190},[208],[48,17192,17194],{"className":17193,"style":6667},[212],[48,17195],{},[48,17197],{"className":17198,"style":6674},[6673],[48,17200,17202],{"className":17201},[6594],[48,17203,17205,17292],{"className":17204},[203,204],[48,17206,17208,17289],{"className":17207},[208],[48,17209,17211,17250],{"className":17210,"style":6604},[212],[48,17212,17213,17216],{"style":6607},[48,17214],{"className":17215,"style":6611},[220],[48,17217,17219,17228,17231,17234,17237,17240],{"className":17218},[163],[48,17220,17222,17225],{"className":17221},[163,4646],[48,17223,9434],{"className":17224},[163,6708],[48,17226,6704],{"className":17227},[163],[48,17229,13159],{"className":17230,"style":242},[163,171],[48,17232,81],{"className":17233},[167],[48,17235,84],{"className":17236},[163,171],[48,17238,87],{"className":17239},[175],[48,17241,17243,17246],{"className":17242},[163,4646],[48,17244,6704],{"className":17245},[163],[48,17247,17249],{"className":17248},[163,6708],"停机",[48,17251,17252,17255],{"style":6624},[48,17253],{"className":17254,"style":6611},[220],[48,17256,17258,17267,17270,17273,17276,17279],{"className":17257},[163],[48,17259,17261,17264],{"className":17260},[163,4646],[48,17262,9434],{"className":17263},[163,6708],[48,17265,6704],{"className":17266},[163],[48,17268,13159],{"className":17269,"style":242},[163,171],[48,17271,81],{"className":17272},[167],[48,17274,84],{"className":17275},[163,171],[48,17277,87],{"className":17278},[175],[48,17280,17282,17285],{"className":17281},[163,4646],[48,17283,6704],{"className":17284},[163],[48,17286,17288],{"className":17287},[163,6708],"不停机",[48,17290,269],{"className":17291},[268],[48,17293,17295],{"className":17294},[208],[48,17296,17298],{"className":17297,"style":6667},[212],[48,17299],{},[48,17301],{"className":17302},[175,6745],[11,17304,17305],{},"这个函数本身必须在有限时间内总是给出正确回答（这就是「通用」的含义）。",[11,17307,17308,17309,17337],{},"构造一个对角线程序 ",[48,17310,17312,17325],{"className":17311,"translate":52},[56],[48,17313,17315],{"className":17314},[60],[62,17316,17317],{"xmlns":64},[67,17318,17319,17323],{},[70,17320,17321],{},[73,17322,1515],{},[142,17324,1515],{"encoding":144},[48,17326,17328],{"className":17327,"ariaHidden":150},[149],[48,17329,17331,17334],{"className":17330},[154],[48,17332],{"className":17333,"style":4878},[158],[48,17335,1515],{"className":17336,"style":1680},[163,171],"，其行为定义如下：",[48,17339,17341],{"className":17340,"translate":52},[51],[48,17342,17344,17435],{"className":17343,"translate":52},[56],[48,17345,17347],{"className":17346},[60],[62,17348,17349],{"xmlns":64,"display":65},[67,17350,17351,17432],{},[70,17352,17353,17355,17357,17359,17361,17363],{},[73,17354,1515],{},[78,17356,81],{"stretchy":80},[73,17358,13159],{},[78,17360,87],{"stretchy":80},[78,17362,90],{},[70,17364,17365,17367],{},[78,17366,5022],{"fence":150},[6463,17368,17369,17401],{"rowspacing":6465,"columnalign":6466,"columnspacing":6467},[6469,17370,17371,17378],{},[6472,17372,17373],{},[6475,17374,17375],{"scriptlevel":2710,"displaystyle":80},[4598,17376,17377],{},"死循环",[6472,17379,17380],{},[6475,17381,17382],{"scriptlevel":2710,"displaystyle":80},[70,17383,17384,17387,17389,17391,17393,17395,17397,17399],{},[4598,17385,17386],{},"若 halts",[78,17388,81],{"stretchy":80},[73,17390,13159],{},[78,17392,874],{"separator":150},[73,17394,13159],{},[78,17396,87],{"stretchy":80},[78,17398,90],{},[4598,17400,17031],{},[6469,17402,17403,17410],{},[6472,17404,17405],{},[6475,17406,17407],{"scriptlevel":2710,"displaystyle":80},[4598,17408,17409],{},"立即停机",[6472,17411,17412],{},[6475,17413,17414],{"scriptlevel":2710,"displaystyle":80},[70,17415,17416,17418,17420,17422,17424,17426,17428,17430],{},[4598,17417,17386],{},[78,17419,81],{"stretchy":80},[73,17421,13159],{},[78,17423,874],{"separator":150},[73,17425,13159],{},[78,17427,87],{"stretchy":80},[78,17429,90],{},[4598,17431,17059],{},[142,17433,17434],{"encoding":144},"D(P) = \\begin{cases} \\text{死循环} & \\text{若 } \\text{halts}(P, P) = \\text{真} \\\\ \\text{立即停机} & \\text{若 } \\text{halts}(P, P) = \\text{假} \\end{cases}",[48,17436,17438,17465],{"className":17437,"ariaHidden":150},[149],[48,17439,17441,17444,17447,17450,17453,17456,17459,17462],{"className":17440},[154],[48,17442],{"className":17443,"style":159},[158],[48,17445,1515],{"className":17446,"style":1680},[163,171],[48,17448,81],{"className":17449},[167],[48,17451,13159],{"className":17452,"style":242},[163,171],[48,17454,87],{"className":17455},[175],[48,17457],{"className":17458,"style":180},[179],[48,17460,90],{"className":17461},[184],[48,17463],{"className":17464,"style":180},[179],[48,17466,17468,17471],{"className":17467},[154],[48,17469],{"className":17470,"style":6571},[158],[48,17472,17474,17480,17675],{"className":17473},[3012],[48,17475,17477],{"className":17476,"style":6579},[167,6578],[48,17478,5022],{"className":17479},[6583,6584],[48,17481,17483],{"className":17482},[163],[48,17484,17486,17537,17540],{"className":17485},[6463],[48,17487,17489],{"className":17488},[6594],[48,17490,17492,17529],{"className":17491},[203,204],[48,17493,17495,17526],{"className":17494},[208],[48,17496,17498,17512],{"className":17497,"style":6604},[212],[48,17499,17500,17503],{"style":6607},[48,17501],{"className":17502,"style":6611},[220],[48,17504,17506],{"className":17505},[163],[48,17507,17509],{"className":17508},[163,4646],[48,17510,17377],{"className":17511},[163,6708],[48,17513,17514,17517],{"style":6624},[48,17515],{"className":17516,"style":6611},[220],[48,17518,17520],{"className":17519},[163],[48,17521,17523],{"className":17522},[163,4646],[48,17524,17409],{"className":17525},[163,6708],[48,17527,269],{"className":17528},[268],[48,17530,17532],{"className":17531},[208],[48,17533,17535],{"className":17534,"style":6667},[212],[48,17536],{},[48,17538],{"className":17539,"style":6674},[6673],[48,17541,17543],{"className":17542},[6594],[48,17544,17546,17667],{"className":17545},[203,204],[48,17547,17549,17664],{"className":17548},[208],[48,17550,17552,17608],{"className":17551,"style":6604},[212],[48,17553,17554,17557],{"style":6607},[48,17555],{"className":17556,"style":6611},[220],[48,17558,17560,17569,17575,17578,17581,17584,17587,17590,17593,17596,17599,17602],{"className":17559},[163],[48,17561,17563,17566],{"className":17562},[163,4646],[48,17564,9434],{"className":17565},[163,6708],[48,17567,6704],{"className":17568},[163],[48,17570,17572],{"className":17571},[163,4646],[48,17573,17004],{"className":17574},[163],[48,17576,81],{"className":17577},[167],[48,17579,13159],{"className":17580,"style":242},[163,171],[48,17582,874],{"className":17583},[944],[48,17585],{"className":17586,"style":282},[179],[48,17588,13159],{"className":17589,"style":242},[163,171],[48,17591,87],{"className":17592},[175],[48,17594],{"className":17595,"style":180},[179],[48,17597,90],{"className":17598},[184],[48,17600],{"className":17601,"style":180},[179],[48,17603,17605],{"className":17604},[163,4646],[48,17606,17031],{"className":17607},[163,6708],[48,17609,17610,17613],{"style":6624},[48,17611],{"className":17612,"style":6611},[220],[48,17614,17616,17625,17631,17634,17637,17640,17643,17646,17649,17652,17655,17658],{"className":17615},[163],[48,17617,17619,17622],{"className":17618},[163,4646],[48,17620,9434],{"className":17621},[163,6708],[48,17623,6704],{"className":17624},[163],[48,17626,17628],{"className":17627},[163,4646],[48,17629,17004],{"className":17630},[163],[48,17632,81],{"className":17633},[167],[48,17635,13159],{"className":17636,"style":242},[163,171],[48,17638,874],{"className":17639},[944],[48,17641],{"className":17642,"style":282},[179],[48,17644,13159],{"className":17645,"style":242},[163,171],[48,17647,87],{"className":17648},[175],[48,17650],{"className":17651,"style":180},[179],[48,17653,90],{"className":17654},[184],[48,17656],{"className":17657,"style":180},[179],[48,17659,17661],{"className":17660},[163,4646],[48,17662,17059],{"className":17663},[163,6708],[48,17665,269],{"className":17666},[268],[48,17668,17670],{"className":17669},[208],[48,17671,17673],{"className":17672,"style":6667},[212],[48,17674],{},[48,17676],{"className":17677},[175,6745],[11,17679,17680,17681,17741,17742,8847,17770,17798],{},"请注意这里的要害操作：把程序本身作为自己的输入喂给自己（",[48,17682,17684,17708],{"className":17683,"translate":52},[56],[48,17685,17687],{"className":17686},[60],[62,17688,17689],{"xmlns":64},[67,17690,17691,17705],{},[70,17692,17693,17695,17697,17699,17701,17703],{},[4598,17694,17004],{},[78,17696,81],{"stretchy":80},[73,17698,13159],{},[78,17700,874],{"separator":150},[73,17702,13159],{},[78,17704,87],{"stretchy":80},[142,17706,17707],{"encoding":144},"\\text{halts}(P, P)",[48,17709,17711],{"className":17710,"ariaHidden":150},[149],[48,17712,17714,17717,17723,17726,17729,17732,17735,17738],{"className":17713},[154],[48,17715],{"className":17716,"style":159},[158],[48,17718,17720],{"className":17719},[163,4646],[48,17721,17004],{"className":17722},[163],[48,17724,81],{"className":17725},[167],[48,17727,13159],{"className":17728,"style":242},[163,171],[48,17730,874],{"className":17731},[944],[48,17733],{"className":17734,"style":282},[179],[48,17736,13159],{"className":17737,"style":242},[163,171],[48,17739,87],{"className":17740},[175],"）。