[{"data":1,"prerenderedAt":21090},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en":3,"i-ci:tag":21084,"i-mingcute:time-line":21088},{"post":4,"nextPost":16215,"prevPost":20911},{"id":5,"title":6,"body":7,"description":13,"draft":16202,"enableComment":7941,"extension":16203,"image":7750,"important":16202,"location":16204,"meta":16205,"navigation":7941,"ogImage":16204,"path":16206,"seo":16207,"stem":16208,"summary":16209,"tags":16210,"time":16213,"__hash__":16214},"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":9,"toc":16191},"minimark",[10,14,17,21,24,189,334,907,1069,1189,1192,2125,2128,2131,2190,2245,2248,2400,3842,4049,4300,4727,4905,4937,5122,5504,5813,5908,6057,6141,6321,6596,7735,7738,7741,7744,8056,8059,8066,8228,8231,8234,8461,8537,8541,8718,8845,9038,9041,9292,9347,10220,10396,10399,11247,11574,11577,11889,11891,11897,11987,11990,11993,12113,12116,12119,12435,12438,12470,12810,12931,12978,13387,13423,13620,14462,14637,14640,14878,14880,14886,14896,14899,14902,14907,14910,15496,15499,15740,15743,15746,15749,16112,16115,16187],[11,12,13],"p",{},"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。",[11,15,16],{},"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。",[18,19,20],"h2",{"id":20},"等势",[11,22,23],{},"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？",[11,25,26,27,188],{},"对于有限集，答案很简单：数元素个数即可。但自然数集 ",[28,29,33,98],"span",{"className":30,"translate":32},[31],"katex","no",[28,34,37],{"className":35},[36],"katex-mathml",[38,39,41],"math",{"xmlns":40},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[42,43,44,93],"semantics",{},[45,46,47,52,56,60,64,68,71,73,76,78,81,83,86,90],"mrow",{},[48,49,51],"mi",{"mathvariant":50},"double-struck","N",[53,54,55],"mo",{},"=",[53,57,59],{"stretchy":58},"false","{",[61,62,63],"mn",{},"0",[53,65,67],{"separator":66},"true",",",[61,69,70],{},"1",[53,72,67],{"separator":66},[61,74,75],{},"2",[53,77,67],{"separator":66},[61,79,80],{},"3",[53,82,67],{"separator":66},[53,84,85],{},"…",[87,88,89],"mtext",{}," ",[53,91,92],{"stretchy":58},"}",[94,95,97],"annotation",{"encoding":96},"application\u002Fx-tex","\\mathbb{N} = \\{0, 1, 2, 3, \\dots\\}",[28,99,102,128],{"className":100,"ariaHidden":66},[101],"katex-html",[28,103,106,111,116,121,125],{"className":104},[105],"base",[28,107],{"className":108,"style":110},[109],"strut","height:0.6889em;",[28,112,51],{"className":113},[114,115],"mord","mathbb",[28,117],{"className":118,"style":120},[119],"mspace","margin-right:0.2778em;",[28,122,55],{"className":123},[124],"mrel",[28,126],{"className":127,"style":120},[119],[28,129,131,135,139,142,146,150,153,156,159,162,165,168,171,174,177,181,184],{"className":130},[105],[28,132],{"className":133,"style":134},[109],"height:1em;vertical-align:-0.25em;",[28,136,59],{"className":137},[138],"mopen",[28,140,63],{"className":141},[114],[28,143,67],{"className":144},[145],"mpunct",[28,147],{"className":148,"style":149},[119],"margin-right:0.1667em;",[28,151,70],{"className":152},[114],[28,154,67],{"className":155},[145],[28,157],{"className":158,"style":149},[119],[28,160,75],{"className":161},[114],[28,163,67],{"className":164},[145],[28,166],{"className":167,"style":149},[119],[28,169,80],{"className":170},[114],[28,172,67],{"className":173},[145],[28,175],{"className":176,"style":149},[119],[28,178,85],{"className":179},[180],"minner",[28,182],{"className":183,"style":149},[119],[28,185,92],{"className":186},[187],"mclose"," 是无穷的，无法「数到底」；因此我们需要一种新的工具。",[11,190,191,192,223,224,254,255,333],{},"定义（双射）。给定两个集合 ",[28,193,195,209],{"className":194,"translate":32},[31],[28,196,198],{"className":197},[36],[38,199,200],{"xmlns":40},[42,201,202,207],{},[45,203,204],{},[48,205,206],{},"A",[94,208,206],{"encoding":96},[28,210,212],{"className":211,"ariaHidden":66},[101],[28,213,215,219],{"className":214},[105],[28,216],{"className":217,"style":218},[109],"height:0.6833em;",[28,220,206],{"className":221},[114,222],"mathnormal"," 与 ",[28,225,227,241],{"className":226,"translate":32},[31],[28,228,230],{"className":229},[36],[38,231,232],{"xmlns":40},[42,233,234,239],{},[45,235,236],{},[48,237,238],{},"B",[94,240,238],{"encoding":96},[28,242,244],{"className":243,"ariaHidden":66},[101],[28,245,247,250],{"className":246},[105],[28,248],{"className":249,"style":218},[109],[28,251,238],{"className":252,"style":253},[114,222],"margin-right:0.0502em;","，若存在函数 ",[28,256,258,283],{"className":257,"translate":32},[31],[28,259,261],{"className":260},[36],[38,262,263],{"xmlns":40},[42,264,265,280],{},[45,266,267,270,273,275,278],{},[48,268,269],{},"f",[53,271,272],{},":",[48,274,206],{},[53,276,277],{},"→",[48,279,238],{},[94,281,282],{"encoding":96},"f: A \\to B",[28,284,286,306,324],{"className":285,"ariaHidden":66},[101],[28,287,289,293,297,300,303],{"className":288},[105],[28,290],{"className":291,"style":292},[109],"height:0.8889em;vertical-align:-0.1944em;",[28,294,269],{"className":295,"style":296},[114,222],"margin-right:0.1076em;",[28,298],{"className":299,"style":120},[119],[28,301,272],{"className":302},[124],[28,304],{"className":305,"style":120},[119],[28,307,309,312,315,318,321],{"className":308},[105],[28,310],{"className":311,"style":218},[109],[28,313,206],{"className":314},[114,222],[28,316],{"className":317,"style":120},[119],[28,319,277],{"className":320},[124],[28,322],{"className":323,"style":120},[119],[28,325,327,330],{"className":326},[105],[28,328],{"className":329,"style":218},[109],[28,331,238],{"className":332,"style":253},[114,222]," 满足：",[335,336,337,728],"ul",{},[338,339,340,341,727],"li",{},"单射：",[28,342,344,405],{"className":343,"translate":32},[31],[28,345,347],{"className":346},[36],[38,348,349],{"xmlns":40},[42,350,351,402],{},[45,352,353,361,365,371,374,376,379,385,388,390,392,394,400],{},[354,355,356,359],"msub",{},[48,357,358],{},"a",[61,360,70],{},[53,362,364],{"mathvariant":363},"normal","≠",[354,366,367,369],{},[48,368,358],{},[61,370,75],{},[53,372,373],{},"⇒",[48,375,269],{},[53,377,378],{"stretchy":58},"(",[354,380,381,383],{},[48,382,358],{},[61,384,70],{},[53,386,387],{"stretchy":58},")",[53,389,364],{"mathvariant":363},[48,391,269],{},[53,393,378],{"stretchy":58},[354,395,396,398],{},[48,397,358],{},[61,399,75],{},[53,401,387],{"stretchy":58},[94,403,404],{"encoding":96},"a_1 \\neq a_2 \\Rightarrow f(a_1) \\neq f(a_2)",[28,406,408,519,575,672],{"className":407,"ariaHidden":66},[101],[28,409,411,414,470,473,516],{"className":410},[105],[28,412],{"className":413,"style":292},[109],[28,415,417,420],{"className":416},[114],[28,418,358],{"className":419},[114,222],[28,421,424],{"className":422},[423],"msupsub",[28,425,429,461],{"className":426},[427,428],"vlist-t","vlist-t2",[28,430,433,456],{"className":431},[432],"vlist-r",[28,434,438],{"className":435,"style":437},[436],"vlist","height:0.3011em;",[28,439,441,446],{"style":440},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[28,442],{"className":443,"style":445},[444],"pstrut","height:2.7em;",[28,447,453],{"className":448},[449,450,451,452],"sizing","reset-size6","size3","mtight",[28,454,70],{"className":455},[114,452],[28,457,460],{"className":458},[459],"vlist-s","​",[28,462,464],{"className":463},[432],[28,465,468],{"className":466,"style":467},[436],"height:0.15em;",[28,469],{},[28,471],{"className":472,"style":120},[119],[28,474,476,509,513],{"className":475},[124],[28,477,479],{"className":478},[124],[28,480,483],{"className":481},[114,482],"vbox",[28,484,487],{"className":485},[486],"thinbox",[28,488,491,494,505],{"className":489},[490],"rlap",[28,492],{"className":493,"style":292},[109],[28,495,498],{"className":496},[497],"inner",[28,499,501],{"className":500},[114],[28,502,504],{"className":503},[124],"",[28,506],{"className":507},[508],"fix",[28,510],{"className":511},[119,512],"nobreak",[28,514,55],{"className":515},[124],[28,517],{"className":518,"style":120},[119],[28,520,522,526,566,569,572],{"className":521},[105],[28,523],{"className":524,"style":525},[109],"height:0.5806em;vertical-align:-0.15em;",[28,527,529,532],{"className":528},[114],[28,530,358],{"className":531},[114,222],[28,533,535],{"className":534},[423],[28,536,538,558],{"className":537},[427,428],[28,539,541,555],{"className":540},[432],[28,542,544],{"className":543,"style":437},[436],[28,545,546,549],{"style":440},[28,547],{"className":548,"style":445},[444],[28,550,552],{"className":551},[449,450,451,452],[28,553,75],{"className":554},[114,452],[28,556,460],{"className":557},[459],[28,559,561],{"className":560},[432],[28,562,564],{"className":563,"style":467},[436],[28,565],{},[28,567],{"className":568,"style":120},[119],[28,570,373],{"className":571},[124],[28,573],{"className":574,"style":120},[119],[28,576,578,581,584,587,627,630,633,669],{"className":577},[105],[28,579],{"className":580,"style":134},[109],[28,582,269],{"className":583,"style":296},[114,222],[28,585,378],{"className":586},[138],[28,588,590,593],{"className":589},[114],[28,591,358],{"className":592},[114,222],[28,594,596],{"className":595},[423],[28,597,599,619],{"className":598},[427,428],[28,600,602,616],{"className":601},[432],[28,603,605],{"className":604,"style":437},[436],[28,606,607,610],{"style":440},[28,608],{"className":609,"style":445},[444],[28,611,613],{"className":612},[449,450,451,452],[28,614,70],{"className":615},[114,452],[28,617,460],{"className":618},[459],[28,620,622],{"className":621},[432],[28,623,625],{"className":624,"style":467},[436],[28,626],{},[28,628,387],{"className":629},[187],[28,631],{"className":632,"style":120},[119],[28,634,636,663,666],{"className":635},[124],[28,637,639],{"className":638},[124],[28,640,642],{"className":641},[114,482],[28,643,645],{"className":644},[486],[28,646,648,651,660],{"className":647},[490],[28,649],{"className":650,"style":292},[109],[28,652,654],{"className":653},[497],[28,655,657],{"className":656},[114],[28,658,504],{"className":659},[124],[28,661],{"className":662},[508],[28,664],{"className":665},[119,512],[28,667,55],{"className":668},[124],[28,670],{"className":671,"style":120},[119],[28,673,675,678,681,684,724],{"className":674},[105],[28,676],{"className":677,"style":134},[109],[28,679,269],{"className":680,"style":296},[114,222],[28,682,378],{"className":683},[138],[28,685,687,690],{"className":686},[114],[28,688,358],{"className":689},[114,222],[28,691,693],{"className":692},[423],[28,694,696,716],{"className":695},[427,428],[28,697,699,713],{"className":698},[432],[28,700,702],{"className":701,"style":437},[436],[28,703,704,707],{"style":440},[28,705],{"className":706,"style":445},[444],[28,708,710],{"className":709},[449,450,451,452],[28,711,75],{"className":712},[114,452],[28,714,460],{"className":715},[459],[28,717,719],{"className":718},[432],[28,720,722],{"className":721,"style":467},[436],[28,723],{},[28,725,387],{"className":726},[187],"（不同的输入产生不同的输出）",[338,729,730,731,785,786,838,839,906],{},"满射：对每个 ",[28,732,734,754],{"className":733,"translate":32},[31],[28,735,737],{"className":736},[36],[38,738,739],{"xmlns":40},[42,740,741,751],{},[45,742,743,746,749],{},[48,744,745],{},"b",[53,747,748],{},"∈",[48,750,238],{},[94,752,753],{"encoding":96},"b \\in B",[28,755,757,776],{"className":756,"ariaHidden":66},[101],[28,758,760,764,767,770,773],{"className":759},[105],[28,761],{"className":762,"style":763},[109],"height:0.7335em;vertical-align:-0.0391em;",[28,765,745],{"className":766},[114,222],[28,768],{"className":769,"style":120},[119],[28,771,748],{"className":772},[124],[28,774],{"className":775,"style":120},[119],[28,777,779,782],{"className":778},[105],[28,780],{"className":781,"style":218},[109],[28,783,238],{"className":784,"style":253},[114,222],"，都存在 ",[28,787,789,807],{"className":788,"translate":32},[31],[28,790,792],{"className":791},[36],[38,793,794],{"xmlns":40},[42,795,796,804],{},[45,797,798,800,802],{},[48,799,358],{},[53,801,748],{},[48,803,206],{},[94,805,806],{"encoding":96},"a \\in A",[28,808,810,829],{"className":809,"ariaHidden":66},[101],[28,811,813,817,820,823,826],{"className":812},[105],[28,814],{"className":815,"style":816},[109],"height:0.5782em;vertical-align:-0.0391em;",[28,818,358],{"className":819},[114,222],[28,821],{"className":822,"style":120},[119],[28,824,748],{"className":825},[124],[28,827],{"className":828,"style":120},[119],[28,830,832,835],{"className":831},[105],[28,833],{"className":834,"style":218},[109],[28,836,206],{"className":837},[114,222]," 使得 ",[28,840,842,866],{"className":841,"translate":32},[31],[28,843,845],{"className":844},[36],[38,846,847],{"xmlns":40},[42,848,849,863],{},[45,850,851,853,855,857,859,861],{},[48,852,269],{},[53,854,378],{"stretchy":58},[48,856,358],{},[53,858,387],{"stretchy":58},[53,860,55],{},[48,862,745],{},[94,864,865],{"encoding":96},"f(a) = b",[28,867,869,896],{"className":868,"ariaHidden":66},[101],[28,870,872,875,878,881,884,887,890,893],{"className":871},[105],[28,873],{"className":874,"style":134},[109],[28,876,269],{"className":877,"style":296},[114,222],[28,879,378],{"className":880},[138],[28,882,358],{"className":883},[114,222],[28,885,387],{"className":886},[187],[28,888],{"className":889,"style":120},[119],[28,891,55],{"className":892},[124],[28,894],{"className":895,"style":120},[119],[28,897,899,903],{"className":898},[105],[28,900],{"className":901,"style":902},[109],"height:0.6944em;",[28,904,745],{"className":905},[114,222],"（每个输出都被覆盖）",[11,908,909,910,938,939,223,967,995,996,1068],{},"则称 ",[28,911,913,926],{"className":912,"translate":32},[31],[28,914,916],{"className":915},[36],[38,917,918],{"xmlns":40},[42,919,920,924],{},[45,921,922],{},[48,923,269],{},[94,925,269],{"encoding":96},[28,927,929],{"className":928,"ariaHidden":66},[101],[28,930,932,935],{"className":931},[105],[28,933],{"className":934,"style":292},[109],[28,936,269],{"className":937,"style":296},[114,222]," 为双射，并称 ",[28,940,942,955],{"className":941,"translate":32},[31],[28,943,945],{"className":944},[36],[38,946,947],{"xmlns":40},[42,948,949,953],{},[45,950,951],{},[48,952,206],{},[94,954,206],{"encoding":96},[28,956,958],{"className":957,"ariaHidden":66},[101],[28,959,961,964],{"className":960},[105],[28,962],{"className":963,"style":218},[109],[28,965,206],{"className":966},[114,222],[28,968,970,983],{"className":969,"translate":32},[31],[28,971,973],{"className":972},[36],[38,974,975],{"xmlns":40},[42,976,977,981],{},[45,978,979],{},[48,980,238],{},[94,982,238],{"encoding":96},[28,984,986],{"className":985,"ariaHidden":66},[101],[28,987,989,992],{"className":988},[105],[28,990],{"className":991,"style":218},[109],[28,993,238],{"className":994,"style":253},[114,222]," 等势（记为 ",[28,997,999,1026],{"className":998,"translate":32},[31],[28,1000,1002],{"className":1001},[36],[38,1003,1004],{"xmlns":40},[42,1005,1006,1023],{},[45,1007,1008,1011,1013,1015,1017,1019,1021],{},[48,1009,1010],{"mathvariant":363},"∣",[48,1012,206],{},[48,1014,1010],{"mathvariant":363},[53,1016,55],{},[48,1018,1010],{"mathvariant":363},[48,1020,238],{},[48,1022,1010],{"mathvariant":363},[94,1024,1025],{"encoding":96},"|A| = |B|",[28,1027,1029,1053],{"className":1028,"ariaHidden":66},[101],[28,1030,1032,1035,1038,1041,1044,1047,1050],{"className":1031},[105],[28,1033],{"className":1034,"style":134},[109],[28,1036,1010],{"className":1037},[114],[28,1039,206],{"className":1040},[114,222],[28,1042,1010],{"className":1043},[114],[28,1045],{"className":1046,"style":120},[119],[28,1048,55],{"className":1049},[124],[28,1051],{"className":1052,"style":120},[119],[28,1054,1056,1059,1062,1065],{"className":1055},[105],[28,1057],{"className":1058,"style":134},[109],[28,1060,1010],{"className":1061},[114],[28,1063,238],{"className":1064,"style":253},[114,222],[28,1066,1010],{"className":1067},[114],"）——即两者「大小相等」。",[11,1070,1071,1072,1100,1101,1130,1131,1159,1160,1188],{},"定义（可数集）。若集合 ",[28,1073,1075,1088],{"className":1074,"translate":32},[31],[28,1076,1078],{"className":1077},[36],[38,1079,1080],{"xmlns":40},[42,1081,1082,1086],{},[45,1083,1084],{},[48,1085,206],{},[94,1087,206],{"encoding":96},[28,1089,1091],{"className":1090,"ariaHidden":66},[101],[28,1092,1094,1097],{"className":1093},[105],[28,1095],{"className":1096,"style":218},[109],[28,1098,206],{"className":1099},[114,222]," 能与自然数集 ",[28,1102,1104,1118],{"className":1103,"translate":32},[31],[28,1105,1107],{"className":1106},[36],[38,1108,1109],{"xmlns":40},[42,1110,1111,1115],{},[45,1112,1113],{},[48,1114,51],{"mathvariant":50},[94,1116,1117],{"encoding":96},"\\mathbb{N}",[28,1119,1121],{"className":1120,"ariaHidden":66},[101],[28,1122,1124,1127],{"className":1123},[105],[28,1125],{"className":1126,"style":110},[109],[28,1128,51],{"className":1129},[114,115]," 建立双射（或 ",[28,1132,1134,1147],{"className":1133,"translate":32},[31],[28,1135,1137],{"className":1136},[36],[38,1138,1139],{"xmlns":40},[42,1140,1141,1145],{},[45,1142,1143],{},[48,1144,206],{},[94,1146,206],{"encoding":96},[28,1148,1150],{"className":1149,"ariaHidden":66},[101],[28,1151,1153,1156],{"className":1152},[105],[28,1154],{"className":1155,"style":218},[109],[28,1157,206],{"className":1158},[114,222]," 为有限集），则称 ",[28,1161,1163,1176],{"className":1162,"translate":32},[31],[28,1164,1166],{"className":1165},[36],[38,1167,1168],{"xmlns":40},[42,1169,1170,1174],{},[45,1171,1172],{},[48,1173,206],{},[94,1175,206],{"encoding":96},[28,1177,1179],{"className":1178,"ariaHidden":66},[101],[28,1180,1182,1185],{"className":1181},[105],[28,1183],{"className":1184,"style":218},[109],[28,1186,206],{"className":1187},[114,222]," 为可数集。",[11,1190,1191],{},"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：",[335,1193,1194,1370,2048],{},[338,1195,1196,1197,1295,1296,1369],{},"偶数集 ",[28,1198,1200,1238],{"className":1199,"translate":32},[31],[28,1201,1203],{"className":1202},[36],[38,1204,1205],{"xmlns":40},[42,1206,1207,1235],{},[45,1208,1209,1211,1213,1215,1217,1219,1222,1224,1227,1229,1231,1233],{},[53,1210,59],{"stretchy":58},[61,1212,63],{},[53,1214,67],{"separator":66},[61,1216,75],{},[53,1218,67],{"separator":66},[61,1220,1221],{},"4",[53,1223,67],{"separator":66},[61,1225,1226],{},"6",[53,1228,67],{"separator":66},[53,1230,85],{},[87,1232,89],{},[53,1234,92],{"stretchy":58},[94,1236,1237],{"encoding":96},"\\{0, 2, 4, 6, \\dots\\}",[28,1239,1241],{"className":1240,"ariaHidden":66},[101],[28,1242,1244,1247,1250,1253,1256,1259,1262,1265,1268,1271,1274,1277,1280,1283,1286,1289,1292],{"className":1243},[105],[28,1245],{"className":1246,"style":134},[109],[28,1248,59],{"className":1249},[138],[28,1251,63],{"className":1252},[114],[28,1254,67],{"className":1255},[145],[28,1257],{"className":1258,"style":149},[119],[28,1260,75],{"className":1261},[114],[28,1263,67],{"className":1264},[145],[28,1266],{"className":1267,"style":149},[119],[28,1269,1221],{"className":1270},[114],[28,1272,67],{"className":1273},[145],[28,1275],{"className":1276,"style":149},[119],[28,1278,1226],{"className":1279},[114],[28,1281,67],{"className":1282},[145],[28,1284],{"className":1285,"style":149},[119],[28,1287,85],{"className":1288},[180],[28,1290],{"className":1291,"style":149},[119],[28,1293,92],{"className":1294},[187]," 可数：双射为 ",[28,1297,1299,1326],{"className":1298,"translate":32},[31],[28,1300,1302],{"className":1301},[36],[38,1303,1304],{"xmlns":40},[42,1305,1306,1323],{},[45,1307,1308,1310,1312,1315,1317,1319,1321],{},[48,1309,269],{},[53,1311,378],{"stretchy":58},[48,1313,1314],{},"n",[53,1316,387],{"stretchy":58},[53,1318,55],{},[61,1320,75],{},[48,1322,1314],{},[94,1324,1325],{"encoding":96},"f(n) = 2n",[28,1327,1329,1356],{"className":1328,"ariaHidden":66},[101],[28,1330,1332,1335,1338,1341,1344,1347,1350,1353],{"className":1331},[105],[28,1333],{"className":1334,"style":134},[109],[28,1336,269],{"className":1337,"style":296},[114,222],[28,1339,378],{"className":1340},[138],[28,1342,1314],{"className":1343},[114,222],[28,1345,387],{"className":1346},[187],[28,1348],{"className":1349,"style":120},[119],[28,1351,55],{"className":1352},[124],[28,1354],{"className":1355,"style":120},[119],[28,1357,1359,1363,1366],{"className":1358},[105],[28,1360],{"className":1361,"style":1362},[109],"height:0.6444em;",[28,1364,75],{"className":1365},[114],[28,1367,1314],{"className":1368},[114,222],"。",[338,1371,1372,1373,1532,1533,1847,1848,1943,1944,1369],{},"整数集 ",[28,1374,1376,1430],{"className":1375,"translate":32},[31],[28,1377,1379],{"className":1378},[36],[38,1380,1381],{"xmlns":40},[42,1382,1383,1427],{},[45,1384,1385,1388,1390,1392,1394,1396,1399,1401,1403,1405,1407,1409,1411,1413,1415,1417,1419,1421,1423,1425],{},[48,1386,1387],{"mathvariant":50},"Z",[53,1389,55],{},[53,1391,59],{"stretchy":58},[53,1393,85],{},[53,1395,67],{"separator":66},[53,1397,1398],{},"−",[61,1400,75],{},[53,1402,67],{"separator":66},[53,1404,1398],{},[61,1406,70],{},[53,1408,67],{"separator":66},[61,1410,63],{},[53,1412,67],{"separator":66},[61,1414,70],{},[53,1416,67],{"separator":66},[61,1418,75],{},[53,1420,67],{"separator":66},[53,1422,85],{},[87,1424,89],{},[53,1426,92],{"stretchy":58},[94,1428,1429],{"encoding":96},"\\mathbb{Z} = \\{\\dots, -2, -1, 0, 1, 