这正是对角线论证中「取第 ",[48,17743,17745,17758],{"className":17744,"translate":52},[56],[48,17746,17748],{"className":17747},[60],[62,17749,17750],{"xmlns":64},[67,17751,17752,17756],{},[70,17753,17754],{},[73,17755,6221],{},[142,17757,6221],{"encoding":144},[48,17759,17761],{"className":17760,"ariaHidden":150},[149],[48,17762,17764,17767],{"className":17763},[154],[48,17765],{"className":17766,"style":578},[158],[48,17768,6221],{"className":17769},[163,171],[48,17771,17773,17786],{"className":17772,"translate":52},[56],[48,17774,17776],{"className":17775},[60],[62,17777,17778],{"xmlns":64},[67,17779,17780,17784],{},[70,17781,17782],{},[73,17783,6221],{},[142,17785,6221],{"encoding":144},[48,17787,17789],{"className":17788,"ariaHidden":150},[149],[48,17790,17792,17795],{"className":17791},[154],[48,17793],{"className":17794,"style":578},[158],[48,17796,6221],{"className":17797},[163,171]," 位」的直接类比——让一个程序「审视」自身的自指动作。",[11,17800,17801,17802,1906],{},"导出矛盾。现在考察具体的执行 ",[48,17803,17805,17825],{"className":17804,"translate":52},[56],[48,17806,17808],{"className":17807},[60],[62,17809,17810],{"xmlns":64},[67,17811,17812,17822],{},[70,17813,17814,17816,17818,17820],{},[73,17815,1515],{},[78,17817,81],{"stretchy":80},[73,17819,1515],{},[78,17821,87],{"stretchy":80},[142,17823,17824],{"encoding":144},"D(D)",[48,17826,17828],{"className":17827,"ariaHidden":150},[149],[48,17829,17831,17834,17837,17840,17843],{"className":17830},[154],[48,17832],{"className":17833,"style":159},[158],[48,17835,1515],{"className":17836,"style":1680},[163,171],[48,17838,81],{"className":17839},[167],[48,17841,1515],{"className":17842,"style":1680},[163,171],[48,17844,87],{"className":17845},[175],[993,17847,17848,18052],{},[996,17849,14223,17850,17935,17936,17979,17980,14303,18008,18051],{},[48,17851,17853,17881],{"className":17852,"translate":52},[56],[48,17854,17856],{"className":17855},[60],[62,17857,17858],{"xmlns":64},[67,17859,17860,17878],{},[70,17861,17862,17864,17866,17868,17870,17872,17874,17876],{},[4598,17863,17004],{},[78,17865,81],{"stretchy":80},[73,17867,1515],{},[78,17869,874],{"separator":150},[73,17871,1515],{},[78,17873,87],{"stretchy":80},[78,17875,90],{},[4598,17877,17031],{},[142,17879,17880],{"encoding":144},"\\text{halts}(D, D) = \\text{真}",[48,17882,17884,17923],{"className":17883,"ariaHidden":150},[149],[48,17885,17887,17890,17896,17899,17902,17905,17908,17911,17914,17917,17920],{"className":17886},[154],[48,17888],{"className":17889,"style":159},[158],[48,17891,17893],{"className":17892},[163,4646],[48,17894,17004],{"className":17895},[163],[48,17897,81],{"className":17898},[167],[48,17900,1515],{"className":17901,"style":1680},[163,171],[48,17903,874],{"className":17904},[944],[48,17906],{"className":17907,"style":282},[179],[48,17909,1515],{"className":17910,"style":1680},[163,171],[48,17912,87],{"className":17913},[175],[48,17915],{"className":17916,"style":180},[179],[48,17918,90],{"className":17919},[184],[48,17921],{"className":17922,"style":180},[179],[48,17924,17926,17929],{"className":17925},[154],[48,17927],{"className":17928,"style":4878},[158],[48,17930,17932],{"className":17931},[163,4646],[48,17933,17031],{"className":17934},[163,6708],"（即判定「",[48,17937,17939,17958],{"className":17938,"translate":52},[56],[48,17940,17942],{"className":17941},[60],[62,17943,17944],{"xmlns":64},[67,17945,17946,17956],{},[70,17947,17948,17950,17952,17954],{},[73,17949,1515],{},[78,17951,81],{"stretchy":80},[73,17953,1515],{},[78,17955,87],{"stretchy":80},[142,17957,17824],{"encoding":144},[48,17959,17961],{"className":17960,"ariaHidden":150},[149],[48,17962,17964,17967,17970,17973,17976],{"className":17963},[154],[48,17965],{"className":17966,"style":159},[158],[48,17968,1515],{"className":17969,"style":1680},[163,171],[48,17971,81],{"className":17972},[167],[48,17974,1515],{"className":17975,"style":1680},[163,171],[48,17977,87],{"className":17978},[175]," 停机」），那么根据 ",[48,17981,17983,17996],{"className":17982,"translate":52},[56],[48,17984,17986],{"className":17985},[60],[62,17987,17988],{"xmlns":64},[67,17989,17990,17994],{},[70,17991,17992],{},[73,17993,1515],{},[142,17995,1515],{"encoding":144},[48,17997,17999],{"className":17998,"ariaHidden":150},[149],[48,18000,18002,18005],{"className":18001},[154],[48,18003],{"className":18004,"style":4878},[158],[48,18006,1515],{"className":18007,"style":1680},[163,171],[48,18009,18011,18030],{"className":18010,"translate":52},[56],[48,18012,18014],{"className":18013},[60],[62,18015,18016],{"xmlns":64},[67,18017,18018,18028],{},[70,18019,18020,18022,18024,18026],{},[73,18021,1515],{},[78,18023,81],{"stretchy":80},[73,18025,1515],{},[78,18027,87],{"stretchy":80},[142,18029,17824],{"encoding":144},[48,18031,18033],{"className":18032,"ariaHidden":150},[149],[48,18034,18036,18039,18042,18045,18048],{"className":18035},[154],[48,18037],{"className":18038,"style":159},[158],[48,18040,1515],{"className":18041,"style":1680},[163,171],[48,18043,81],{"className":18044},[167],[48,18046,1515],{"className":18047,"style":1680},[163,171],[48,18049,87],{"className":18050},[175]," 应该死循环。这与「它停机」矛盾。",[996,18053,14223,18054,17935,18139,18182,18183,14303,18211,18254],{},[48,18055,18057,18085],{"className":18056,"translate":52},[56],[48,18058,18060],{"className":18059},[60],[62,18061,18062],{"xmlns":64},[67,18063,18064,18082],{},[70,18065,18066,18068,18070,18072,18074,18076,18078,18080],{},[4598,18067,17004],{},[78,18069,81],{"stretchy":80},[73,18071,1515],{},[78,18073,874],{"separator":150},[73,18075,1515],{},[78,18077,87],{"stretchy":80},[78,18079,90],{},[4598,18081,17059],{},[142,18083,18084],{"encoding":144},"\\text{halts}(D, D) = \\text{假}",[48,18086,18088,18127],{"className":18087,"ariaHidden":150},[149],[48,18089,18091,18094,18100,18103,18106,18109,18112,18115,18118,18121,18124],{"className":18090},[154],[48,18092],{"className":18093,"style":159},[158],[48,18095,18097],{"className":18096},[163,4646],[48,18098,17004],{"className":18099},[163],[48,18101,81],{"className":18102},[167],[48,18104,1515],{"className":18105,"style":1680},[163,171],[48,18107,874],{"className":18108},[944],[48,18110],{"className":18111,"style":282},[179],[48,18113,1515],{"className":18114,"style":1680},[163,171],[48,18116,87],{"className":18117},[175],[48,18119],{"className":18120,"style":180},[179],[48,18122,90],{"className":18123},[184],[48,18125],{"className":18126,"style":180},[179],[48,18128,18130,18133],{"className":18129},[154],[48,18131],{"className":18132,"style":4878},[158],[48,18134,18136],{"className":18135},[163,4646],[48,18137,17059],{"className":18138},[163,6708],[48,18140,18142,18161],{"className":18141,"translate":52},[56],[48,18143,18145],{"className":18144},[60],[62,18146,18147],{"xmlns":64},[67,18148,18149,18159],{},[70,18150,18151,18153,18155,18157],{},[73,18152,1515],{},[78,18154,81],{"stretchy":80},[73,18156,1515],{},[78,18158,87],{"stretchy":80},[142,18160,17824],{"encoding":144},[48,18162,18164],{"className":18163,"ariaHidden":150},[149],[48,18165,18167,18170,18173,18176,18179],{"className":18166},[154],[48,18168],{"className":18169,"style":159},[158],[48,18171,1515],{"className":18172,"style":1680},[163,171],[48,18174,81],{"className":18175},[167],[48,18177,1515],{"className":18178,"style":1680},[163,171],[48,18180,87],{"className":18181},[175]," 不停机」），那么根据 ",[48,18184,18186,18199],{"className":18185,"translate":52},[56],[48,18187,18189],{"className":18188},[60],[62,18190,18191],{"xmlns":64},[67,18192,18193,18197],{},[70,18194,18195],{},[73,18196,1515],{},[142,18198,1515],{"encoding":144},[48,18200,18202],{"className":18201,"ariaHidden":150},[149],[48,18203,18205,18208],{"className":18204},[154],[48,18206],{"className":18207,"style":4878},[158],[48,18209,1515],{"className":18210,"style":1680},[163,171],[48,18212,18214,18233],{"className":18213,"translate":52},[56],[48,18215,18217],{"className":18216},[60],[62,18218,18219],{"xmlns":64},[67,18220,18221,18231],{},[70,18222,18223,18225,18227,18229],{},[73,18224,1515],{},[78,18226,81],{"stretchy":80},[73,18228,1515],{},[78,18230,87],{"stretchy":80},[142,18232,17824],{"encoding":144},[48,18234,18236],{"className":18235,"ariaHidden":150},[149],[48,18237,18239,18242,18245,18248,18251],{"className":18238},[154],[48,18240],{"className":18241,"style":159},[158],[48,18243,1515],{"className":18244,"style":1680},[163,171],[48,18246,81],{"className":18247},[167],[48,18249,1515],{"className":18250,"style":1680},[163,171],[48,18252,87],{"className":18253},[175]," 应该立即停机。