2, \\dots\\}",[28,1431,1433,1451],{"className":1432,"ariaHidden":66},[101],[28,1434,1436,1439,1442,1445,1448],{"className":1435},[105],[28,1437],{"className":1438,"style":110},[109],[28,1440,1387],{"className":1441},[114,115],[28,1443],{"className":1444,"style":120},[119],[28,1446,55],{"className":1447},[124],[28,1449],{"className":1450,"style":120},[119],[28,1452,1454,1457,1460,1463,1466,1469,1472,1475,1478,1481,1484,1487,1490,1493,1496,1499,1502,1505,1508,1511,1514,1517,1520,1523,1526,1529],{"className":1453},[105],[28,1455],{"className":1456,"style":134},[109],[28,1458,59],{"className":1459},[138],[28,1461,85],{"className":1462},[180],[28,1464],{"className":1465,"style":149},[119],[28,1467,67],{"className":1468},[145],[28,1470],{"className":1471,"style":149},[119],[28,1473,1398],{"className":1474},[114],[28,1476,75],{"className":1477},[114],[28,1479,67],{"className":1480},[145],[28,1482],{"className":1483,"style":149},[119],[28,1485,1398],{"className":1486},[114],[28,1488,70],{"className":1489},[114],[28,1491,67],{"className":1492},[145],[28,1494],{"className":1495,"style":149},[119],[28,1497,63],{"className":1498},[114],[28,1500,67],{"className":1501},[145],[28,1503],{"className":1504,"style":149},[119],[28,1506,70],{"className":1507},[114],[28,1509,67],{"className":1510},[145],[28,1512],{"className":1513,"style":149},[119],[28,1515,75],{"className":1516},[114],[28,1518,67],{"className":1519},[145],[28,1521],{"className":1522,"style":149},[119],[28,1524,85],{"className":1525},[180],[28,1527],{"className":1528,"style":149},[119],[28,1530,92],{"className":1531},[187]," 可数：可以用 ",[28,1534,1536,1633],{"className":1535,"translate":32},[31],[28,1537,1539],{"className":1538},[36],[38,1540,1541],{"xmlns":40},[42,1542,1543,1630],{},[45,1544,1545,1547,1549,1551,1553,1555],{},[48,1546,269],{},[53,1548,378],{"stretchy":58},[48,1550,1314],{},[53,1552,387],{"stretchy":58},[53,1554,55],{},[45,1556,1557,1559],{},[53,1558,59],{"fence":66},[1560,1561,1565,1594],"mtable",{"rowspacing":1562,"columnalign":1563,"columnspacing":1564},"0.36em","left left","1em",[1566,1567,1568,1583],"mtr",{},[1569,1570,1571],"mtd",{},[1572,1573,1574],"mstyle",{"scriptlevel":63,"displaystyle":58},[45,1575,1576,1578,1581],{},[48,1577,1314],{},[48,1579,1580],{"mathvariant":363},"\u002F",[61,1582,75],{},[1569,1584,1585],{},[1572,1586,1587],{"scriptlevel":63,"displaystyle":58},[45,1588,1589,1591],{},[48,1590,1314],{},[87,1592,1593],{}," 为偶数",[1566,1595,1596,1619],{},[1569,1597,1598],{},[1572,1599,1600],{"scriptlevel":63,"displaystyle":58},[45,1601,1602,1604,1606,1608,1611,1613,1615,1617],{},[53,1603,1398],{},[53,1605,378],{"stretchy":58},[48,1607,1314],{},[53,1609,1610],{},"+",[61,1612,70],{},[53,1614,387],{"stretchy":58},[48,1616,1580],{"mathvariant":363},[61,1618,75],{},[1569,1620,1621],{},[1572,1622,1623],{"scriptlevel":63,"displaystyle":58},[45,1624,1625,1627],{},[48,1626,1314],{},[87,1628,1629],{}," 为奇数",[94,1631,1632],{"encoding":96},"f(n) = \\begin{cases} n\u002F2 & n \\text{ 为偶数} \\\\ -(n+1)\u002F2 & n \\text{ 为奇数} \\end{cases}",[28,1634,1636,1663],{"className":1635,"ariaHidden":66},[101],[28,1637,1639,1642,1645,1648,1651,1654,1657,1660],{"className":1638},[105],[28,1640],{"className":1641,"style":134},[109],[28,1643,269],{"className":1644,"style":296},[114,222],[28,1646,378],{"className":1647},[138],[28,1649,1314],{"className":1650},[114,222],[28,1652,387],{"className":1653},[187],[28,1655],{"className":1656,"style":120},[119],[28,1658,55],{"className":1659},[124],[28,1661],{"className":1662,"style":120},[119],[28,1664,1666,1670],{"className":1665},[105],[28,1667],{"className":1668,"style":1669},[109],"height:3em;vertical-align:-1.25em;",[28,1671,1673,1683,1843],{"className":1672},[180],[28,1674,1678],{"className":1675,"style":1677},[138,1676],"delimcenter","top:0em;",[28,1679,59],{"className":1680},[1681,1682],"delimsizing","size4",[28,1684,1686],{"className":1685},[114],[28,1687,1689,1770,1775],{"className":1688},[1560],[28,1690,1693],{"className":1691},[1692],"col-align-l",[28,1694,1696,1761],{"className":1695},[427,428],[28,1697,1699,1758],{"className":1698},[432],[28,1700,1703,1720],{"className":1701,"style":1702},[436],"height:1.69em;",[28,1704,1706,1710],{"style":1705},"top:-3.69em;",[28,1707],{"className":1708,"style":1709},[444],"height:3.008em;",[28,1711,1713,1716],{"className":1712},[114],[28,1714,1314],{"className":1715},[114,222],[28,1717,1719],{"className":1718},[114],"\u002F2",[28,1721,1723,1726],{"style":1722},"top:-2.25em;",[28,1724],{"className":1725,"style":1709},[444],[28,1727,1729,1732,1735,1738,1742,1746,1749,1752,1755],{"className":1728},[114],[28,1730,1398],{"className":1731},[114],[28,1733,378],{"className":1734},[138],[28,1736,1314],{"className":1737},[114,222],[28,1739],{"className":1740,"style":1741},[119],"margin-right:0.2222em;",[28,1743,1610],{"className":1744},[1745],"mbin",[28,1747],{"className":1748,"style":1741},[119],[28,1750,70],{"className":1751},[114],[28,1753,387],{"className":1754},[187],[28,1756,1719],{"className":1757},[114],[28,1759,460],{"className":1760},[459],[28,1762,1764],{"className":1763},[432],[28,1765,1768],{"className":1766,"style":1767},[436],"height:1.19em;",[28,1769],{},[28,1771],{"className":1772,"style":1774},[1773],"arraycolsep","width:1em;",[28,1776,1778],{"className":1777},[1692],[28,1779,1781,1835],{"className":1780},[427,428],[28,1782,1784,1832],{"className":1783},[432],[28,1785,1787,1811],{"className":1786,"style":1702},[436],[28,1788,1789,1792],{"style":1705},[28,1790],{"className":1791,"style":1709},[444],[28,1793,1795,1798],{"className":1794},[114],[28,1796,1314],{"className":1797},[114,222],[28,1799,1802,1806],{"className":1800},[114,1801],"text",[28,1803,1805],{"className":1804},[114]," ",[28,1807,1810],{"className":1808},[114,1809],"cjk_fallback","为偶数",[28,1812,1813,1816],{"style":1722},[28,1814],{"className":1815,"style":1709},[444],[28,1817,1819,1822],{"className":1818},[114],[28,1820,1314],{"className":1821},[114,222],[28,1823,1825,1828],{"className":1824},[114,1801],[28,1826,1805],{"className":1827},[114],[28,1829,1831],{"className":1830},[114,1809],"为奇数",[28,1833,460],{"className":1834},[459],[28,1836,1838],{"className":1837},[432],[28,1839,1841],{"className":1840,"style":1767},[436],[28,1842],{},[28,1844],{"className":1845},[187,1846],"nulldelimiter"," 把 ",[28,1849,1851,1885],{"className":1850,"translate":32},[31],[28,1852,1854],{"className":1853},[36],[38,1855,1856],{"xmlns":40},[42,1857,1858,1882],{},[45,1859,1860,1862,1864,1866,1868,1870,1872,1874,1876,1878,1880],{},[61,1861,63],{},[53,1863,67],{"separator":66},[61,1865,70],{},[53,1867,67],{"separator":66},[61,1869,75],{},[53,1871,67],{"separator":66},[61,1873,80],{},[53,1875,67],{"separator":66},[61,1877,1221],{},[53,1879,67],{"separator":66},[53,1881,85],{},[94,1883,1884],{"encoding":96},"0,1,2,3,4,\\dots",[28,1886,1888],{"className":1887,"ariaHidden":66},[101],[28,1889,1891,1895,1898,1901,1904,1907,1910,1913,1916,1919,1922,1925,1928,1931,1934,1937,1940],{"className":1890},[105],[28,1892],{"className":1893,"style":1894},[109],"height:0.8389em;vertical-align:-0.1944em;",[28,1896,63],{"className":1897},[114],[28,1899,67],{"className":1900},[145],[28,1902],{"className":1903,"style":149},[119],[28,1905,70],{"className":1906},[114],[28,1908,67],{"className":1909},[145],[28,1911],{"className":1912,"style":149},[119],[28,1914,75],{"className":1915},[114],[28,1917,67],{"className":1918},[145],[28,1920],{"className":1921,"style":149},[119],[28,1923,80],{"className":1924},[114],[28,1926,67],{"className":1927},[145],[28,1929],{"className":1930,"style":149},[119],[28,1932,1221],{"className":1933},[114],[28,1935,67],{"className":1936},[145],[28,1938],{"className":1939,"style":149},[119],[28,1941,85],{"className":1942},[180]," 映射到 ",[28,1945,1947,1985],{"className":1946,"translate":32},[31],[28,1948,1950],{"className":1949},[36],[38,1951,1952],{"xmlns":40},[42,1953,1954,1982],{},[45,1955,1956,1958,1960,1962,1964,1966,1968,1970,1972,1974,1976,1978,1980],{},[61,1957,63],{},[53,1959,67],{"separator":66},[53,1961,1398],{},[61,1963,70],{},[53,1965,67],{"separator":66},[61,1967,70],{},[53,1969,67],{"separator":66},[53,1971,1398],{},[61,1973,75],{},[53,1975,67],{"separator":66},[61,1977,75],{},[53,1979,67],{"separator":66},[53,1981,85],{},[94,1983,1984],{"encoding":96},"0,-1,1,-2,2,\\dots",[28,1986,1988],{"className":1987,"ariaHidden":66},[101],[28,1989,1991,1994,1997,2000,2003,2006,2009,2012,2015,2018,2021,2024,2027,2030,2033,2036,2039,2042,2045],{"className":1990},[105],[28,1992],{"className":1993,"style":1894},[109],[28,1995,63],{"className":1996},[114],[28,1998,67],{"className":1999},[145],[28,2001],{"className":2002,"style":149},[119],[28,2004,1398],{"className":2005},[114],[28,2007,70],{"className":2008},[114],[28,2010,67],{"className":2011},[145],[28,2013],{"className":2014,"style":149},[119],[28,2016,70],{"className":2017},[114],[28,2019,67],{"className":2020},[145],[28,2022],{"className":2023,"style":149},[119],[28,2025,1398],{"className":2026},[114],[28,2028,75],{"className":2029},[114],[28,2031,67],{"className":2032},[145],[28,2034],{"className":2035,"style":149},[119],[28,2037,75],{"className":2038},[114],[28,2040,67],{"className":2041},[145],[28,2043],{"className":2044,"style":149},[119],[28,2046,85],{"className":2047},[180],[338,2049,2050,2051,2082,2083,2124],{},"有理数集 ",[28,2052,2054,2069],{"className":2053,"translate":32},[31],[28,2055,2057],{"className":2056},[36],[38,2058,2059],{"xmlns":40},[42,2060,2061,2066],{},[45,2062,2063],{},[48,2064,2065],{"mathvariant":50},"Q",[94,2067,2068],{"encoding":96},"\\mathbb{Q}",[28,2070,2072],{"className":2071,"ariaHidden":66},[101],[28,2073,2075,2079],{"className":2074},[105],[28,2076],{"className":2077,"style":2078},[109],"height:0.8556em;vertical-align:-0.1667em;",[28,2080,2065],{"className":2081},[114,115]," 同样可数（可以用「对角线枚举法」把所有分数 ",[28,2084,2086,2105],{"className":2085,"translate":32},[31],[28,2087,2089],{"className":2088},[36],[38,2090,2091],{"xmlns":40},[42,2092,2093,2102],{},[45,2094,2095,2097,2099],{},[48,2096,11],{},[48,2098,1580],{"mathvariant":363},[48,2100,2101],{},"q",[94,2103,2104],{"encoding":96},"p\u002Fq",[28,2106,2108],{"className":2107,"ariaHidden":66},[101],[28,2109,2111,2114,2117,2120],{"className":2110},[105],[28,2112],{"className":2113,"style":134},[109],[28,2115,11],{"className":2116},[114,222],[28,2118,1580],{"className":2119},[114],[28,2121,2101],{"className":2122,"style":2123},[114,222],"margin-right:0.0359em;"," 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）。",[11,2126,2127],{},"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。",[18,2129,2130],{"id":2130},"实数的不可数",[2132,2133,2134],"blockquote",{},[11,2135,2136,2137,2189],{},"定理（康托尔，1891）。