这与「它不停机」矛盾。",[11,18256,18257,18258,18290],{},"两种情况都导致自相矛盾。因此，「存在通用停机判定器 ",[48,18259,18261,18275],{"className":18260,"translate":52},[56],[48,18262,18264],{"className":18263},[60],[62,18265,18266],{"xmlns":64},[67,18267,18268,18272],{},[70,18269,18270],{},[4598,18271,17004],{},[142,18273,18274],{"encoding":144},"\\text{halts}",[48,18276,18278],{"className":18277,"ariaHidden":150},[149],[48,18279,18281,18284],{"className":18280},[154],[48,18282],{"className":18283,"style":5810},[158],[48,18285,18287],{"className":18286},[163,4646],[48,18288,17004],{"className":18289},[163],"」的假设是假的。停机问题不可判定。",[11,18292,18293,18294,18487],{},"若按某种规则把所有程序枚举出来（",[48,18295,18297,18335],{"className":18296,"translate":52},[56],[48,18298,18300],{"className":18299},[60],[62,18301,18302],{"xmlns":64},[67,18303,18304,18332],{},[70,18305,18306,18312,18314,18320,18322,18328,18330],{},[115,18307,18308,18310],{},[73,18309,13159],{},[1520,18311,1522],{},[78,18313,874],{"separator":150},[115,18315,18316,18318],{},[73,18317,13159],{},[1520,18319,2868],{},[78,18321,874],{"separator":150},[115,18323,18324,18326],{},[73,18325,13159],{},[1520,18327,5037],{},[78,18329,874],{"separator":150},[78,18331,2934],{},[142,18333,18334],{"encoding":144},"P_1, P_2, P_3, \\dots",[48,18336,18338],{"className":18337,"ariaHidden":150},[149],[48,18339,18341,18345,18386,18389,18392,18432,18435,18438,18478,18481,18484],{"className":18340},[154],[48,18342],{"className":18343,"style":18344},[158],"height:0.8778em;vertical-align:-0.1944em;",[48,18346,18348,18351],{"className":18347},[163],[48,18349,13159],{"className":18350,"style":242},[163,171],[48,18352,18354],{"className":18353},[292],[48,18355,18357,18378],{"className":18356},[203,204],[48,18358,18360,18375],{"className":18359},[208],[48,18361,18363],{"className":18362,"style":2750},[212],[48,18364,18366,18369],{"style":18365},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[48,18367],{"className":18368,"style":309},[220],[48,18370,18372],{"className":18371},[225,226,227,228],[48,18373,1522],{"className":18374},[163,228],[48,18376,269],{"className":18377},[268],[48,18379,18381],{"className":18380},[208],[48,18382,18384],{"className":18383,"style":2771},[212],[48,18385],{},[48,18387,874],{"className":18388},[944],[48,18390],{"className":18391,"style":282},[179],[48,18393,18395,18398],{"className":18394},[163],[48,18396,13159],{"className":18397,"style":242},[163,171],[48,18399,18401],{"className":18400},[292],[48,18402,18404,18424],{"className":18403},[203,204],[48,18405,18407,18421],{"className":18406},[208],[48,18408,18410],{"className":18409,"style":2750},[212],[48,18411,18412,18415],{"style":18365},[48,18413],{"className":18414,"style":309},[220],[48,18416,18418],{"className":18417},[225,226,227,228],[48,18419,2868],{"className":18420},[163,228],[48,18422,269],{"className":18423},[268],[48,18425,18427],{"className":18426},[208],[48,18428,18430],{"className":18429,"style":2771},[212],[48,18431],{},[48,18433,874],{"className":18434},[944],[48,18436],{"className":18437,"style":282},[179],[48,18439,18441,18444],{"className":18440},[163],[48,18442,13159],{"className":18443,"style":242},[163,171],[48,18445,18447],{"className":18446},[292],[48,18448,18450,18470],{"className":18449},[203,204],[48,18451,18453,18467],{"className":18452},[208],[48,18454,18456],{"className":18455,"style":2750},[212],[48,18457,18458,18461],{"style":18365},[48,18459],{"className":18460,"style":309},[220],[48,18462,18464],{"className":18463},[225,226,227,228],[48,18465,5037],{"className":18466},[163,228],[48,18468,269],{"className":18469},[268],[48,18471,18473],{"className":18472},[208],[48,18474,18476],{"className":18475,"style":2771},[212],[48,18477],{},[48,18479,874],{"className":18480},[944],[48,18482],{"className":18483,"style":282},[179],[48,18485,2934],{"className":18486},[3012],"——这是可行的，因为程序本质上就是有限长的字符串），并想象一张无穷表格：",[48,18489,18491],{"className":18490,"translate":52},[51],[48,18492,18494,18659],{"className":18493,"translate":52},[56],[48,18495,18497],{"className":18496},[60],[62,18498,18499],{"xmlns":64,"display":65},[67,18500,18501,18656],{},[70,18502,18503,18505,18654],{},[78,18504,81],{"fence":150},[6463,18506,18508,18568,18628],{"rowspacing":11498,"columnalign":18507,"columnspacing":6467},"center center center",[6469,18509,18510,18536,18562],{},[6472,18511,18512],{},[6475,18513,18514],{"scriptlevel":2710,"displaystyle":80},[70,18515,18516,18518,18520,18526,18528,18534],{},[4598,18517,17004],{},[78,18519,81],{"stretchy":80},[115,18521,18522,18524],{},[73,18523,13159],{},[1520,18525,1522],{},[78,18527,874],{"separator":150},[115,18529,18530,18532],{},[73,18531,13159],{},[1520,18533,1522],{},[78,18535,87],{"stretchy":80},[6472,18537,18538],{},[6475,18539,18540],{"scriptlevel":2710,"displaystyle":80},[70,18541,18542,18544,18546,18552,18554,18560],{},[4598,18543,17004],{},[78,18545,81],{"stretchy":80},[115,18547,18548,18550],{},[73,18549,13159],{},[1520,18551,1522],{},[78,18553,874],{"separator":150},[115,18555,18556,18558],{},[73,18557,13159],{},[1520,18559,2868],{},[78,18561,87],{"stretchy":80},[6472,18563,18564],{},[6475,18565,18566],{"scriptlevel":2710,"displaystyle":80},[78,18567,11548],{"lspace":3327,"rspace":3327},[6469,18569,18570,18596,18622],{},[6472,18571,18572],{},[6475,18573,18574],{"scriptlevel":2710,"displaystyle":80},[70,18575,18576,18578,18580,18586,18588,18594],{},[4598,18577,17004],{},[78,18579,81],{"stretchy":80},[115,18581,18582,18584],{},[73,18583,13159],{},[1520,18585,2868],{},[78,18587,874],{"separator":150},[115,18589,18590,18592],{},[73,18591,13159],{},[1520,18593,1522],{},[78,18595,87],{"stretchy":80},[6472,18597,18598],{},[6475,18599,18600],{"scriptlevel":2710,"displaystyle":80},[70,18601,18602,18604,18606,18612,18614,18620],{},[4598,18603,17004],{},[78,18605,81],{"stretchy":80},[115,18607,18608,18610],{},[73,18609,13159],{},[1520,18611,2868],{},[78,18613,874],{"separator":150},[115,18615,18616,18618],{},[73,18617,13159],{},[1520,18619,2868],{},[78,18621,87],{"stretchy":80},[6472,18623,18624],{},[6475,18625,18626],{"scriptlevel":2710,"displaystyle":80},[78,18627,11548],{"lspace":3327,"rspace":3327},[6469,18629,18630,18642,18648],{},[6472,18631,18632],{},[6475,18633,18634],{"scriptlevel":2710,"displaystyle":80},[70,18635,18636,18638],{},[73,18637,7578],{"mathvariant":75},[7580,18639,18640],{"height":3327,"voffset":3327},[179,18641],{"mathbackground":7584,"width":3327,"height":7585},[6472,18643,18644],{},[6475,18645,18646],{"scriptlevel":2710,"displaystyle":80},[70,18647],{},[6472,18649,18650],{},[6475,18651,18652],{"scriptlevel":2710,"displaystyle":80},[78,18653,11685],{"lspace":3327,"rspace":3327},[78,18655,87],{"fence":150},[142,18657,18658],{"encoding":144},"\\begin{pmatrix}\n\\text{halts}(P_1, P_1) & \\text{halts}(P_1, P_2) & \\cdots \\\\\n\\text{halts}(P_2, P_1) & \\text{halts}(P_2, P_2) & \\cdots \\\\\n\\vdots & & \\ddots\n\\end{pmatrix}",[48,18660,18662],{"className":18661,"ariaHidden":150},[149],[48,18663,18665,18669],{"className":18664},[154],[48,18666],{"className":18667,"style":18668},[158],"height:4.26em;vertical-align:-1.88em;",[48,18670,18672,18717,19292],{"className":18671},[3012],[48,18673,18675],{"className":18674},[167],[48,18676,18678],{"className":18677},[6583,11710],[48,18679,18681,18708],{"className":18680},[203,204],[48,18682,18684,18705],{"className":18683},[208],[48,18685,18688],{"className":18686,"style":18687},[212],"height:2.35em;",[48,18689,18691,18695],{"style":18690},"top:-4.35em;",[48,18692],{"className":18693,"style":18694},[220],"height:6.2em;",[48,18696,18698],{"style":18697},"width:0.875em;height:4.2em;",[4327,18699,18702],{"xmlns":4329,"width":11733,"height":18700,"viewBox":18701},"4.2em","0 0 875 4200",[4335,18703],{"d":18704},"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,684c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-692c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z",[48,18706,269],{"className":18707},[268],[48,18709,18711],{"className":18710},[208],[48,18712,18715],{"className":18713,"style":18714},[212],"height:1.85em;",[48,18716],{},[48,18718,18720],{"className":18719},[163],[48,18721,18723,18979,18982,18985,19230,19233,19236],{"className":18722},[6463],[48,18724,18726],{"className":18725},[11760],[48,18727,18729,18970],{"className":18728},[203,204],[48,18730,18732,18967],{"className":18731},[208],[48,18733,18736,18843,18949],{"className":18734,"style":18735},[212],"height:2.38em;",[48,18737,18739,18742],{"style":18738},"top:-5.2275em;",[48,18740],{"className":18741,"style":7855},[220],[48,18743,18745,18751,18754,18794,18797,18800,18840],{"className":18744},[163],[48,18746,18748],{"className":18747},[163,4646],[48,18749,17004],{"className":18750},[163],[48,18752,81],{"className":18753},[167],[48,18755,18757,18760],{"className":18756},[163],[48,18758,13159],{"className":18759,"style":242},[163,171],[48,18761,18763],{"className":18762},[292],[48,18764,18766,18786],{"className":18765},[203,204],[48,18767,18769,18783],{"className":18768},[208],[48,18770,18772],{"className":18771,"style":2750},[212],[48,18773,18774,18777],{"style":18365},[48,18775],{"className":18776,"style":309},[220],[48,18778,18780],{"className":18779},[225,226,227,228],[48,18781,1522],{"className":18782},[163,228],[48,18784,269],{"className":18785},[268],[48,18787,18789],{"className":18788},[208],[48,18790,18792],{"className":18791,"style":2771},[212],[48,18793],{},[48,18795,874],{"className":18796},[944],[48,18798],{"className":18799,"style":282},[179],[48,18801,18803,18806],{"className":18802},[163],[48,18804,13159],{"className":18805,"style":242},[163,171],[48,18807,18809],{"className":18808},[292],[48,18810,18812,18832],{"className":18811},[203,204],[48,18813,18815,18829],{"className":18814},[208],[48,18816,18818],{"className":18817,"style":2750},[212],[48,18819,18820,18823],{"style":18365},[48,18821],{"className":18822,"style":309},[220],[48,18824,18826],{"className":18825},[225,226,227,228],[48,18827,1522],{"className":18828},[163,228],[48,18830,269],{"className":18831},[268],[48,18833,18835],{"className":18834},[208],[48,18836,18838],{"className":18837,"style":2771},[212],[48,18839],{},[48,18841,87],{"className":18842},[175],[48,18844,18845,18848],{"style":8274},[48,18846],{"className":18847,"style":7855},[220],[48,18849,18851,18857,18860,18900,18903,18906,18946],{"className":18850},[163],[48,18852,18854],{"className":18853},[163,4646],[48,18855,17004],{"className":18856},[163],[48,18858,81],{"className":18859},[167],[48,18861,18863,18866],{"className":18862},[163],[48,18864,13159],{"className":18865,"style":242},[163,171],[48,18867,18869],{"className":18868},[292],[48,18870,18872,18892],{"className":18871},[203,204],[48,18873,18875,18889],{"className":18874},[208],[48,18876,18878],{"className":18877,"style":2750},[212],[48,18879,18880,18883],{"style":18365},[48,18881],{"className":18882,"style":309},[220],[48,18884,18886],{"className":18885},[225,226,227,228],[48,18887,2868],{"className":18888},[163,228],[48,18890,269],{"className":18891},[268],[48,18893,18895],{"className":18894},[208],[48,18896,18898],{"className":18897,"style":2771},[212],[48,18899],{},[48,18901,874],{"className":18902},[944],[48,18904],{"className":18905,"style":282},[179],[48,18907,18909,18912],{"className":18908},[163],[48,18910,13159],{"className":18911,"style":242},[163,171],[48,18913,18915],{"className":18914},[292],[48,18916,18918,18938],{"className":18917},[203,204],[48,18919,18921,18935],{"className":18920},[208],[48,18922,18924],{"className":18923,"style":2750},[212],[48,18925,18926,18929],{"style":18365},[48,18927],{"className":18928,"style":309},[220],[48,18930,18932],{"className":18931},[225,226,227,228],[48,18933,1522],{"className":18934},[163,228],[48,18936,269],{"className":18937},[268],[48,18939,18941],{"className":18940},[208],[48,18942,18944],{"className":18943,"style":2771},[212],[48,18945],{},[48,18947,87],{"className":18948},[175],[48,18950,18952,18955],{"style":18951},"top:-2.1675em;",[48,18953],{"className":18954,"style":7855},[220],[48,18956,18958],{"className":18957},[163],[48,18959,18961,18964],{"className":18960},[163],[48,18962,7578],{"className":18963},[163],[48,18965],{"className":18966,"style":8722},[163,8721],[48,18968,269],{"className":18969},[268],[48,18971,18973],{"className":18972},[208],[48,18974,18977],{"className":18975,"style":18976},[212],"height:1.88em;",[48,18978],{},[48,18980],{"className":18981,"style":12014},[6673],[48,18983],{"className":18984,"style":12014},[6673],[48,18986,18988],{"className":18987},[11760],[48,18989,18991,19222],{"className":18990},[203,204],[48,18992,18994,19219],{"className":18993},[208],[48,18995,18997,19104,19210],{"className":18996,"style":18735},[212],[48,18998,19000,19003],{"style":18999},"top:-5.04em;",[48,19001],{"className":19002,"style":7625},[220],[48,19004,19006,19012,19015,19055,19058,19061,19101],{"className":19005},[163],[48,19007,19009],{"className":19008},[163,4646],[48,19010,17004],{"className":19011},[163],[48,19013,81],{"className":19014},[167],[48,19016,19018,19021],{"className":19017},[163],[48,19019,13159],{"className":19020,"style":242},[163,171],[48,19022,19024],{"className":19023},[292],[48,19025,19027,19047],{"className":19026},[203,204],[48,19028,19030,19044],{"className":19029},[208],[48,19031,19033],{"className":19032,"style":2750},[212],[48,19034,19035,19038],{"style":18365},[48,19036],{"className":19037,"style":309},[220],[48,19039,19041],{"className":19040},[225,226,227,228],[48,19042,1522],{"className":19043},[163,228],[48,19045,269],{"className":19046},[268],[48,19048,19050],{"className":19049},[208],[48,19051,19053],{"className":19052,"style":2771},[212],[48,19054],{},[48,19056,874],{"className":19057},[944],[48,19059],{"className":19060,"style":282},[179],[48,19062,19064,19067],{"className":19063},[163],[48,19065,13159],{"className":19066,"style":242},[163,171],[48,19068,19070],{"className":19069},[292],[48,19071,19073,19093],{"className":19072},[203,204],[48,19074,19076,19090],{"className":19075},[208],[48,19077,19079],{"className":19078,"style":2750},[212],[48,19080,19081,19084],{"style":18365},[48,19082],{"className":19083,"style":309},[220],[48,19085,19087],{"className":19086},[225,226,227,228],[48,19088,2868],{"className":19089},[163,228],[48,19091,269],{"className":19092},[268],[48,19094,19096],{"className":19095},[208],[48,19097,19099],{"className":19098,"style":2771},[212],[48,19100],{},[48,19102,87],{"className":19103},[175],[48,19105,19106,19109],{"style":7720},[48,19107],{"className":19108,"style":7625},[220],[48,19110,19112,19118,19121,19161,19164,19167,19207],{"className":19111},[163],[48,19113,19115],{"className":19114},[163,4646],[48,19116,17004],{"className":19117},[163],[48,19119,81],{"className":19120},[167],[48,19122,19124,19127],{"className":19123},[163],[48,19125,13159],{"className":19126,"style":242},[163,171],[48,19128,19130],{"className":19129},[292],[48,19131,19133,19153],{"className":19132},[203,204],[48,19134,19136,19150],{"className":19135},[208],[48,19137,19139],{"className":19138,"style":2750},[212],[48,19140,19141,19144],{"style":18365},[48,19142],{"className":19143,"style":309},[220],[48,19145,19147],{"className":19146},[225,226,227,228],[48,19148,2868],{"className":19149},[163,228],[48,19151,269],{"className":19152},[268],[48,19154,19156],{"className":19155},[208],[48,19157,19159],{"className":19158,"style":2771},[212],[48,19160],{},[48,19162,874],{"className":19163},[944],[48,19165],{"className":19166,"style":282},[179],[48,19168,19170,19173],{"className":19169},[163],[48,19171,13159],{"className":19172,"style":242},[163,171],[48,19174,19176],{"className":19175},[292],[48,19177,19179,19199],{"className":19178},[203,204],[48,19180,19182,19196],{"className":19181},[208],[48,19183,19185],{"className":19184,"style":2750},[212],[48,19186,19187,19190],{"style":18365},[48,19188],{"className":19189,"style":309},[220],[48,19191,19193],{"className":19192},[225,226,227,228],[48,19194,2868],{"className":19195},[163,228],[48,19197,269],{"className":19198},[268],[48,19200,19202],{"className":19201},[208],[48,19203,19205],{"className":19204,"style":2771},[212],[48,19206],{},[48,19208,87],{"className":19209},[175],