区间 ",[28,2138,2140,2162],{"className":2139,"translate":32},[31],[28,2141,2143],{"className":2142},[36],[38,2144,2145],{"xmlns":40},[42,2146,2147,2159],{},[45,2148,2149,2151,2153,2155,2157],{},[53,2150,378],{"stretchy":58},[61,2152,63],{},[53,2154,67],{"separator":66},[61,2156,70],{},[53,2158,387],{"stretchy":58},[94,2160,2161],{"encoding":96},"(0,1)",[28,2163,2165],{"className":2164,"ariaHidden":66},[101],[28,2166,2168,2171,2174,2177,2180,2183,2186],{"className":2167},[105],[28,2169],{"className":2170,"style":134},[109],[28,2172,378],{"className":2173},[138],[28,2175,63],{"className":2176},[114],[28,2178,67],{"className":2179},[145],[28,2181],{"className":2182,"style":149},[119],[28,2184,70],{"className":2185},[114],[28,2187,387],{"className":2188},[187]," 中的实数是不可数的。",[11,2191,2192,2193,2244],{},"这意味着：不存在任何方法能把 ",[28,2194,2196,2217],{"className":2195,"translate":32},[31],[28,2197,2199],{"className":2198},[36],[38,2200,2201],{"xmlns":40},[42,2202,2203,2215],{},[45,2204,2205,2207,2209,2211,2213],{},[53,2206,378],{"stretchy":58},[61,2208,63],{},[53,2210,67],{"separator":66},[61,2212,70],{},[53,2214,387],{"stretchy":58},[94,2216,2161],{"encoding":96},[28,2218,2220],{"className":2219,"ariaHidden":66},[101],[28,2221,2223,2226,2229,2232,2235,2238,2241],{"className":2222},[105],[28,2224],{"className":2225,"style":134},[109],[28,2227,378],{"className":2228},[138],[28,2230,63],{"className":2231},[114],[28,2233,67],{"className":2234},[145],[28,2236],{"className":2237,"style":149},[119],[28,2239,70],{"className":2240},[114],[28,2242,387],{"className":2243},[187]," 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。",[11,2246,2247],{},"我们采用反证法。",[11,2249,2250,2251,2302,2303,2399],{},"假设：",[28,2252,2254,2275],{"className":2253,"translate":32},[31],[28,2255,2257],{"className":2256},[36],[38,2258,2259],{"xmlns":40},[42,2260,2261,2273],{},[45,2262,2263,2265,2267,2269,2271],{},[53,2264,378],{"stretchy":58},[61,2266,63],{},[53,2268,67],{"separator":66},[61,2270,70],{},[53,2272,387],{"stretchy":58},[94,2274,2161],{"encoding":96},[28,2276,2278],{"className":2277,"ariaHidden":66},[101],[28,2279,2281,2284,2287,2290,2293,2296,2299],{"className":2280},[105],[28,2282],{"className":2283,"style":134},[109],[28,2285,378],{"className":2286},[138],[28,2288,63],{"className":2289},[114],[28,2291,67],{"className":2292},[145],[28,2294],{"className":2295,"style":149},[119],[28,2297,70],{"className":2298},[114],[28,2300,387],{"className":2301},[187]," 中的实数可数。即存在双射 ",[28,2304,2306,2336],{"className":2305,"translate":32},[31],[28,2307,2309],{"className":2308},[36],[38,2310,2311],{"xmlns":40},[42,2312,2313,2333],{},[45,2314,2315,2317,2319,2321,2323,2325,2327,2329,2331],{},[48,2316,269],{},[53,2318,272],{},[48,2320,51],{"mathvariant":50},[53,2322,277],{},[53,2324,378],{"stretchy":58},[61,2326,63],{},[53,2328,67],{"separator":66},[61,2330,70],{},[53,2332,387],{"stretchy":58},[94,2334,2335],{"encoding":96},"f: \\mathbb{N} \\to (0,1)",[28,2337,2339,2357,2375],{"className":2338,"ariaHidden":66},[101],[28,2340,2342,2345,2348,2351,2354],{"className":2341},[105],[28,2343],{"className":2344,"style":292},[109],[28,2346,269],{"className":2347,"style":296},[114,222],[28,2349],{"className":2350,"style":120},[119],[28,2352,272],{"className":2353},[124],[28,2355],{"className":2356,"style":120},[119],[28,2358,2360,2363,2366,2369,2372],{"className":2359},[105],[28,2361],{"className":2362,"style":110},[109],[28,2364,51],{"className":2365},[114,115],[28,2367],{"className":2368,"style":120},[119],[28,2370,277],{"className":2371},[124],[28,2373],{"className":2374,"style":120},[119],[28,2376,2378,2381,2384,2387,2390,2393,2396],{"className":2377},[105],[28,2379],{"className":2380,"style":134},[109],[28,2382,378],{"className":2383},[138],[28,2385,63],{"className":2386},[114],[28,2388,67],{"className":2389},[145],[28,2391],{"className":2392,"style":149},[119],[28,2394,70],{"className":2395},[114],[28,2397,387],{"className":2398},[187],"，使得我们可以把这些实数全部排成一个无穷列表：",[28,2401,2404],{"className":2402,"translate":32},[2403],"katex-display",[28,2405,2407,2697],{"className":2406,"translate":32},[31],[28,2408,2410],{"className":2409},[36],[38,2411,2413],{"xmlns":40,"display":2412},"block",[42,2414,2415,2694],{},[1560,2416,2420,2483,2543,2603,2663],{"rowspacing":2417,"columnalign":2418,"columnspacing":2419},"0.25em","right left","0em",[1566,2421,2422,2433],{},[1569,2423,2424],{},[1572,2425,2426],{"scriptlevel":63,"displaystyle":66},[354,2427,2428,2431],{},[48,2429,2430],{},"x",[61,2432,70],{},[1569,2434,2435],{},[1572,2436,2437],{"scriptlevel":63,"displaystyle":66},[45,2438,2439,2441,2443,2446,2454,2456,2463,2465,2472,2474,2481],{},[45,2440],{},[53,2442,55],{},[61,2444,2445],{},"0.",[354,2447,2448,2451],{},[48,2449,2450],{},"d",[61,2452,2453],{},"11",[87,2455,89],{},[354,2457,2458,2460],{},[48,2459,2450],{},[61,2461,2462],{},"12",[87,2464,89],{},[354,2466,2467,2469],{},[48,2468,2450],{},[61,2470,2471],{},"13",[87,2473,89],{},[354,2475,2476,2478],{},[48,2477,2450],{},[61,2479,2480],{},"14",[53,2482,85],{},[1566,2484,2485,2495],{},[1569,2486,2487],{},[1572,2488,2489],{"scriptlevel":63,"displaystyle":66},[354,2490,2491,2493],{},[48,2492,2430],{},[61,2494,75],{},[1569,2496,2497],{},[1572,2498,2499],{"scriptlevel":63,"displaystyle":66},[45,2500,2501,2503,2505,2507,2514,2516,2523,2525,2532,2534,2541],{},[45,2502],{},[53,2504,55],{},[61,2506,2445],{},[354,2508,2509,2511],{},[48,2510,2450],{},[61,2512,2513],{},"21",[87,2515,89],{},[354,2517,2518,2520],{},[48,2519,2450],{},[61,2521,2522],{},"22",[87,2524,89],{},[354,2526,2527,2529],{},[48,2528,2450],{},[61,2530,2531],{},"23",[87,2533,89],{},[354,2535,2536,2538],{},[48,2537,2450],{},[61,2539,2540],{},"24",[53,2542,85],{},[1566,2544,2545,2555],{},[1569,2546,2547],{},[1572,2548,2549],{"scriptlevel":63,"displaystyle":66},[354,2550,2551,2553],{},[48,2552,2430],{},[61,2554,80],{},[1569,2556,2557],{},[1572,2558,2559],{"scriptlevel":63,"displaystyle":66},[45,2560,2561,2563,2565,2567,2574,2576,2583,2585,2592,2594,2601],{},[45,2562],{},[53,2564,55],{},[61,2566,2445],{},[354,2568,2569,2571],{},[48,2570,2450],{},[61,2572,2573],{},"31",[87,2575,89],{},[354,2577,2578,2580],{},[48,2579,2450],{},[61,2581,2582],{},"32",[87,2584,89],{},[354,2586,2587,2589],{},[48,2588,2450],{},[61,2590,2591],{},"33",[87,2593,89],{},[354,2595,2596,2598],{},[48,2597,2450],{},[61,2599,2600],{},"34",[53,2602,85],{},[1566,2604,2605,2615],{},[1569,2606,2607],{},[1572,2608,2609],{"scriptlevel":63,"displaystyle":66},[354,2610,2611,2613],{},[48,2612,2430],{},[61,2614,1221],{},[1569,2616,2617],{},[1572,2618,2619],{"scriptlevel":63,"displaystyle":66},[45,2620,2621,2623,2625,2627,2634,2636,2643,2645,2652,2654,2661],{},[45,2622],{},[53,2624,55],{},[61,2626,2445],{},[354,2628,2629,2631],{},[48,2630,2450],{},[61,2632,2633],{},"41",[87,2635,89],{},[354,2637,2638,2640],{},[48,2639,2450],{},[61,2641,2642],{},"42",[87,2644,89],{},[354,2646,2647,2649],{},[48,2648,2450],{},[61,2650,2651],{},"43",[87,2653,89],{},[354,2655,2656,2658],{},[48,2657,2450],{},[61,2659,2660],{},"44",[53,2662,85],{},[1566,2664,2665,2671],{},[1569,2666,2667],{},[1572,2668,2669],{"scriptlevel":63,"displaystyle":66},[45,2670],{},[1569,2672,2673],{},[1572,2674,2675],{"scriptlevel":63,"displaystyle":66},[45,2676,2677,2679,2682],{},[45,2678],{},[87,2680,2681],{},"  ",[45,2683,2684,2687],{},[48,2685,2686],{"mathvariant":363},"⋮",[2688,2689,2690],"mpadded",{"height":2419,"voffset":2419},[119,2691],{"mathbackground":2692,"width":2419,"height":2693},"black","1.5em",[94,2695,2696],{"encoding":96},"\\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&\\ \\ \\vdots\n\\end{aligned}",[28,2698,2700],{"className":2699,"ariaHidden":66},[101],[28,2701,2703,2707],{"className":2702},[105],[28,2704],{"className":2705,"style":2706},[109],"height:7.86em;vertical-align:-3.68em;",[28,2708,2710],{"className":2709},[114],[28,2711,2713,2945],{"className":2712},[1560],[28,2714,2717],{"className":2715},[2716],"col-align-r",[28,2718,2720,2936],{"className":2719},[427,428],[28,2721,2723,2933],{"className":2722},[432],[28,2724,2727,2777,2826,2875,2924],{"className":2725,"style":2726},[436],"height:4.18em;",[28,2728,2730,2734],{"style":2729},"top:-6.84em;",[28,2731],{"className":2732,"style":2733},[444],"height:3.5em;",[28,2735,2737],{"className":2736},[114],[28,2738,2740,2743],{"className":2739},[114],[28,2741,2430],{"className":2742},[114,222],[28,2744,2746],{"className":2745},[423],[28,2747,2749,2769],{"className":2748},[427,428],[28,2750,2752,2766],{"className":2751},[432],[28,2753,2755],{"className":2754,"style":437},[436],[28,2756,2757,2760],{"style":440},[28,2758],{"className":2759,"style":445},[444],[28,2761,2763],{"className":2762},[449,450,451,452],[28,2764,70],{"className":2765},[114,452],[28,2767,460],{"className":2768},[459],[28,2770,2772],{"className":2771},[432],[28,2773,2775],{"className":2774,"style":467},[436],[28,2776],{},[28,2778,2780,2783],{"style":2779},"top:-5.34em;",[28,2781],{"className":2782,"style":2733},[444],[28,2784,2786],{"className":2785},[114],[28,2787,2789,2792],{"className":2788},[114],[28,2790,2430],{"className":2791},[114,222],[28,2793,2795],{"className":2794},[423],[28,2796,2798,2818],{"className":2797},[427,428],[28,2799,2801,2815],{"className":2800},[432],[28,2802,2804],{"className":2803,"style":437},[436],[28,2805,2806,2809],{"style":440},[28,2807],{"className":2808,"style":445},[444],[28,2810,2812],{"className":2811},[449,450,451,452],[28,2813,75],{"className":2814},[114,452],[28,2816,460],{"className":2817},[459],[28,2819,2821],{"className":2820},[432],[28,2822,2824],{"className":2823,"style":467},[436],[28,2825],{},[28,2827,2829,2832],{"style":2828},"top:-3.84em;",[28,2830],{"className":2831,"style":2733},[444],[28,2833,2835],{"className":2834},[114],[28,2836,2838,2841],{"className":2837},[114],[28,2839,2430],{"className":2840},[114,222],[28,2842,2844],{"className":2843},[423],[28,2845,2847,2867],{"className":2846},[427,428],[28,2848,2850,2864],{"className":2849},[432],[28,2851,2853],{"className":2852,"style":437},[436],[28,2854,2855,2858],{"style":440},[28,2856],{"className":2857,"style":445},[444],[28,2859,2861],{"className":2860},[449,450,451,452],[28,2862,80],{"className":2863},[114,452],[28,2865,460],{"className":2866},[459],[28,2868,2870],{"className":2869},[432],[28,2871,2873],{"className":2872,"style":467},[436],[28,2874],{},[28,2876,2878,2881],{"style":2877},"top:-2.34em;",[28,2879],{"className":2880,"style":2733},[444],[28,2882,2884],{"className":2883},[114],[28,2885,2887,2890],{"className":2886},[114],[28,2888,2430],{"className":2889},[114,222],[28,2891,2893],{"className":2892},[423],[28,2894,2896,2916],{"className":2895},[427,428],[28,2897,2899,2913],{"className":2898},[432],[28,2900,2902],{"className":2901,"style":437},[436],[28,2903,2904,2907],{"style":440},[28,2905],{"className":2906,"style":445},[444],[28,2908,2910],{"className":2909},[449,450,451,452],[28,2911,1221