[48,19211,19213,19216],{"style":19212},"top:-1.98em;",[48,19214],{"className":19215,"style":7625},[220],[48,19217],{"className":19218},[163],[48,19220,269],{"className":19221},[268],[48,19223,19225],{"className":19224},[208],[48,19226,19228],{"className":19227,"style":18976},[212],[48,19229],{},[48,19231],{"className":19232,"style":12014},[6673],[48,19234],{"className":19235,"style":12014},[6673],[48,19237,19239],{"className":19238},[11760],[48,19240,19242,19284],{"className":19241},[203,204],[48,19243,19245,19281],{"className":19244},[208],[48,19246,19248,19259,19270],{"className":19247,"style":18735},[212],[48,19249,19250,19253],{"style":18999},[48,19251],{"className":19252,"style":7625},[220],[48,19254,19256],{"className":19255},[163],[48,19257,11548],{"className":19258},[3012],[48,19260,19261,19264],{"style":7720},[48,19262],{"className":19263,"style":7625},[220],[48,19265,19267],{"className":19266},[163],[48,19268,11548],{"className":19269},[3012],[48,19271,19272,19275],{"style":19212},[48,19273],{"className":19274,"style":7625},[220],[48,19276,19278],{"className":19277},[163],[48,19279,11685],{"className":19280},[3012],[48,19282,269],{"className":19283},[268],[48,19285,19287],{"className":19286},[208],[48,19288,19290],{"className":19289,"style":18976},[212],[48,19291],{},[48,19293,19295],{"className":19294},[175],[48,19296,19298],{"className":19297},[6583,11710],[48,19299,19301,19322],{"className":19300},[203,204],[48,19302,19304,19319],{"className":19303},[208],[48,19305,19307],{"className":19306,"style":18687},[212],[48,19308,19309,19312],{"style":18690},[48,19310],{"className":19311,"style":18694},[220],[48,19313,19314],{"style":18697},[4327,19315,19316],{"xmlns":4329,"width":11733,"height":18700,"viewBox":18701},[4335,19317],{"d":19318},"M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,\n63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5\nc11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,609\nc-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-744c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z",[48,19320,269],{"className":19321},[268],[48,19323,19325],{"className":19324},[208],[48,19326,19328],{"className":19327,"style":18714},[212],[48,19329],{},[11,19331,19332,19333,19361,19362,19504],{},"对角线程序 ",[48,19334,19336,19349],{"className":19335,"translate":52},[56],[48,19337,19339],{"className":19338},[60],[62,19340,19341],{"xmlns":64},[67,19342,19343,19347],{},[70,19344,19345],{},[73,19346,1515],{},[142,19348,1515],{"encoding":144},[48,19350,19352],{"className":19351,"ariaHidden":150},[149],[48,19353,19355,19358],{"className":19354},[154],[48,19356],{"className":19357,"style":4878},[158],[48,19359,1515],{"className":19360,"style":1680},[163,171]," 的行为恰恰取决于这张表格的主对角线 ",[48,19363,19365,19397],{"className":19364,"translate":52},[56],[48,19366,19368],{"className":19367},[60],[62,19369,19370],{"xmlns":64},[67,19371,19372,19394],{},[70,19373,19374,19376,19378,19384,19386,19392],{},[4598,19375,17004],{},[78,19377,81],{"stretchy":80},[115,19379,19380,19382],{},[73,19381,13159],{},[73,19383,2631],{},[78,19385,874],{"separator":150},[115,19387,19388,19390],{},[73,19389,13159],{},[73,19391,2631],{},[78,19393,87],{"stretchy":80},[142,19395,19396],{"encoding":144},"\\text{halts}(P_i, P_i)",[48,19398,19400],{"className":19399,"ariaHidden":150},[149],[48,19401,19403,19406,19412,19415,19455,19458,19461,19501],{"className":19402},[154],[48,19404],{"className":19405,"style":159},[158],[48,19407,19409],{"className":19408},[163,4646],[48,19410,17004],{"className":19411},[163],[48,19413,81],{"className":19414},[167],[48,19416,19418,19421],{"className":19417},[163],[48,19419,13159],{"className":19420,"style":242},[163,171],[48,19422,19424],{"className":19423},[292],[48,19425,19427,19447],{"className":19426},[203,204],[48,19428,19430,19444],{"className":19429},[208],[48,19431,19433],{"className":19432,"style":8789},[212],[48,19434,19435,19438],{"style":18365},[48,19436],{"className":19437,"style":309},[220],[48,19439,19441],{"className":19440},[225,226,227,228],[48,19442,2631],{"className":19443},[163,171,228],[48,19445,269],{"className":19446},[268],[48,19448,19450],{"className":19449},[208],[48,19451,19453],{"className":19452,"style":2771},[212],[48,19454],{},[48,19456,874],{"className":19457},[944],[48,19459],{"className":19460,"style":282},[179],[48,19462,19464,19467],{"className":19463},[163],[48,19465,13159],{"className":19466,"style":242},[163,171],[48,19468,19470],{"className":19469},[292],[48,19471,19473,19493],{"className":19472},[203,204],[48,19474,19476,19490],{"className":19475},[208],[48,19477,19479],{"className":19478,"style":8789},[212],[48,19480,19481,19484],{"style":18365},[48,19482],{"className":19483,"style":309},[220],[48,19485,19487],{"className":19486},[225,226,227,228],[48,19488,2631],{"className":19489},[163,171,228],[48,19491,269],{"className":19492},[268],[48,19494,19496],{"className":19495},[208],[48,19497,19499],{"className":19498,"style":2771},[212],[48,19500],{},[48,19502,87],{"className":19503},[175],"，并且刻意表现得相反。这与康托尔构造「每一位都与相应对角线数字不同的实数」在结构上完全一致。",[11,19506,19507],{},"我们无法真正实现一个通用停机判定器（因为它根本不存在！），但可以通过代码演示这种逻辑矛盾是如何具体产生的：",[12625,19509,19511],{"className":12627,"code":19510,"language":12629,"meta":4787,"style":4787},"import sys\nsys.setrecursionlimit(50)\n\ndef hypothetical_halts(f, x, fuel=10):\n    \"\"\"假想的停机判定器（仅用于悖论演示；并非真正可行的实现）\"\"\"\n    try:\n        result = f(x, fuel)\n        return result is not None\n    except RecursionError:\n        return False\n\ndef D(x, fuel=10):\n    if fuel \u003C= 0:\n        raise RecursionError(\"燃料耗尽，模拟死循环\")\n    will_halt = hypothetical_halts(x, x, fuel - 1)\n    if will_halt:\n        raise RecursionError(\"模拟死循环：halts 判定其停机，故 D 刻意不停机\")\n    else:\n        return \"D 停机\"\n\noutcome = D(D, fuel=8)\nprint(\"D(D) 的执行结果：\", outcome)\n",[2561,19512,19513,19521,19531,19535,19551,19556,19564,19574,19591,19601,19608,19612,19628,19643,19657,19675,19682,19695,19702,19709,19713,19734],{"__ignoreMap":4787},[48,19514,19515,19518],{"class":12634,"line":12635},[48,19516,19517],{"class":12638},"import",[48,19519,19520],{"class":12646}," sys\n",[48,19522,19523,19526,19529],{"class":12634,"line":4788},[48,19524,19525],{"class":12646},"sys.setrecursionlimit(",[48,19527,19528],{"class":12652},"50",[48,19530,12736],{"class":12646},[48,19532,19533],{"class":12634,"line":12665},[48,19534,12818],{"emptyLinePlaceholder":4800},[48,19536,19537,19539,19542,19545,19547,19549],{"class":12634,"line":12671},[48,19538,12639],{"class":12638},[48,19540,19541],{"class":12642}," hypothetical_halts",[48,19543,19544],{"class":12646},"(f, x, fuel",[48,19546,90],{"class":12638},[48,19548,12653],{"class":12652},[48,19550,12656],{"class":12646},[48,19552,19553],{"class":12634,"line":12677},[48,19554,19555],{"class":12661},"    \"\"\"假想的停机判定器（仅用于悖论演示；并非真正可行的实现）\"\"\"\n",[48,19557,19558,19561],{"class":12634,"line":12682},[48,19559,19560],{"class":12638},"    try",[48,19562,19563],{"class":12646},":\n",[48,19565,19566,19569,19571],{"class":12634,"line":12693},[48,19567,19568],{"class":12646},"        result ",[48,19570,90],{"class":12638},[48,19572,19573],{"class":12646}," f(x, fuel)\n",[48,19575,19576,19579,19582,19585,19588],{"class":12634,"line":12711},[48,19577,19578],{"class":12638},"        return",[48,19580,19581],{"class":12646}," result ",[48,19583,19584],{"class":12638},"is",[48,19586,19587],{"class":12638}," not",[48,19589,19590],{"class":12652}," None\n",[48,19592,19593,19596,19599],{"class":12634,"line":12739},[48,19594,19595],{"class":12638},"    except",[48,19597,19598],{"class":12652}," RecursionError",[48,19600,19563],{"class":12646},[48,19602,19603,19605],{"class":12634,"line":12757},[48,19604,19578],{"class":12638},[48,19606,19607],{"class":12652}," False\n",[48,19609,19610],{"class":12634,"line":12785},[48,19611,12818],{"emptyLinePlaceholder":4800},[48,19613,19614,19616,19619,19622,19624,19626],{"class":12634,"line":12797},[48,19615,12639],{"class":12638},[48,19617,19618],{"class":12642}," D",[48,19620,19621],{"class":12646},"(x, fuel",[48,19623,90],{"class":12638},[48,19625,12653],{"class":12652},[48,19627,12656],{"class":12646},[48,19629,19630,19633,19636,19639,19641],{"class":12634,"line":12815},[48,19631,19632],{"class":12638},"    if",[48,19634,19635],{"class":12646}," fuel ",[48,19637,19638],{"class":12638},"\u003C=",[48,19640,16533],{"class":12652},[48,19642,19563],{"class":12646},[48,19644,19645,19648,19650,19652,19655],{"class":12634,"line":12821},[48,19646,19647],{"class":12638},"        