],{"className":2912},[114,452],[28,2914,460],{"className":2915},[459],[28,2917,2919],{"className":2918},[432],[28,2920,2922],{"className":2921,"style":467},[436],[28,2923],{},[28,2925,2927,2930],{"style":2926},"top:-0.18em;",[28,2928],{"className":2929,"style":2733},[444],[28,2931],{"className":2932},[114],[28,2934,460],{"className":2935},[459],[28,2937,2939],{"className":2938},[432],[28,2940,2943],{"className":2941,"style":2942},[436],"height:3.68em;",[28,2944],{},[28,2946,2948],{"className":2947},[1692],[28,2949,2951,3834],{"className":2950},[427,428],[28,2952,2954,3831],{"className":2953},[432],[28,2955,2957,3169,3380,3591,3802],{"className":2956,"style":2726},[436],[28,2958,2960,2964],{"style":2959},"top:-7.0275em;",[28,2961],{"className":2962,"style":2963},[444],"height:3.6875em;",[28,2965,2967,2970,2973,2976,2979,2982,3025,3028,3071,3074,3117,3120,3163,3166],{"className":2966},[114],[28,2968],{"className":2969},[114],[28,2971],{"className":2972,"style":120},[119],[28,2974,55],{"className":2975},[124],[28,2977],{"className":2978,"style":120},[119],[28,2980,2445],{"className":2981},[114],[28,2983,2985,2988],{"className":2984},[114],[28,2986,2450],{"className":2987},[114,222],[28,2989,2991],{"className":2990},[423],[28,2992,2994,3017],{"className":2993},[427,428],[28,2995,2997,3014],{"className":2996},[432],[28,2998,3000],{"className":2999,"style":437},[436],[28,3001,3002,3005],{"style":440},[28,3003],{"className":3004,"style":445},[444],[28,3006,3008],{"className":3007},[449,450,451,452],[28,3009,3011],{"className":3010},[114,452],[28,3012,2453],{"className":3013},[114,452],[28,3015,460],{"className":3016},[459],[28,3018,3020],{"className":3019},[432],[28,30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",[28,3846,3848,3872],{"className":3847,"translate":32},[31],[28,3849,3851],{"className":3850},[36],[38,3852,3853],{"xmlns":40},[42,3854,3855,3869],{},[45,3856,3857],{},[354,3858,3859,3861],{},[48,3860,2450],{},[45,3862,3863,3866],{},[48,3864,3865],{},"i",[48,3867,3868],{},"j",[94,3870,3871],{"encoding":96},"d_{ij}",[28,3873,3875],{"className":3874,"ariaHidden":66},[101],[28,3876,3878,3882],{"className":3877},[105],[28,3879],{"className":3880,"style":3881},[109],"height:0.9805em;vertical-align:-0.2861em;",[28,3883,3885,3888],{"className":3884},[114],[28,3886,2450],{"className":3887},[114,222],[28,3889,3891],{"className":3890},[423],[28,3892,3894,3920],{"className":3893},[427,428],[28,3895,3897,3917],{"className":3896},[432],[28,3898,3901],{"className":3899,"style":3900},[436],"height:0.3117em;",[28,3902,3903,3906],{"style":440},[28,3904],{"className":3905,"style":445},[444],[28,3907,3909],{"className":3908},[449,450,451,452],[28,3910,3912],{"className":3911},[114,452],[28,3913,3916],{"className":3914,"style":3915},[114,222,452],"margin-right:0.0572em;","ij",[28,3918,460],{"className":3919},[459],[28,3921,3923],{"className":3922},[432],[28,3924,3927],{"className":3925,"style":3926},[436],"height:0.2861em;",[28,3928],{}," 表示第 ",[28,3931,3933,3946],{"className":3932,"translate":32},[31],[28,3934,3936],{"className":3935},[36],[38,3937,3938],{"xmlns":40},[42,3939,3940,3944],{},[45,3941,3942],{},[48,3943,3865],{},[94,3945,3865],{"encoding":96},[28,3947,3949],{"className":3948,"ariaHidden":66},[101],[28,3950,3952,3956],{"className":3951},[105],[28,3953],{"className":3954,"style":3955},[109],"height:0.6595em;",[28,3957,3865],{"className":3958},[114,222]," 个数的第 ",[28,3961,3963,3976],{"className":3962,"translate":32},[31],[28,3964,3966],{"className":3965},[36],[38,3967,3968],{"xmlns":40},[42,3969,3970,3974],{},[45,3971,3972],{},[48,3973,3868],{},[94,3975,3868],{"encoding":96},[28,3977,3979],{"className":3978,"ariaHidden":66},[101],[28,3980,3982,3986],{"className":3981},[105],[28,3983],{"className":3984,"style":3985},[109],"height:0.854em;vertical-align:-0.1944em;",[28,3987,3868],{"className":3988,"style":3915},[114,222]," 位小数（取值 ",[28,3991,3993,4006],{"className":3992,"translate":32},[31],[28,3994,3996],{"className":3995},[36],[38,3997,3998],{"xmlns":40},[42,3999,4000,4004],{},[45,4001,4002],{},[61,4003,63],{},[94,4005,63],{"encoding":96},[28,4007,4009],{"className":4008,"ariaHidden":66},[101],[28,4010,4012,4015],{"className":4011},[105],[28,4013],{"className":4014,"style":1362},[109],[28,4016,63],{"className":4017},[114]," 到 ",[28,4020,4022,4036],{"className":4021,"translate":32},[31],[28,4023,4025],{"className":4024},[36],[38,4026,4027],{"xmlns":40},[42,4028,4029,4034],{},[45,4030,4031],{},[61,4032,4033],{},"9",[94,4035,4033],{"encoding":96},[28,4037,4039],{"className":4038,"ariaHidden":66},[101],[28,4040,4042,4045],{"className":4041},[105],[28,4043],{"className":4044,"style":1362},[109],[28,4046,4033],{"className":4047},[114],"）。",[11,4050,4051,4052,4299],{},"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 ",[28,4053,4055,4101],{"className":4054,"translate":32},[31],[28,4056,4058],{"className":4057},[36],[38,4059,4060],{"xmlns":40},[42,4061,4062,4098],{},[45,4063,4064,4067,4069,4071,4078,4084,4090,4096],{},[48,4065,4066],{},"y",[53,4068,55],{},[61,4070,2445],{},[354,4072,4073,4076],{},[48,4074,4075],{},"e",[61,4077,70],{},[354,4079,4080,4082],{},[48,4081,4075],{},[61,4083,75],{},[354,4085,4086,4088],{},[48,4087,4075],{},[61,4089,80],{},[354,4091,4092,4094],{},[48,4093,4075],{},[61,4095,1221],{},[53,4097,85],{},[94,4099,4100],{"encoding":96},"y = 0.e_1 e_2 e_3 e_4 \\dots",[28,4102,4104,4123],{"className":4103,"ariaHidden":66},[101],[28,4105,4107,4111,4114,4117,4120],{"className":4106},[105],[28,4108],{"className":4109,"style":4110},[109],"height:0.625em;vertical-align:-0.1944em;",[28,4112,4066],{"className":4113,"style":2123},[114,222],[28,4115],{"className":4116,"style":120},[119],[28,4118,55],{"className":4119},[124],[28,4121],{"className":4122,"style":120},[119],[28,4124,4126,4130,4133,4173,4213,4253,4293,4296],{"className":4125},[105],[28,4127],{"className":4128,"style":4129},[109],"height:0.7944em;vertical-align:-0.15em;",[28,4131,2445],{"className":4132},[114],[28,4134,4136,4139],{"className":4135},[114],[28,4137,4075],{"className":4138},[114,222],[28,4140,4142],{"className":4141},[423],[28,4143,4145,4165],{"className":4144},[427,428],[28,4146,4148,4162],{"className":4147},[432],[28,4149,4151],{"className":4150,"style":437},[436],[28,4152,4153,4156],{"style":440},[28,4154],{"className":4155,"style":445},[444],[28,4157,4159],{"className":4158},[449,450,451,452],[28,4160,70],{"className":4161},[114,452],[28,4163,460],{"className":4164},[459],[28,4166,4168],{"className":4167},[432],[28,4169,4171],{"className":4170,"style":467},[436],[28,4172],{},[28,4174,4176,4179],{"className":4175},[114],[28,4177,4075],{"className":4178},[114,222],[28,4180,4182],{"className":4181},[423],[28,4183,4185,4205],{"className":4184},[427,428],[28,4186,4188,4202],{"className":4187},[432],[28,4189,4191],{"className":4190,"style":437},[436],[28,4192,4193,4196],{"style":440},[28,4194],{"className":4195,"style":445},[444],[28,4197,4199],{"className":4198},[449,450,451,452],[28,4200,75],{"className":4201},[114,452],[28,4203,460],{"className":4204},[459],[28,4206,4208],{"className":4207},[432],[28,4209,4211],{"className":4210,"style":467},[436],[28,4212],{},[28,4214,4216,4219],{"className":4215},[114],[28,4217,4075],{"className":4218},[114,222],[28,4220,4222],{"className":4221},[423],[28,4223,4225,4245],{"className":4224},[427,428],[28,4226,4228,4242],{"className":4227},[432],[28,4229,4231],{"className":4230,"style":437},[436],[28,4232,4233,4236],{"style":440},[28,4234],{"className":4235,"style":445},[444],[28,4237,4239],{"className":4238},[449,450,451,452],[28,4240,80],{"className":4241},[114,452],[28,4243,460],{"className":4244},[459],[28,4246,4248],{"className":4247},[432],[28,4249,4251],{"className":4250,"style":467},[436],[28,4252],{},[28,4254,4256,4259],{"className":4255},[114],[28,4257,4075],{"className":4258},[114,222],[28,4260,4262],{"className":4261},[423],[28,4263,4265,4285],{"className":4264},[427,428],[28,4266,4268,4282],{"className":4267},[432],[28,4269,4271],{"className":4270,"style":437},[436],[28,4272,4273,4276],{"style":440},[28,4274],{"className":4275,"style":445},[444],[28,4277,4279],{"className":4278},[449,450,451,452],[28,4280,1221],{"className":4281},[114,452],[28,4283,460],{"className":4284},[459],[28,4286,4288],{"className":4287},[432],[28,4289,4291],{"className":4290,"style":467},[436],[28,4292],{},[28,4294],{"className":4295,"style":149},[119],[28,4297,85],{"className":4298},[180],"：",[28,4301,4303],{"className":4302,"translate":32},[2403],[28,4304,4306,4394],{"className":4305,"translate":32},[31],[28,4307,4309],{"className":4308},[36],[38,4310,4311],{"xmlns":40,"display":2412},[42,4312,4313,4391],{},[45,4314,4315,4321,4323],{},[354,4316,4317,4319],{},[48,4318,4075],{},[48,4320,1314],{},[53,4322,55],{},[45,4324,4325,4327],{},[53,4326,59],{"fence":66},[1560,4328,4329,4361],{"rowspacing":1562,"columnalign":1563,"columnspacing":1564},[1566,4330,4331,4338],{},[1569,4332,4333],{},[1572,4334,4335],{"scriptlevel":63,"displaystyle":58},[61,4336,4337],{},"5",[1569,4339,4340],{},[1572,4341,4342],{"scriptlevel":63,"displaystyle":58},[45,4343,4344,4347,4357,4359],{},[87,4345,4346],{},"若 ",[354,4348,4349,4351],{},[48,4350,2450],{},[45,4352,4353,4355],{},[48,4354,1314],{},[48,4356,1314],{},[53,4358,364],{"mathvariant":363},[61,4360,4337],{},[1566,4362,4363,4369],{},[1569,4364,4365],{},[1572,4366,4367],{"scriptlevel":63,"displaystyle":58},[61,4368,1226],{},[1569,4370,4371],{},[1572,4372,4373],{"scriptlevel":63,"displaystyle":58},[45,4374,4375,4377,4387,4389],{},[87,4376,4346],{},[354,4378,4379,4381],{},[48,4380,2450],{},[45,4382,4383,4385],{},[48,4384,1314],{},[48,4386,1314],{},[53,4388,55],{},[61,4390,4337],{},[94,4392,4393],{"encoding":96},"e_n = \\begin{cases} 5 & \\text{若 } d_{nn} \\neq 5 \\\\ 6 & \\text{若 } d_{nn} = 5 \\end{cases}",[28,4395,4397,4453],{"className":4396,"ariaHidden":66},[101],[28,4398,4400,4403,4444,4447,4450],{"className":4399},[105],[28,4401],{"className":4402,"style":525},[109],[28,4404,4406,4409],{"className":4405},[114],[28,4407,4075],{"className":4408},[114,222],[28,4410,4412],{"className":4411},[423],[28,4413,4415,4436],{"className":4414},[427,428],[28,4416,4418,4433],{"className":4417},[432],[28,4419,4422],{"className":4420,"style":4421},[436],"height:0.1514em;",[28,4423,4424,4427],{"style":440},[28,4425],{"className":4426,"style":445},[444],[28,4428,4430],{"className":4429},[449,450,451,452],[28,4431,1314],{"className":4432},[114,222,452],[28,4434,460],{"className":4435},[459],[28,4437,4439],{"className":4438},[432],[28,4440,4442],{"className":4441,"style":467},[436],[28,4443],{},[28,4445],{"className":4446,"style":120},[119],[28,4448,55],{"className":4449},[124],[28,4451],{"className":4452,"style":120},[119],[28,4454,4456,4459],{"className":4455},[105],[28,4457],{"className":4458,"style":1669},[109],[28,4460,4462,4468,4724],{"className":4461},[180],[28,4463,4465],{"className":4464,"style":1677},[138,1676],[28,4466,59],{"className":4467},[1681,1682],[28,4469,4471],{"className":4470},[114],[28,4472,4474,4519,4522],{"className":4473},[1560],[28,4475,4477],{"className":4476},[1692],[28,4478,4480,4511],{"className":4479},[427,428],[28,4481,4483,4508],{"className":4482},[432],[28,4484,4486,4497],{"className":4485,"style":1702},[436],[28,4487,4488,4491],{"style":1705},[28,4489],{"className":4490,"style":1709},[444],[28,4492,4494],{"className":4493},[114],[28,4495,4337],{"className":4496},[114],[28,4498,4499,4502],{"style":1722},[28,4500],{"className":4501,"style":1709},[444],[28,4503,4505],{"className":4504},[114],[28,4506,1226],{"className":4507},[114],[28,4509,460],{"className":4510},[459],[28,4512,4514],{"className":4513},[432],[28,4515,4517],{"className":4516,"style":1767},[436],[28,4518],{},[28,4520],{"className":4521,"style":1774},[1773],[28,4523,4525],{"className":4524},[1692],[28,4526,4528,4716],{"className":4527},[427,428],[28,4529,4531,4713],{"className":4530},[432],[28,4532,4534,4641],{"className":4533,"style":1702},[436],[28,4535,4536,4539],{"style":1705},[28,4537],{"className":4538,"style":1709},[444],[28,4540,4542,4552,4596,4599,4635,4638],{"className":4541},[114],[28,4543,4545,4549],{"className":4544},[114,1801],[28,4546,4548],{"className":4547},[114,1809],"若",[28,4550,1805],{"className":4551},[114],[28,4553,4555,4558],{"className":4554},[114],[28,4556,2450],{"className":4557},[114,222],[28,4559,4561],{"className":4560},[423],[28,4562,4564,4588],{"className":4563},[427,428],[28,4565,4567,4585],{"className":4566},[432],[28,4568,4570],{"className":4569,"style":4421},[436],[28,4571,4572,4575],{"style":440},[28,4573],{"className":4574,"style":445},[444],[28,4576,4578],{"className":4577},[449,450,451,452],[28,4579,4581],{"className":4580},[114,452],[28,4582,4584],{"className":4583},[114,222,452],"nn",[28,4586,460],{"className":4587},[459],[28,4589,4591],{"className":4590},[432],[28,4592,4594],{"className":4593,"style":467},[436],[28,4595],{},[28,4597],{"className":4598,"style":120},[119],[28,4600,4602,4629,4632],{"className":4601},[124],[28,4603,4605],{"className":4604},[124],[28,4606,4608],{"className":4607},[114,482],[28,4609,4611],{"className":4610},[486],[28,4612,4614,4617,4626],{"className":4613},[490],[28,4615],{"className":4616,"style":292},[109],[28,4618,4620],{"className":4619},[497],[28,4621,4623],{"className":4622},[114],[28,4624,504],{"className":4625},[124],[28,4627],{"className":4628},[508],[28,4630],{"className":4631},[119,512],[28,4633,55],{"className":4634},[124],[28,4636],{"className":4637,"style":120},[119],[28,4639,4337],{"className":4640},[114],[28,4642,4643,4646],{"style":1722},[28,4644],{"className":4645,"style":1709},[444],[28,4647,4649,4658,4701,4704,4707,4710],{"className":4648},[114],[28,4650,4652,4655],{"className":4651},[114,1801],[28,4653,4548],{"className":4654},[114,1809],[28,4656,1805],{"className":4657},[114],[28,4659,4661,4664],{"className":4660},[114],[28,4662,2450],{"className":4663},[114,222],[28,4665,4667],{"className":4666},[423],[28,4668,4670,4693],{"className":4669},[427,428],[28,4671,4673,4690],{"className":4672},[432],[28,4674,4676],{"className":4675,"style":4421},[436],[28,4677,4678,4681],{"style":440},[28,4679],{"className":4680,"style":445},[444],[28,4682,4684],{"className":4683},[449,450,451,452],[28,4685,4687],{"className":4686},[114,452],[28,4688,4584],{"className":4689},[114,222,452],[28,4691,460],{"className":4692},[459],[28,4694,4696],{"className":4695},[432],[28,4697,4699],{"className":4698,"style":467},[436],[28,4700],{},[28,4702],{"className":4703,"style":120},[119],[28,4705,55],{"className":4706},[124],[28,4708],{"className":4709,"style":120},[119],[28,4711,4337],{"className":4712},[114],[28,4714,460],{"className":4715},[459],[28,4717,4719],{"className":4718},[432],[28,4720,4722],{"className":4721,"style":1767},[436],[28,4723],{},[28,4725],{"className":4726},[187,1846],[11,4728,4729,4730,4904],{},"（这条规则的唯一目的是保证 ",[28,4731,4733,4763],{"className":4732,"translate":32},[31],[28,4734,4736],{"className":4735},[36],[38,4737,4738],{"xmlns":40},[42,4739,4740,4760],{},[45,4741,4742,4748,4750],{},[354,4743,4744,4746],{},[48,4745,4075],{},[48,4747,1314],{},[53,4749,364],{"mathvariant":363},[354,4751,4752,4754],{},[48,4753,2450],{},[45,4755,4756,4758],{},[48,4757,1314],{},[48,4759,1314],{},[94,4761,4762],{"encoding":96},"e_n \\neq d_{nn}",[28,4764,4766,4854],{"className":4765,"ariaHidden":66},[101],[28,4767,4769,4772,4812,4815,4851],{"className":4768},[105],[28,4770],{"className":4771,"style":292},[109],[28,4773,4775,4778],{"className":4774},[114],[28,4776,4075],{"className":4777},[114,222],[28,4779,4781],{"className":4780},[423],[28,4782,4784,4804],{"className":4783},[427,428],[28,4785,4787,4801],{"className":4786},[432],[28,4788,4790],{"className":4789,"style":4421},[436],[28,4791,4792,4795],{"style":440},[28,4793],{"className":4794,"style":445},[444],[28,4796,4798],{"className":4797},[449,450,451,452],[28,4799,1314],{"className":4800},[114,222,452],[28,4802,460],{"className":4803},[459],[28,4805,4807],{"className":4806},[432],[28,4808,4810],{"className":4809,"style":467},[436],[28,4811],{},[28,4813],{"className":4814,"style":120},[119],[28,4816,4818,4845,4848],{"className":4817},[124],[28,4819,4821],{"className":4820},[124],[28,4822,4824],{"className":4823},[114,482],[28,4825,4827],{"className":4826},[486],[28,4828,4830,4833,4842],{"className":4829},[490],[28,4831],{"className":4832,"style":292},[109],[28,4834,4836],{"className":4835},[497],[28,4837,4839],{"className":4838},[114],[28,4840,504],{"className":4841},[124],[28,4843],{"className":4844},[508],[28,4846],{"className":4847},[119,512],[28,4849,55],{"className":4850},[124],[28,4852],{"className":4853,"style":120},[119],[28,4855,4857,4861],{"className":4856},[105],[28,4858],{"className":4859,"style":4860},[109],"height:0.8444em;vertical-align:-0.15em;",[28,4862,4864,4867],{"className":4863},[114],[28,4865,2450],{"className":4866},[114,222],[28,4868,4870],{"className":4869},[423],[28,4871,4873,4896],{"className":4872},[427,428],[28,4874,4876,4893],{"className":4875},[432],[28,4877,4879],{"className":4878,"style":4421},[436],[28,4880,4881,4884],{"style":440},[28,4882],{"className":4883,"style":445},[444],[28,4885,4887],{"className":4886},[449,450,451,452],[28,4888,4890],{"className":4889},[114,452],[28,4891,4584],{"className":4892},[114,222,452],[28,4894,460],{"className":4895},[459],[28,4897,4899],{"className":4898},[432],[28,4900,4902],{"className":4901,"style":467},[436],[28,4903],{},"；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。）",[11,4906,4907,4908,4936],{},"现在关键问题出现了：",[28,4909,4911,4924],{"className":4910,"translate":32},[31],[28,4912,4914],{"className":4913},[36],[38,4915,4916],{"xmlns":40},[42,4917,4918,4922],{},[45,4919,4920],{},[48,4921,4066],{},[94,4923,4066],{"encoding":96},[28,4925,4927],{"className":4926,"ariaHidden":66},[101],[28,4928,4930,4933],{"className":4929},[105],[28,4931],{"className":4932,"style":4110},[109],[28,4934,4066],{"className":4935,"style":2123},[114,222]," 会出现在原来的列表里吗？",[11,4938,4939,4940,4968,4969,4997,4998,5093,5094,4048],{},"假设 ",[28,4941,4943,4956],{"className":4942,"translate":32},[31],[28,4944,4946],{"className":4945},[36],[38,4947,4948],{"xmlns":40},[42,4949,4950,4954],{},[45,4951,4952],{},[48,4953,4066],{},[94,4955,4066],{"encoding":96},[28,4957,4959],{"className":4958,"ariaHidden":66},[101],[28,4960,4962,4965],{"className":4961},[105],[28,4963],{"className":4964,"style":4110},[109],[28,4966,4066],{"className":4967,"style":2123},[114,222]," 出现在列表中，那么 ",[28,4970,4972,4985],{"className":4971,"translate":32},[31],[28,4973,4975],{"className":4974},[36],[38,4976,4977],{"xmlns":40},[42,4978,4979,4983],{},[45,4980,4981],{},[48,4982,4066],{},[94,4984,4066],{"encoding":96},[28,4986,4988],{"className":4987,"ariaHidden":66},[101],[28,4989,4991,4994],{"className":4990},[105],[28,4992],{"className":4993,"style":4110},[109],[28,4995,4066],{"className":4996,"style":2123},[114,222]," 必定等于被枚举的某个数，不妨设 ",[28,4999,5001,5024],{"className":5000,"translate":32},[31],[28,5002,5004],{"className":5003},[36],[38,5005,5006],{"xmlns":40},[42,5007,5008,5021],{},[45,5009,5010,5012,5014],{},[48,5011,4066],{},[53,5013,55],{},[354,5015,5016,5018],{},[48,5017,2430],{},[48,5019,5020],{},"k",[94,5022,5023],{"encoding":96},"y = x_k",[28,5025,5027,5045],{"className":5026,"ariaHidden":66},[101],[28,5028,5030,5033,5036,5039,5042],{"className":5029},[105],[28,5031],{"className":5032,"style":4110},[109],[28,5034,4066],{"className":5035,"style":2123},[114,222],[28,5037],{"className":5038,"style":120},[119],[28,5040,55],{"className":5041},[124],[28,5043],{"className":5044,"style":120},[119],[28,5046,5048,5051],{"className":5047},[105],[28,5049],{"className":5050,"style":525},[109],[28,5052,5054,5057],{"className":5053},[114],[28,5055,2430],{"className":5056},[114,222],[28,5058,5060],{"className":5059},[423],[28,5061,5063,5085],{"className":5062},[427,428],[28,5064,5066,5082],{"className":5065},[432],[28,5067,5070],{"className":5068,"style":5069},[436],"height:0.3361em;",[28,5071,5072,5075],{"style":440},[28,5073],{"className":5074,"style":445},[444],[28,5076,5078],{"className":5077},[449,450,451,452],[28,5079,5020],{"className":5080,"style":5081},[114,222,452],"margin-right:0.0315em;",[28,5083,460],{"className":5084},[459],[28,5086,5088],{"className":5087},[432],[28,5089,5091],{"className":5090,"style":467},[436],[28,5092],{},"（对某个确定的指标 ",[28,5095,5097,5110],{"className":5096,"translate":32},[31],[28,5098,5100],{"className":5099},[36],[38,5101,5102],{"xmlns":40},[42,5103,5104,5108],{},[45,5105,5106],{},[48,5107,5020],{},[94,5109,5020],{"encoding":96},[28,5111,5113],{"className":5112,"ariaHidden":66},[101],[28,5114,5116,5119],{"className":5115},[105],[28,5117],{"className":5118,"style":902},[109],[28,5120,5020],{"className":5121,"style":5081},[114,222],[11,5123,5124,5125,5153,5154,5182,5183,5253,5254,5323,5324,5404,5405,5153,5475,5503],{},"但根据我们的构造规则，",[28,5126,5128,5141],{"className":5127,"translate":32},[31],[28,5129,5131],{"className":5130},[36],[38,5132,5133],{"xmlns":40},[42,5134,5135,5139],{},[45,5136,5137],{},[48,5138,4066],{},[94,5140,4066],{"encoding":96},[28,5142,5144],{"className":5143,"ariaHidden":66},[101],[28,5145,5147,5150],{"className":5146},[105],[28,5148],{"className":5149,"style":4110},[109],[28,5151,4066],{"className":5152,"style":2123},[114,222]," 