raise",[48,19649,19598],{"class":12652},[48,19651,81],{"class":12646},[48,19653,19654],{"class":12661},"\"燃料耗尽，模拟死循环\"",[48,19656,12736],{"class":12646},[48,19658,19659,19662,19664,19667,19670,19673],{"class":12634,"line":12826},[48,19660,19661],{"class":12646},"    will_halt ",[48,19663,90],{"class":12638},[48,19665,19666],{"class":12646}," hypothetical_halts(x, x, fuel ",[48,19668,19669],{"class":12638},"-",[48,19671,19672],{"class":12652}," 1",[48,19674,12736],{"class":12646},[48,19676,19677,19679],{"class":12634,"line":12837},[48,19678,19632],{"class":12638},[48,19680,19681],{"class":12646}," will_halt:\n",[48,19683,19684,19686,19688,19690,19693],{"class":12634,"line":12862},[48,19685,19647],{"class":12638},[48,19687,19598],{"class":12652},[48,19689,81],{"class":12646},[48,19691,19692],{"class":12661},"\"模拟死循环：halts 判定其停机，故 D 刻意不停机\"",[48,19694,12736],{"class":12646},[48,19696,19697,19700],{"class":12634,"line":12885},[48,19698,19699],{"class":12638},"    else",[48,19701,19563],{"class":12646},[48,19703,19704,19706],{"class":12634,"line":12898},[48,19705,19578],{"class":12638},[48,19707,19708],{"class":12661}," \"D 停机\"\n",[48,19710,19711],{"class":12634,"line":12904},[48,19712,12818],{"emptyLinePlaceholder":4800},[48,19714,19715,19718,19720,19723,19727,19729,19732],{"class":12634,"line":12909},[48,19716,19717],{"class":12646},"outcome ",[48,19719,90],{"class":12638},[48,19721,19722],{"class":12646}," D(D, ",[48,19724,19726],{"class":19725},"s9osk","fuel",[48,19728,90],{"class":12638},[48,19730,19731],{"class":12652},"8",[48,19733,12736],{"class":12646},[48,19735,19736,19738,19740,19743],{"class":12634,"line":12920},[48,19737,12923],{"class":12652},[48,19739,81],{"class":12646},[48,19741,19742],{"class":12661},"\"D(D) 的执行结果：\"",[48,19744,19745],{"class":12646},", outcome)\n",[11,19747,12934],{},[12625,19749,19752],{"className":19750,"code":19751,"language":4646},[12938],"=== 对角线悖论推导 ===\n\n假设 halts(f, x) 是一个完美的停机判定器（能正确判断任意程序）\n构造 D(x)：若 halts(x,x)==停机 则 D 死循环；若 halts(x,x)==不停机 则 D 立即停机\n\n情形 A：若 halts(D,D) 判定为「停机」 -> 根据 D 的定义，D 应死循环 -> 矛盾\n情形 B：若 halts(D,D) 判定为「不停机」 -> 根据 D 的定义，D 应立即停机 -> 矛盾\n\n=> 无论哪种判定都会自相矛盾，证明通用 halts 函数不可能存在。\n\n=== 代码实例化（低燃料以便快速演示） ===\n触发递归异常，模拟自相矛盾的执行状态：模拟死循环：halts 判定其停机，故 D 刻意不停机\n",[2561,19753,19751],{"__ignoreMap":4787},[11,19755,19756,19757,19760,19761,19763],{},"这段代码里的 ",[2561,19758,19759],{},"hypothetical_halts"," 显然不是一个真正的「通用」判定器——它只能在一个有限的 ",[2561,19762,19726],{},"（步数预算）内运行，这正是我们能在一台真实计算机上执行它的唯一原因。而这恰恰佐证了图灵的结论：一个总能给出正确答案的真正通用的停机判定器不可能存在。我们所能写的任何「模拟版本」都必然带有这样或那样的局限（比如这里采用的步数上限）。",[35,19765,19766],{"id":19766},"哥德尔不完备定理",[11,19768,19769],{},"1931 年，库尔特·哥德尔运用对角线论证的一个更加精巧的变体，证明了数理逻辑史上最深刻的成果之一：",[1295,19771,19772],{},[11,19773,19774],{},"哥德尔第一不完备定理。任何包含初等算术的相容（不自相矛盾）形式系统，都存在一个在该系统内既不能被证明也不能被否证的命题。",[11,19776,19777],{},"哥德尔的证明大体分为三步：",[2555,19779,19780,19783,20139],{},[996,19781,19782],{},"哥德尔编号。把形式系统中的每个公式和每个证明都编码成一个自然数（这种编码技巧本身就是天才之举，如今被称为「哥德尔数」）。这样一来，「某个证明是否证明了某个公式」就变成了自然数上的一个算术关系，可以在系统内部表达出来。",[996,19784,19785,19786,19815,19816,19899,19902,19903,19947,19948,19978,19979,20065,20066,20109,20110,20138],{},"构造自指命题。利用编码技巧，构造一个命题 ",[48,19787,19789,19803],{"className":19788,"translate":52},[56],[48,19790,19792],{"className":19791},[60],[62,19793,19794],{"xmlns":64},[67,19795,19796,19801],{},[70,19797,19798],{},[73,19799,19800],{},"G",[142,19802,19800],{"encoding":144},[48,19804,19806],{"className":19805,"ariaHidden":150},[149],[48,19807,19809,19812],{"className":19808},[154],[48,19810],{"className":19811,"style":4878},[158],[48,19813,19800],{"className":19814},[163,171],"，其核心内容是：",[48,19817,19819],{"className":19818,"translate":52},[51],[48,19820,19822,19848],{"className":19821,"translate":52},[56],[48,19823,19825],{"className":19824},[60],[62,19826,19827],{"xmlns":64,"display":65},[67,19828,19829,19845],{},[70,19830,19831,19833,19835,19837,19840,19842],{},[73,19832,19800],{},[78,19834,1495],{},[78,19836,90],{},[4598,19838,19839],{},"「命题 ",[73,19841,19800],{},[4598,19843,19844],{}," 本身在该系统内不可证明」",[142,19846,19847],{"encoding":144},"G := \\text{「命题 } G \\text{ 本身在该系统内不可证明」}",[48,19849,19851,19870],{"className":19850,"ariaHidden":150},[149],[48,19852,19854,19857,19860,19863,19867],{"className":19853},[154],[48,19855],{"className":19856,"style":4878},[158],[48,19858,19800],{"className":19859},[163,171],[48,19861],{"className":19862,"style":180},[179],[48,19864,19866],{"className":19865},[184],":=",[48,19868],{"className":19869,"style":180},[179],[48,19871,19873,19876,19886,19889],{"className":19872},[154],[48,19874],{"className":19875,"style":4878},[158],[48,19877,19879,19883],{"className":19878},[163,4646],[48,19880,19882],{"className":19881},[163,6708],"「命题",[48,19884,6704],{"className":19885},[163],[48,19887,19800],{"className":19888},[163,171],[48,19890,19892,19895],{"className":19891},[163,4646],[48,19893,6704],{"className":19894},[163],[48,19896,19898],{"className":19897},[163,6708],"本身在该系统内不可证明」",[19900,19901],"br",{},"这是一个高度自指的构造，其数学基础被称为对角线引理（又称不动点引理）：对任意只含一个自由变量的公式 ",[48,19904,19906,19926],{"className":19905,"translate":52},[56],[48,19907,19909],{"className":19908},[60],[62,19910,19911],{"xmlns":64},[67,19912,19913,19923],{},[70,19914,19915,19917,19919,19921],{},[73,19916,1490],{},[78,19918,81],{"stretchy":80},[73,19920,84],{},[78,19922,87],{"stretchy":80},[142,19924,19925],{"encoding":144},"\\phi(x)",[48,19927,19929],{"className":19928,"ariaHidden":150},[149],[48,19930,19932,19935,19938,19941,19944],{"className":19931},[154],[48,19933],{"className":19934,"style":159},[158],[48,19936,1490],{"className":19937},[163,171],[48,19939,81],{"className":19940},[167],[48,19942,84],{"className":19943},[163,171],[48,19945,87],{"className":19946},[175],"，都能构造一个句子 ",[48,19949,19951,19966],{"className":19950,"translate":52},[56],[48,19952,19954],{"className":19953},[60],[62,19955,19956],{"xmlns":64},[67,19957,19958,19963],{},[70,19959,19960],{},[73,19961,19962],{},"ψ",[142,19964,19965],{"encoding":144},"\\psi",[48,19967,19969],{"className":19968,"ariaHidden":150},[149],[48,19970,19972,19975],{"className":19971},[154],[48,19973],{"className":19974,"style":766},[158],[48,19976,19962],{"className":19977,"style":343},[163,171],"，使系统证明 ",[48,19980,19982,20017],{"className":19981,"translate":52},[56],[48,19983,19985],{"className":19984},[60],[62,19986,19987],{"xmlns":64},[67,19988,19989,20014],{},[70,19990,19991,19993,19996,19998,20000,20005,20007,20012],{},[73,19992,19962],{},[78,19994,19995],{},"↔",[73,19997,1490],{},[78,19999,81],{"stretchy":80},[78,20001,20002],{},[73,20003,20004],{"mathvariant":75},"⌜",[73,20006,19962],{},[78,20008,20009],{},[73,20010,20011],{"mathvariant":75},"⌝",[78,20013,87],{"stretchy":80},[142,20015,20016],{"encoding":144},"\\psi \\leftrightarrow \\phi(\\ulcorner \\psi \\urcorner)",[48,20018,20020,20038],{"className":20019,"ariaHidden":150},[149],[48,20021,20023,20026,20029,20032,20035],{"className":20022},[154],[48,20024],{"className":20025,"style":766},[158],[48,20027,19962],{"className":20028,"style":343},[163,171],[48,20030],{"className":20031,"style":180},[179],[48,20033,19995],{"className":20034},[184],[48,20036],{"className":20037,"style":180},[179],[48,20039,20041,20044,20047,20050,20055,20058,20062],{"className":20040},[154],[48,20042],{"className":20043,"style":159},[158],[48,20045,1490],{"className":20046},[163,171],[48,20048,81],{"className":20049},[167],[48,20051,20054],{"className":20052},[167,20053],"amsrm","┌",[48,20056,19962],{"className":20057,"style":343},[163,171],[48,20059,20061],{"className":20060},[175,20053],"┐",[48,20063,87],{"className":20064},[175],"（其中 ",[48,20067,20069,20091],{"className":20068,"translate":52},[56],[48,20070,20072],{"className":20071},[60],[62,20073,20074],{"xmlns":64},[67,20075,20076,20088],{},[70,20077,20078,20082,20084],{},[78,20079,20080],{},[73,20081,20004],{"mathvariant":75},[73,20083,19962],{},[78,20085,20086],{},[73,20087,20011],{"mathvariant":75},[142,20089,20090],{"encoding":144},"\\ulcorner \\psi \\urcorner",[48,20092,20094],{"className":20093,"ariaHidden":150},[149],[48,20095,20097,20100,20103,20106],{"className":20096},[154],[48,20098],{"className":20099,"style":766},[158],[48,20101,20054],{"className":20102},[167,20053],[48,20104,19962],{"className":20105,"style":343},[163,171],[48,20107,20061],{"className":20108},[175,20053]," 表示 ",[48,20111,20113,20126],{"className":20112,"translate":52},[56],[48,20114,20116],{"className":20115},[60],[62,20117,20118],{"xmlns":64},[67,20119,20120,20124],{},[70,20121,20122],{},[73,20123,19962],{},[142,20125,19965],{"encoding":144},[48,20127,20129],{"className":20128,"ariaHidden":150},[149],[48,20130,20132,20135],{"className":20131},[154],[48,20133],{"className":20134,"style":766},[158],[48,20136,19962],{"className":20137,"style":343},[163,171]," 的哥德尔数）。