的第 ",[28,5155,5157,5170],{"className":5156,"translate":32},[31],[28,5158,5160],{"className":5159},[36],[38,5161,5162],{"xmlns":40},[42,5163,5164,5168],{},[45,5165,5166],{},[48,5167,5020],{},[94,5169,5020],{"encoding":96},[28,5171,5173],{"className":5172,"ariaHidden":66},[101],[28,5174,5176,5179],{"className":5175},[105],[28,5177],{"className":5178,"style":902},[109],[28,5180,5020],{"className":5181,"style":5081},[114,222]," 位小数是 ",[28,5184,5186,5204],{"className":5185,"translate":32},[31],[28,5187,5189],{"className":5188},[36],[38,5190,5191],{"xmlns":40},[42,5192,5193,5201],{},[45,5194,5195],{},[354,5196,5197,5199],{},[48,5198,4075],{},[48,5200,5020],{},[94,5202,5203],{"encoding":96},"e_k",[28,5205,5207],{"className":5206,"ariaHidden":66},[101],[28,5208,5210,5213],{"className":5209},[105],[28,5211],{"className":5212,"style":525},[109],[28,5214,5216,5219],{"className":5215},[114],[28,5217,4075],{"className":5218},[114,222],[28,5220,5222],{"className":5221},[423],[28,5223,5225,5245],{"className":5224},[427,428],[28,5226,5228,5242],{"className":5227},[432],[28,5229,5231],{"className":5230,"style":5069},[436],[28,5232,5233,5236],{"style":440},[28,5234],{"className":5235,"style":445},[444],[28,5237,5239],{"className":5238},[449,450,451,452],[28,5240,5020],{"className":5241,"style":5081},[114,222,452],[28,5243,460],{"className":5244},[459],[28,5246,5248],{"className":5247},[432],[28,5249,5251],{"className":5250,"style":467},[436],[28,5252],{},"，而 ",[28,5255,5257,5274],{"className":5256,"translate":32},[31],[28,5258,5260],{"className":5259},[36],[38,5261,5262],{"xmlns":40},[42,5263,5264,5272],{},[45,5265,5266],{},[354,5267,5268,5270],{},[48,5269,4075],{},[48,5271,5020],{},[94,5273,5203],{"encoding":96},[28,5275,5277],{"className":5276,"ariaHidden":66},[101],[28,5278,5280,5283],{"className":5279},[105],[28,5281],{"className":5282,"style":525},[109],[28,5284,5286,5289],{"className":5285},[114],[28,5287,4075],{"className":5288},[114,222],[28,5290,5292],{"className":5291},[423],[28,5293,5295,5315],{"className":5294},[427,428],[28,5296,5298,5312],{"className":5297},[432],[28,5299,5301],{"className":5300,"style":5069},[436],[28,5302,5303,5306],{"style":440},[28,5304],{"className":5305,"style":445},[444],[28,5307,5309],{"className":5308},[449,450,451,452],[28,5310,5020],{"className":5311,"style":5081},[114,222,452],[28,5313,460],{"className":5314},[459],[28,5316,5318],{"className":5317},[432],[28,5319,5321],{"className":5320,"style":467},[436],[28,5322],{}," 被刻意构造得与 ",[28,5325,5327,5349],{"className":5326,"translate":32},[31],[28,5328,5330],{"className":5329},[36],[38,5331,5332],{"xmlns":40},[42,5333,5334,5346],{},[45,5335,5336],{},[354,5337,5338,5340],{},[48,5339,2450],{},[45,5341,5342,5344],{},[48,5343,5020],{},[48,5345,5020],{},[94,5347,5348],{"encoding":96},"d_{kk}",[28,5350,5352],{"className":5351,"ariaHidden":66},[101],[28,5353,5355,5358],{"className":5354},[105],[28,5356],{"className":5357,"style":4860},[109],[28,5359,5361,5364],{"className":5360},[114],[28,5362,2450],{"className":5363},[114,222],[28,5365,5367],{"className":5366},[423],[28,5368,5370,5396],{"className":5369},[427,428],[28,5371,5373,5393],{"className":5372},[432],[28,5374,5376],{"className":5375,"style":5069},[436],[28,5377,5378,5381],{"style":440},[28,5379],{"className":5380,"style":445},[444],[28,5382,5384],{"className":5383},[449,450,451,452],[28,5385,5387,5390],{"className":5386},[114,452],[28,5388,5020],{"className":5389,"style":5081},[114,222,452],[28,5391,5020],{"className":5392,"style":5081},[114,222,452],[28,5394,460],{"className":5395},[459],[28,5397,5399],{"className":5398},[432],[28,5400,5402],{"className":5401,"style":467},[436],[28,5403],{},"（即 ",[28,5406,5408,5426],{"className":5407,"translate":32},[31],[28,5409,5411],{"className":5410},[36],[38,5412,5413],{"xmlns":40},[42,5414,5415,5423],{},[45,5416,5417],{},[354,5418,5419,5421],{},[48,5420,2430],{},[48,5422,5020],{},[94,5424,5425],{"encoding":96},"x_k",[28,5427,5429],{"className":5428,"ariaHidden":66},[101],[28,5430,5432,5435],{"className":5431},[105],[28,5433],{"className":5434,"style":525},[109],[28,5436,5438,5441],{"className":5437},[114],[28,5439,2430],{"className":5440},[114,222],[28,5442,5444],{"className":5443},[423],[28,5445,5447,5467],{"className":5446},[427,428],[28,5448,5450,5464],{"className":5449},[432],[28,5451,5453],{"className":5452,"style":5069},[436],[28,5454,5455,5458],{"style":440},[28,5456],{"className":5457,"style":445},[444],[28,5459,5461],{"className":5460},[449,450,451,452],[28,5462,5020],{"className":5463,"style":5081},[114,222,452],[28,5465,460],{"className":5466},[459],[28,5468,5470],{"className":5469},[432],[28,5471,5473],{"className":5472,"style":467},[436],[28,5474],{},[28,5476,5478,5491],{"className":5477,"translate":32},[31],[28,5479,5481],{"className":5480},[36],[38,5482,5483],{"xmlns":40},[42,5484,5485,5489],{},[45,5486,5487],{},[48,5488,5020],{},[94,5490,5020],{"encoding":96},[28,5492,5494],{"className":5493,"ariaHidden":66},[101],[28,5495,5497,5500],{"className":5496},[105],[28,5498],{"className":5499,"style":902},[109],[28,5501,5020],{"className":5502,"style":5081},[114,222]," 位小数）不同：",[28,5505,5507],{"className":5506,"translate":32},[2403],[28,5508,5510,5558],{"className":5509,"translate":32},[31],[28,5511,5513],{"className":5512},[36],[38,5514,5515],{"xmlns":40,"display":2412},[42,5516,5517,5555],{},[45,5518,5519,5525,5527,5537,5540,5543,5545,5547,5549],{},[354,5520,5521,5523],{},[48,5522,4075],{},[48,5524,5020],{},[53,5526,364],{"mathvariant":363},[354,5528,5529,5531],{},[48,5530,2450],{},[45,5532,5533,5535],{},[48,5534,5020],{},[48,5536,5020],{},[87,5538,5539],{},"  ",[53,5541,5542],{},"⟹",[87,5544,5539],{},[48,5546,4066],{},[53,5548,364],{"mathvariant":363},[354,5550,5551,5553],{},[48,5552,2430],{},[48,5554,5020],{},[94,5556,5557],{"encoding":96},"e_k \\neq d_{kk} \\implies y \\neq x_k",[28,5559,5561,5649,5716,5767],{"className":5560,"ariaHidden":66},[101],[28,5562,5564,5567,5607,5610,5646],{"className":5563},[105],[28,5565],{"className":5566,"style":292},[109],[28,5568,5570,5573],{"className":5569},[114],[28,5571,4075],{"className":5572},[114,222],[28,5574,5576],{"className":5575},[423],[28,5577,5579,5599],{"className":5578},[427,428],[28,5580,5582,5596],{"className":5581},[432],[28,5583,5585],{"className":5584,"style":5069},[436],[28,5586,5587,5590],{"style":440},[28,5588],{"className":5589,"style":445},[444],[28,5591,5593],{"className":5592},[449,450,451,452],[28,5594,5020],{"className":5595,"style":5081},[114,222,452],[28,5597,460],{"className":5598},[459],[28,5600,5602],{"className":5601},[432],[28,5603,5605],{"className":5604,"style":467},[436],[28,5606],{},[28,5608],{"className":5609,"style":120},[119],[28,5611,5613,5640,5643],{"className":5612},[124],[28,5614,5616],{"className":5615},[124],[28,5617,5619],{"className":5618},[114,482],[28,5620,5622],{"className":5621},[486],[28,5623,5625,5628,5637],{"className":5624},[490],[28,5626],{"className":5627,"style":292},[109],[28,5629,5631],{"className":5630},[497],[28,5632,5634],{"className":5633},[114],[28,5635,504],{"className":5636},[124],[28,5638],{"className":5639},[508],[28,5641],{"className":5642},[119,512],[28,5644,55],{"className":5645},[124],[28,5647],{"className":5648,"style":120},[119],[28,5650,5652,5655,5701,5704,5707,5710,5713],{"className":5651},[105],[28,5653],{"className":5654,"style":4860},[109],[28,5656,5658,5661],{"className":5657},[114],[28,5659,2450],{"className":5660},[114,222],[28,5662,5664],{"className":5663},[423],[28,5665,5667,5693],{"className":5666},[427,428],[28,5668,5670,5690],{"className":5669},[432],[28,5671,5673],{"className":5672,"style":5069},[436],[28,5674,5675,5678],{"style":440},[28,5676],{"className":5677,"style":445},[444],[28,5679,5681],{"className":5680},[449,450,451,452],[28,5682,5684,5687],{"className":5683},[114,452],[28,5685,5020],{"className":5686,"style":5081},[114,222,452],[28,5688,5020],{"className":5689,"style":5081},[114,222,452],[28,5691,460],{"className":5692},[459],[28,5694,5696],{"className":5695},[432],[28,5697,5699],{"className":5698,"style":467},[436],[28,5700],{},[28,5702],{"className":5703,"style":120},[119],[28,5705],{"className":5706,"style":120},[119],[28,5708,5542],{"className":5709},[124],[28,5711],{"className":5712,"style":120},[119],[28,5714],{"className":5715,"style":120},[119],[28,5717,5719,5722,5725,5728,5764],{"className":5718},[105],[28,5720],{"className":5721,"style":292},[109],[28,5723,4066],{"className":5724,"style":2123},[114,222],[28,5726],{"className":5727,"style":120},[119],[28,5729,5731,5758,5761],{"className":5730},[124],[28,5732,5734],{"className":5733},[124],[28,5735,5737],{"className":5736},[114,482],[28,5738,5740],{"className":5739},[486],[28,5741,5743,5746,5755],{"className":5742},[490],[28,5744],{"className":5745,"style":292},[109],[28,5747,5749],{"className":5748},[497],[28,5750,5752],{"className":5751},[114],[28,5753,504],{"className":5754},[124],[28,5756],{"className":5757},[508],[28,5759],{"className":5760},[119,512],[28,5762,55],{"className":5763},[124],[28,5765],{"className":5766,"style":120},[119],[28,5768,5770,5773],{"className":5769},[105],[28,5771],{"className":5772,"style":525},[109],[28,5774,5776,5779],{"className":5775},[114],[28,5777,2430],{"className":5778},[114,222],[28,5780,5782],{"className":5781},[423],[28,5783,5785,5805],{"className":5784},[427,428],[28,5786,5788,5802],{"className":5787},[432],[28,5789,5791],{"className":5790,"style":5069},[436],[28,5792,5793,5796],{"style":440},[28,5794],{"className":5795,"style":445},[444],[28,5797,5799],{"className":5798},[449,450,451,452],[28,5800,5020],{"className":5801,"style":5081},[114,222,452],[28,5803,460],{"className":5804},[459],[28,5806,5808],{"className":5807},[432],[28,5809,5811],{"className":5810,"style":467},[436],[28,5812],{},[11,5814,5815,5816,5907],{},"这与 ",[28,5817,5819,5840],{"className":5818,"translate":32},[31],[28,5820,5822],{"className":5821},[36],[38,5823,5824],{"xmlns":40},[42,5825,5826,5838],{},[45,5827,5828,5830,5832],{},[48,5829,4066],{},[53,5831,55],{},[354,5833,5834,5836],{},[48,5835,2430],{},[48,5837,5020],{},[94,5839,5023],{"encoding":96},[28,5841,5843,5861],{"className":5842,"ariaHidden":66},[101],[28,5844,5846,5849,5852,5855,5858],{"className":5845},[105],[28,5847],{"className":5848,"style":4110},[109],[28,5850,4066],{"className":5851,"style":2123},[114,222],[28,5853],{"className":5854,"style":120},[119],[28,5856,55],{"className":5857},[124],[28,5859],{"className":5860,"style":120},[119],[28,5862,5864,5867],{"className":5863},[105],[28,5865],{"className":5866,"style":525},[109],[28,5868,5870,5873],{"className":5869},[114],[28,5871,2430],{"className":5872},[114,222],[28,5874,5876],{"className":5875},[423],[28,5877,5879,5899],{"className":5878},[427,428],[28,5880,5882,5896],{"className":5881},[432],[28,5883,5885],{"className":5884,"style":5069},[436],[28,5886,5887,5890],{"style":440},[28,5888],{"className":5889,"style":445},[444],[28,5891,5893],{"className":5892},[449,450,451,452],[28,5894,5020],{"className":5895,"style":5081},[114,222,452],[28,5897,460],{"className":5898},[459],[28,5900,5902],{"className":5901},[432],[28,5903,5905],{"className":5904,"style":467},[436],[28,5906],{}," 