这个引理正是「对角线」思想在逻辑中的抽象：它让一个命题能够「谈论」关于其自身代码的某种性质。",[996,20140,20141,20142],{},"导出矛盾的两难。",[993,20143,20144,20234,20330],{},[996,20145,20146,20147,20175,20176,20204,20205,20233],{},"若系统能证明 ",[48,20148,20150,20163],{"className":20149,"translate":52},[56],[48,20151,20153],{"className":20152},[60],[62,20154,20155],{"xmlns":64},[67,20156,20157,20161],{},[70,20158,20159],{},[73,20160,19800],{},[142,20162,19800],{"encoding":144},[48,20164,20166],{"className":20165,"ariaHidden":150},[149],[48,20167,20169,20172],{"className":20168},[154],[48,20170],{"className":20171,"style":4878},[158],[48,20173,19800],{"className":20174},[163,171],"，那么根据 ",[48,20177,20179,20192],{"className":20178,"translate":52},[56],[48,20180,20182],{"className":20181},[60],[62,20183,20184],{"xmlns":64},[67,20185,20186,20190],{},[70,20187,20188],{},[73,20189,19800],{},[142,20191,19800],{"encoding":144},[48,20193,20195],{"className":20194,"ariaHidden":150},[149],[48,20196,20198,20201],{"className":20197},[154],[48,20199],{"className":20200,"style":4878},[158],[48,20202,19800],{"className":20203},[163,171]," 的含义（「",[48,20206,20208,20221],{"className":20207,"translate":52},[56],[48,20209,20211],{"className":20210},[60],[62,20212,20213],{"xmlns":64},[67,20214,20215,20219],{},[70,20216,20217],{},[73,20218,19800],{},[142,20220,19800],{"encoding":144},[48,20222,20224],{"className":20223,"ariaHidden":150},[149],[48,20225,20227,20230],{"className":20226},[154],[48,20228],{"className":20229,"style":4878},[158],[48,20231,19800],{"className":20232},[163,171]," 不可证明」），系统就同时证明了一个假命题——与系统的相容性矛盾。",[996,20235,20236,20237,20265,20266,20300,20301,20329],{},"若系统能否证 ",[48,20238,20240,20253],{"className":20239,"translate":52},[56],[48,20241,20243],{"className":20242},[60],[62,20244,20245],{"xmlns":64},[67,20246,20247,20251],{},[70,20248,20249],{},[73,20250,19800],{},[142,20252,19800],{"encoding":144},[48,20254,20256],{"className":20255,"ariaHidden":150},[149],[48,20257,20259,20262],{"className":20258},[154],[48,20260],{"className":20261,"style":4878},[158],[48,20263,19800],{"className":20264},[163,171],"（即证明 ",[48,20267,20269,20285],{"className":20268,"translate":52},[56],[48,20270,20272],{"className":20271},[60],[62,20273,20274],{"xmlns":64},[67,20275,20276,20282],{},[70,20277,20278,20280],{},[73,20279,14778],{"mathvariant":75},[73,20281,19800],{},[142,20283,20284],{"encoding":144},"\\neg G",[48,20286,20288],{"className":20287,"ariaHidden":150},[149],[48,20289,20291,20294,20297],{"className":20290},[154],[48,20292],{"className":20293,"style":4878},[158],[48,20295,14778],{"className":20296},[163],[48,20298,19800],{"className":20299},[163,171],"，相当于证明「",[48,20302,20304,20317],{"className":20303,"translate":52},[56],[48,20305,20307],{"className":20306},[60],[62,20308,20309],{"xmlns":64},[67,20310,20311,20315],{},[70,20312,20313],{},[73,20314,19800],{},[142,20316,19800],{"encoding":144},[48,20318,20320],{"className":20319,"ariaHidden":150},[149],[48,20321,20323,20326],{"className":20322},[154],[48,20324],{"className":20325,"style":4878},[158],[48,20327,19800],{"className":20328},[163,171]," 可证明」），那么在合理的补充条件下，这同样会导致矛盾。",[996,20331,20332,20333,20361],{},"因此 ",[48,20334,20336,20349],{"className":20335,"translate":52},[56],[48,20337,20339],{"className":20338},[60],[62,20340,20341],{"xmlns":64},[67,20342,20343,20347],{},[70,20344,20345],{},[73,20346,19800],{},[142,20348,19800],{"encoding":144},[48,20350,20352],{"className":20351,"ariaHidden":150},[149],[48,20353,20355,20358],{"className":20354},[154],[48,20356],{"className":20357,"style":4878},[158],[48,20359,19800],{"className":20360},[163,171]," 既不能被证明也不能被否证——它是系统内的一个「不可判定命题」。",[11,20363,20364],{},"与前面两个证明的共通之处：",[15271,20366,20367,20497],{},[15274,20368,20369],{},[15277,20370,20371,20373,20404,20435,20466],{},[15280,20372],{},[15280,20374,20375,20376],{},"对角数 ",[48,20377,20379,20392],{"className":20378,"translate":52},[56],[48,20380,20382],{"className":20381},[60],[62,20383,20384],{"xmlns":64},[67,20385,20386,20390],{},[70,20387,20388],{},[73,20389,4526],{},[142,20391,4526],{"encoding":144},[48,20393,20395],{"className":20394,"ariaHidden":150},[149],[48,20396,20398,20401],{"className":20397},[154],[48,20399],{"className":20400,"style":2731},[158],[48,20402,4526],{"className":20403,"style":343},[163,171],[15280,20405,20406,20407],{},"对角集 ",[48,20408,20410,20423],{"className":20409,"translate":52},[56],[48,20411,20413],{"className":20412},[60],[62,20414,20415],{"xmlns":64},[67,20416,20417,20421],{},[70,20418,20419],{},[73,20420,1515],{},[142,20422,1515],{"encoding":144},[48,20424,20426],{"className":20425,"ariaHidden":150},[149],[48,20427,20429,20432],{"className":20428},[154],[48,20430],{"className":20431,"style":4878},[158],[48,20433,1515],{"className":20434,"style":1680},[163,171],[15280,20436,20437,20438],{},"对角程序 ",[48,20439,20441,20454],{"className":20440,"translate":52},[56],[48,20442,20444],{"className":20443},[60],[62,20445,20446],{"xmlns":64},[67,20447,20448,20452],{},[70,20449,20450],{},[73,20451,1515],{},[142,20453,1515],{"encoding":144},[48,20455,20457],{"className":20456,"ariaHidden":150},[149],[48,20458,20460,20463],{"className":20459},[154],[48,20461],{"className":20462,"style":4878},[158],[48,20464,1515],{"className":20465,"style":1680},[163,171],[15280,20467,20468,20469],{},"哥德尔句 ",[48,20470,20472,20485],{"className":20471,"translate":52},[56],[48,20473,20475],{"className":20474},[60],[62,20476,20477],{"xmlns":64},[67,20478,20479,20483],{},[70,20480,20481],{},[73,20482,19800],{},[142,20484,19800],{"encoding":144},[48,20486,20488],{"className":20487,"ariaHidden":150},[149],[48,20489,20491,20494],{"className":20490},[154],[48,20492],{"className":20493,"style":4878},[158],[48,20495,19800],{"className":20496},[163,171],[15286,20498,20499,20516,20533],{},[15277,20500,20501,20504,20507,20510,20513],{},[15291,20502,20503],{},"自指的对象",[15291,20505,20506],{},"每一位「避开」自身",[15291,20508,20509],{},"收集「不属于自己」的元素",[15291,20511,20512],{},"用自己的行为否定对自身的判定",[15291,20514,20515],{},"断言「我不可证明」",[15277,20517,20518,20521,20524,20527,20530],{},[15291,20519,20520],{},"依赖的机制",[15291,20522,20523],{},"十进制展开的对角线",[15291,20525,20526],{},"集合属于关系的自指",[15291,20528,20529],{},"程序以自身为输入",[15291,20531,20532],{},"通过哥德尔编号实现的自指",[15277,20534,20535,20538,20569,20600,20603],{},[15291,20536,20537],{},"矛盾的来源",[15291,20539,20540,20541],{},"假定的双射无法覆盖 ",[48,20542,20544,20557],{"className":20543,"translate":52},[56],[48,20545,20547],{"className":20546},[60],[62,20548,20549],{"xmlns":64},[67,20550,20551,20555],{},[70,20552,20553],{},[73,20554,4526],{},[142,20556,4526],{"encoding":144},[48,20558,20560],{"className":20559,"ariaHidden":150},[149],[48,20561,20563,20566],{"className":20562},[154],[48,20564],{"className":20565,"style":2731},[158],[48,20567,4526],{"className":20568,"style":343},[163,171],[15291,20570,20571,20572],{},"假定的满射无法覆盖 ",[48,20573,20575,20588],{"className":20574,"translate":52},[56],[48,20576,20578],{"className":20577},[60],[62,20579,20580],{"xmlns":64},[67,20581,20582,20586],{},[70,20583,20584],{},[73,20585,1515],{},[142,20587,1515],{"encoding":144},[48,20589,20591],{"className":20590,"ariaHidden":150},[149],[48,20592,20594,20597],{"className":20593},[154],[48,20595],{"className":20596,"style":4878},[158],[48,20598,1515],{"className":20599,"style":1680},[163,171],[15291,20601,20602],{},"假定的判定器导致悖论",[15291,20604,20605],{},"假定的可证明性导致悖论",[11,20607,20608],{},"这四个证明，本质上都在做同一件事：利用「自指」构造出一个必然与假定的「完备覆盖」相矛盾的对象。正是这一点，使对角线论证——横跨集合论、可计算性理论与数理逻辑三大领域——成为二十世纪最具穿透力的证明思想之一。",[35,20610,20611],{"id":20611},"延伸",[11,20613,20614],{},"除了上面讨论的三个经典应用，对角线论证及其背后的思想还渗透进计算机科学与数学的许多角落：",[993,20616,20617,20620,20681,20684],{},[996,20618,20619],{},"柯尔莫哥洛夫复杂度。对角线论证可用于证明「不可压缩」字符串的存在——即无法由任何比字符串本身更短的程序生成的字符串——而且这样的字符串构成绝对多数。",[996,20621,20622,20623,20680],{},"复杂度理论中的时间与空间谱系定理。对角线论证被用来证明，赋予图灵机更多的时间或空间预算，确实能让它解决严格更多的问题（这是 ",[48,20624,20626,20646],{"className":20625,"translate":52},[56],[48,20627,20629],{"className":20628},[60],[62,20630,20631],{"xmlns":64},[67,20632,20633,20643],{},[70,20634,20635,20637,20640],{},[73,20636,13159],{},[78,20638,20639],{},"⊊",[4598,20641,20642],{},"EXP",[142,20644,20645],{"encoding":144},"P \\subsetneq \\text{EXP}",[48,20647,20649,20668],{"className":20648,"ariaHidden":150},[149],[48,20650,20652,20656,20659,20662,20665],{"className":20651},[154],[48,20653],{"className":20654,"style":20655},[158],"height:0.8193em;vertical-align:-0.136em;",[48,20657,13159],{"className":20658,"style":242},[163,171],[48,20660],{"className":20661,"style":180},[179],[48,20663,20639],{"className":20664},[184,20053],[48,20666],{"className":20667,"style":180},[179],[48,20669,20671,20674],{"className":20670},[154],[48,20672],{"className":20673,"style":4878},[158],[48,20675,20677],{"className":20676},[163,4646],[48,20678,20642],{"className":20679},[163]," 等谱系关系的基础）。",[996,20682,20683],{},"塔斯基不可定义定理。该结果证明，在足够强的形式系统内，「真」这个概念无法被完整定义；其证明结构与哥德尔定理非常相似。",[996,20685,20686,20687,20827,20828,20977],{},"罗素悖论。虽然从历史上看它早于对角线论证在现代形式下的普遍认知，但「所有不包含自身的集合构成的集合」这个悖论，",[48,20688,20690,20721],{"className":20689,"translate":52},[56],[48,20691,20693],{"className":20692},[60],[62,20694,20695],{"xmlns":64},[67,20696,20697,20718],{},[70,20698,20699,20702,20704,20706,20708,20710,20712,20714,20716],{},[73,20700,20701],{},"R",[78,20703,90],{},[78,20705,5022],{"stretchy":80},[73,20707,84],{},[78,20709,1495],{},[73,20711,84],{},[78,20713,13755],{"mathvariant":75},[73,20715,84],{},[78,20717,5047],{"stretchy":80},[142,20719,20720],{"encoding":144},"R = \\{x : x \\notin x\\}",[48,20722,20724,20743,20764,20815],{"className":20723,"ariaHidden":150},[149],[48,20725,20727,20730,20734,20737,20740],{"className":20726},[154],[48,20728],{"className":20729,"style":4878},[158],[48,20731,20701],{"className":20732,"style":20733},[163,171],"margin-right:0.0077em;",[48,20735],{"className":20736,"style":180},[179],[48,20738,90],{"className":20739},[184],[48,20741],{"className":20742,"style":180},[179],[48,20744,20746,20749,20752,20755,20758,20761],{"className":20745},[154],[48,20747],{"className":20748,"style":159},[158],[48,20750,5022],{"className":20751},[167],[48,20753,84],{"className":20754},[163,171],[48,20756],{"className":20757,"style":180},[179],[48,20759,1495],{"className":20760},[184],[48,20762],{"className":20763,"style":180},[179],[48,20765,20767,20770,20773,20776,20812],{"className":20766},[154],[48,20768],{"className":20769,"style":159},[158],[48,20771,84],{"className":20772},[163,171],[48,20774],{"className":20775,"style":180},[179],[48,20777,20779,20785],{"className":20778},[184],[48,20780,20782],{"className":20781},[163],[48,20783,104],{"className":20784},[184],[48,20786,20788],{"className":20787},[163,5391],[48,20789,20791],{"className":20790},[5395],[48,20792,20794,20797,20809],{"className":20793},[13864],[48,20795],{"className":20796,"style":159},[158],[48,20798,20800],{"className":20799},[5406],[48,20801,20803,20806],{"className":20802},[163],[48,20804,6483],{"className":20805},[163],[48,20807],{"className":20808,"style":4043},[179],[48,20810],{"className":20811},[5417],[48,20813],{"className":20814,"style":180},[179],[48,20816,20818,20821,20824],{"className":20817},[154],[48,20819],{"className":20820,"style":159},[158],[48,20822,84],{"className":20823},[163,171],[48,20825,5047],{"className":20826},[175],"，与康托尔定理证明中的集合 ",[48,20829,20831,20866],{"className":20830,"translate":52},[56],[48,20832,20834],{"className":20833},[60],[62,20835,20836],{"xmlns":64},[67,20837,20838,20864],{},[70,20839,20840,20842,20844,20846,20848,20850,20852,20854,20856,20858,20860,20862],{},[73,20841,1515],{},[78,20843,90],{},[78,20845,5022],{"stretchy":80},[73,20847,15],{},[78,20849,1495],{},[73,20851,15],{},[78,20853,13755],{"mathvariant":75},[73,20855,101],{},[78,20857,81],{"stretchy":80},[73,20859,15],{},[78,20861,87],{"stretchy":80},[78,20863,5047],{"stretchy":80},[142,20865,15653],{"encoding":144},[48,20867,20869,20887,20908,20959],{"className":20868,"ariaHidden":150},[149],[48,20870,20872,20875,20878,20881,20884],{"className":20871},[154],[48,20873],{"className":20874,"style":4878},[158],[48,20876,1515],{"className":20877,"style":1680},[163,171],[48,20879],{"className":20880,"style":180},[179],[48,20882,90],{"className":20883},[184],[48,20885],{"className":20886,"style":180},[179],[48,20888,20890,20893,20896,20899,20902,20905],{"className":20889},[154],[48,20891],{"className":20892,"style":159},[158],[48,20894,5022],{"className":20895},[167],[48,20897,15],{"className":20898},[163,171],[48,20900],{"className":20901,"style":180},[179],[48,20903,1495],{"className":20904},[184],[48,20906],{"className":20907,"style":180},[179],[48,20909,20911,20914,20917,20920,20956],{"className":20910},[154],[48,20912],{"className":20913,"style":159},[158],[48,20915,15],{"className":20916},[163,171],[48,20918],{"className":20919,"style":180},[179],[48,20921,20923,20929],{"className":20922},[184],[48,20924,20926],{"className":20925},[163],[48,20927,104],{"className":20928},[184],[48,20930,20932],{"className":20931},[163,5391],[48,20933,20935],{"className":20934},[5395],[48,20936,20938,20941,20953],{"className":20937},[13864],[48,20939],{"className":20940,"style":159},[158],[48,20942,20944],{"className":20943},[5406],[48,20945,20947,20950],{"className":20946},[163],[48,20948,6483],{"className":20949},[163],[48,20951],{"className":20952,"style":4043},[179],[48,20954],{"className":20955},[5417],[48,20957],{"className":20958,"style":180},[179],[48,20960,20962,20965,20968,20971,20974],{"className":20961},[154],[48,20963],{"className":20964,"style":159},[158],[48,20966,101],{"className":20967,"style":235},[163,171],[48,20969,81],{"className":20970},[167],[48,20972,15],{"className":20973},[163,171],[48,20975,15765],{"className":20976},[175]," 具有完全相同的结构——两者都是「自指 + 自我排除」的产物。",[11,20979,20980],{},"若把对角线论证提炼成一个可复用的「思维模板」，其轮廓大致如下：",[2555,20982,20983,20986,20989,21050],{},[996,20984,20985],{},"出发点（反证）。假设存在一种「完美覆盖」或「普适判定」——一个双射、一个满射、一个判定器、一个证明系统。",[996,20987,20988],{},"构造自指的对象。利用假设本身所提供的「编号」或「映射」能力，构造一个新对象，它刻意与假定的覆盖中的每一个条目「作对」（对角数、对角集、对角程序、自指命题）。",[996,20990,20991,20992,21020,21021,21049],{},"验证其逃逸性。证明这个新对象与假定覆盖范围内的每一个条目都系统性地不同（通常通过「第 ",[48,20993,20995,21008],{"className":20994,"translate":52},[56],[48,20996,20998],{"className":20997},[60],[62,20999,21000],{"xmlns":64},[67,21001,21002,21006],{},[70,21003,21004],{},[73,21005,6221],{},[142,21007,6221],{"encoding":144},[48,21009,21011],{"className":21010,"ariaHidden":150},[149],[48,21012,21014,21017],{"className":21013},[154],[48,21015],{"className":21016,"style":578},[158],[48,21018,6221],{"className":21019},[163,171]," 个条目在第 ",[48,21022,21024,21037],{"className":21023,"translate":52},[56],[48,21025,21027],{"className":21026},[60],[62,21028,21029],{"xmlns":64},[67,21030,21031,21035],{},[70,21032,21033],{},[73,21034,6221],{},[142,21036,6221],{"encoding":144},[48,21038,21040],{"className":21039,"ariaHidden":150},[149],[48,21041,21043,21046],{"className":21042},[154],[48,21044],{"className":21045,"style":578},[158],[48,21047,6221],{"className":21048},[163,171]," 个位置不同」这一模式）。",[996,21051,21052],{},"导出矛盾。按假设，这个新对象应当被覆盖——但构造过程保证了它必然逃脱覆盖。矛盾产生；原假设被推翻。",[21054,21055,21056],"style",{},"html pre.shiki code .snl16, html code.shiki .snl16{--shiki-default:#F97583}html pre.shiki code .svObZ, html code.shiki .svObZ{--shiki-default:#B392F0}html pre.shiki code .s95oV, html code.shiki .s95oV{--shiki-default:#E1E4E8}html pre.shiki code .sDLfK, html code.shiki .sDLfK{--shiki-default:#79B8FF}html pre.shiki code .sU2Wk, html code.shiki .sU2Wk{--shiki-default:#9ECBFF}html pre.shiki code .sAwPA, html code.shiki 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.s9osk{--shiki-default:#FFAB70}",{"title":4787,"searchDepth":4788,"depth":4788,"links":21058},[21059,21060,21061,21062,21065,21066,21067],{"id":4995,"depth":4788,"text":4995},{"id":7028,"depth":4788,"text":7028},{"id":12620,"depth":4788,"text":12620},{"id":13106,"depth":4788,"text":13106,"children":21063},[21064],{"id":13415,"depth":12665,"text":13415},{"id":16858,"depth":4788,"text":16858},{"id":19766,"depth":4788,"text":19766},{"id":20611,"depth":4788,"text":20611},{},"\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en",{"title":4984,"description":4989},"blog\u002F2026\u002F2026-07-31-the-diagonal-argument-en","对角线论证由康托尔于 1891 年提出，通过构造一个能逃脱任何「完备列表或映射」的对象，证明某些无穷严格大于另一些无穷。这一自指技巧同样蕴含于康托尔定理、图灵停机问题与哥德尔不完备定理之中——一个统一了集合论、可计算性与逻辑学的思想。",[21074,21075],"mathematics","computation","2026-07-31T00:00:00+08:00","ZcA-tfQ1zVCr2HL5QHuWWDVkSCGS-RbN98cAuxLrnw0",{"left":21079,"top":21079,"width":21080,"height":21080,"rotate":21079,"vFlip":4799,"hFlip":4799,"body":21081},0,24,"\u003Cg 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