的假设矛盾！",[11,5909,5910,5911,5939,5940,5968,5969,5998,5999,6027,6028,6056],{},"而且，这一矛盾对每一个 ",[28,5912,5914,5927],{"className":5913,"translate":32},[31],[28,5915,5917],{"className":5916},[36],[38,5918,5919],{"xmlns":40},[42,5920,5921,5925],{},[45,5922,5923],{},[48,5924,5020],{},[94,5926,5020],{"encoding":96},[28,5928,5930],{"className":5929,"ariaHidden":66},[101],[28,5931,5933,5936],{"className":5932},[105],[28,5934],{"className":5935,"style":902},[109],[28,5937,5020],{"className":5938,"style":5081},[114,222]," 都成立——",[28,5941,5943,5956],{"className":5942,"translate":32},[31],[28,5944,5946],{"className":5945},[36],[38,5947,5948],{"xmlns":40},[42,5949,5950,5954],{},[45,5951,5952],{},[48,5953,4066],{},[94,5955,4066],{"encoding":96},[28,5957,5959],{"className":5958,"ariaHidden":66},[101],[28,5960,5962,5965],{"className":5961},[105],[28,5963],{"className":5964,"style":4110},[109],[28,5966,4066],{"className":5967,"style":2123},[114,222]," 在第 1 位与第 1 个数不同，在第 2 位与第 2 个数不同，在第 ",[28,5970,5972,5985],{"className":5971,"translate":32},[31],[28,5973,5975],{"className":5974},[36],[38,5976,5977],{"xmlns":40},[42,5978,5979,5983],{},[45,5980,5981],{},[48,5982,1314],{},[94,5984,1314],{"encoding":96},[28,5986,5988],{"className":5987,"ariaHidden":66},[101],[28,5989,5991,5995],{"className":5990},[105],[28,5992],{"className":5993,"style":5994},[109],"height:0.4306em;",[28,5996,1314],{"className":5997},[114,222]," 位与第 ",[28,6000,6002,6015],{"className":6001,"translate":32},[31],[28,6003,6005],{"className":6004},[36],[38,6006,6007],{"xmlns":40},[42,6008,6009,6013],{},[45,6010,6011],{},[48,6012,1314],{},[94,6014,1314],{"encoding":96},[28,6016,6018],{"className":6017,"ariaHidden":66},[101],[28,6019,6021,6024],{"className":6020},[105],[28,6022],{"className":6023,"style":5994},[109],[28,6025,1314],{"className":6026},[114,222]," 个数不同……因此 ",[28,6029,6031,6044],{"className":6030,"translate":32},[31],[28,6032,6034],{"className":6033},[36],[38,6035,6036],{"xmlns":40},[42,6037,6038,6042],{},[45,6039,6040],{},[48,6041,4066],{},[94,6043,4066],{"encoding":96},[28,6045,6047],{"className":6046,"ariaHidden":66},[101],[28,6048,6050,6053],{"className":6049},[105],[28,6051],{"className":6052,"style":4110},[109],[28,6054,4066],{"className":6055,"style":2123},[114,222]," 与列表中的每一个数都不同。",[11,6058,6059,6060,6088,6089,6140],{},"然而 ",[28,6061,6063,6076],{"className":6062,"translate":32},[31],[28,6064,6066],{"className":6065},[36],[38,6067,6068],{"xmlns":40},[42,6069,6070,6074],{},[45,6071,6072],{},[48,6073,4066],{},[94,6075,4066],{"encoding":96},[28,6077,6079],{"className":6078,"ariaHidden":66},[101],[28,6080,6082,6085],{"className":6081},[105],[28,6083],{"className":6084,"style":4110},[109],[28,6086,4066],{"className":6087,"style":2123},[114,222]," 无疑是 ",[28,6090,6092,6113],{"className":6091,"translate":32},[31],[28,6093,6095],{"className":6094},[36],[38,6096,6097],{"xmlns":40},[42,6098,6099,6111],{},[45,6100,6101,6103,6105,6107,6109],{},[53,6102,378],{"stretchy":58},[61,6104,63],{},[53,6106,67],{"separator":66},[61,6108,70],{},[53,6110,387],{"stretchy":58},[94,6112,2161],{"encoding":96},[28,6114,6116],{"className":6115,"ariaHidden":66},[101],[28,6117,6119,6122,6125,6128,6131,6134,6137],{"className":6118},[105],[28,6120],{"className":6121,"style":134},[109],[28,6123,378],{"className":6124},[138],[28,6126,63],{"className":6127},[114],[28,6129,67],{"className":6130},[145],[28,6132],{"className":6133,"style":149},[119],[28,6135,70],{"className":6136},[114],[28,6138,387],{"className":6139},[187]," 中的实数（它的每一位都是合法的十进制数字）。",[11,6142,6143,6144,6172,6173,6268,6269,6320],{},"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 ",[28,6145,6147,6160],{"className":6146,"translate":32},[31],[28,6148,6150],{"className":6149},[36],[38,6151,6152],{"xmlns":40},[42,6153,6154,6158],{},[45,6155,6156],{},[48,6157,4066],{},[94,6159,4066],{"encoding":96},[28,6161,6163],{"className":6162,"ariaHidden":66},[101],[28,6164,6166,6169],{"className":6165},[105],[28,6167],{"className":6168,"style":4110},[109],[28,6170,4066],{"className":6171,"style":2123},[114,222],"。这就说明最初的假设——存在双射 ",[28,6174,6176,6205],{"className":6175,"translate":32},[31],[28,6177,6179],{"className":6178},[36],[38,6180,6181],{"xmlns":40},[42,6182,6183,6203],{},[45,6184,6185,6187,6189,6191,6193,6195,6197,6199,6201],{},[48,6186,269],{},[53,6188,272],{},[48,6190,51],{"mathvariant":50},[53,6192,277],{},[53,6194,378],{"stretchy":58},[61,6196,63],{},[53,6198,67],{"separator":66},[61,6200,70],{},[53,6202,387],{"stretchy":58},[94,6204,2335],{"encoding":96},[28,6206,6208,6226,6244],{"className":6207,"ariaHidden":66},[101],[28,6209,6211,6214,6217,6220,6223],{"className":6210},[105],[28,6212],{"className":6213,"style":292},[109],[28,6215,269],{"className":6216,"style":296},[114,222],[28,6218],{"className":6219,"style":120},[119],[28,6221,272],{"className":6222},[124],[28,6224],{"className":6225,"style":120},[119],[28,6227,6229,6232,6235,6238,6241],{"className":6228},[105],[28,6230],{"className":6231,"style":110},[109],[28,6233,51],{"className":6234},[114,115],[28,6236],{"className":6237,"style":120},[119],[28,6239,277],{"className":6240},[124],[28,6242],{"className":6243,"style":120},[119],[28,6245,6247,6250,6253,6256,6259,6262,6265],{"className":6246},[105],[28,6248],{"className":6249,"style":134},[109],[28,6251,378],{"className":6252},[138],[28,6254,63],{"className":6255},[114],[28,6257,67],{"className":6258},[145],[28,6260],{"className":6261,"style":149},[119],[28,6263,70],{"className":6264},[114],[28,6266,387],{"className":6267},[187],"——是假的。",[28,6270,6272,6293],{"className":6271,"translate":32},[31],[28,6273,6275],{"className":6274},[36],[38,6276,6277],{"xmlns":40},[42,6278,6279,6291],{},[45,6280,6281,6283,6285,6287,6289],{},[53,6282,378],{"stretchy":58},[61,6284,63],{},[53,6286,67],{"separator":66},[61,6288,70],{},[53,6290,387],{"stretchy":58},[94,6292,2161],{"encoding":96},[28,6294,6296],{"className":6295,"ariaHidden":66},[101],[28,6297,6299,6302,6305,6308,6311,6314,6317],{"className":6298},[105],[28,6300],{"className":6301,"style":134},[109],[28,6303,378],{"className":6304},[138],[28,6306,63],{"className":6307},[114],[28,6309,67],{"className":6310},[145],[28,6312],{"className":6313,"style":149},[119],[28,6315,70],{"className":6316},[114],[28,6318,387],{"className":6319},[187]," 中的实数不可数。",[11,6322,6323,6324,6524,6525,6595],{},"为什么叫「对角线」？这个名称源于证明中选取数字 ",[28,6325,6327,6365],{"className":6326,"translate":32},[31],[28,6328,6330],{"className":6329},[36],[38,6331,6332],{"xmlns":40},[42,6333,6334,6362],{},[45,6335,6336,6342,6344,6350,6352,6358,6360],{},[354,6337,6338,6340],{},[48,6339,2450],{},[61,6341,2453],{},[53,6343,67],{"separator":66},[354,6345,6346,6348],{},[48,6347,2450],{},[61,6349,2522],{},[53,6351,67],{"separator":66},[354,6353,6354,6356],{},[48,6355,2450],{},[61,6357,2591],{},[53,6359,67],{"separator":66},[53,6361,85],{},[94,6363,6364],{"encoding":96},"d_{11}, d_{22}, d_{33}, \\dots",[28,6366,6368],{"className":6367,"ariaHidden":66},[101],[28,6369,6371,6374,6417,6420,6423,6466,6469,6472,6515,6518,6521],{"className":6370},[105],[28,6372],{"className":6373,"style":292},[109],[28,6375,6377,6380],{"className":6376},[114],[28,6378,2450],{"className":6379},[114,222],[28,6381,6383],{"className":6382},[423],[28,6384,6386,6409],{"className":6385},[427,428],[28,6387,6389,6406],{"className":6388},[432],[28,6390,6392],{"className":6391,"style":437},[436],[28,6393,6394,6397],{"style":440},[28,6395],{"className":6396,"style":445},[444],[28,6398,6400],{"className":6399},[449,450,451,452],[28,6401,6403],{"className":6402},[114,452],[28,6404,2453],{"className":6405},[114,452],[28,6407,460],{"className":6408},[459],[28,6410,6412],{"className":6411},[432],[28,6413,6415],{"className":6414,"style":467},[436],[28,6416],{},[28,6418,67],{"className":6419},[145],[28,6421],{"className":6422,"style":149},[119],[28,6424,6426,6429],{"className":6425},[114],[28,6427,2450],{"className":6428},[114,222],[28,6430,6432],{"className":6431},[423],[28,6433,6435,6458],{"className":6434},[427,428],[28,6436,6438,6455],{"className":6437},[432],[28,6439,6441],{"className":6440,"style":437},[436],[28,6442,6443,6446],{"style":440},[28,6444],{"className":6445,"style":445},[444],[28,6447,6449],{"className":6448},[449,450,451,452],[28,6450,6452],{"className":6451},[114,452],[28,6453,2522],{"className":6454},[114,452],[28,6456,460],{"className":6457},[459],[28,6459,6461],{"className":6460},[432],[28,6462,6464],{"className":6463,"style":467},[436],[28,6465],{},[28,6467,67],{"className":6468},[145],[28,6470],{"className":6471,"style":149},[119],[28,6473,6475,6478],{"className":6474},[114],[28,6476,2450],{"className":6477},[114,222],[28,6479,6481],{"className":6480},[423],[28,6482,6484,6507],{"className":6483},[427,428],[28,6485,6487,6504],{"className":6486},[432],[28,6488,6490],{"className":6489,"style":437},[436],[28,6491,6492,6495],{"style":440},[28,6493],{"className":6494,"style":445},[444],[28,6496,6498],{"className":6497},[449,450,451,452],[28,6499,6501],{"className":6500},[114,452],[28,6502,2591],{"className":6503},[114,452],[28,6505,460],{"className":6506},[459],[28,6508,6510],{"className":6509},[432],[28,6511,6513],{"className":6512,"style":467},[436],[28,6514],{},[28,6516,67],{"className":6517},[145],[28,6519],{"className":6520,"style":149},[119],[28,6522,85],{"className":6523},[180]," 的方式——若把所有 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纵向排成无穷矩阵，这些位置恰好构成矩阵的主对角线：",[28,6597,6599],{"className":6598,"translate":32},[2403],[28,6600,6602,6808],{"className":6601,"translate":32},[31],[28,6603,6605],{"className":6604},[36],[38,6606,6607],{"xmlns":40,"display":2412},[42,6608,6609,6805],{},[45,6610,6611,6613,6803],{},[53,6612,378],{"fence":66},[1560,6614,6617,6666,6712,6758],{"rowspacing":6615,"columnalign":6616,"columnspacing":1564},"0.16em","center center center center",[1566,6618,6619,6639,6649,6659],{},[1569,6620,6621],{},[1572,6622,6623],{"scriptlevel":63,"displaystyle":58},[6624,6625,6627],"menclose",{"notation":6626},"box",[1572,6628,6629],{"scriptlevel":63,"displaystyle":58},[1572,6630,6631],{"scriptlevel":63,"displaystyle":58},[1572,6632,6633],{"scriptlevel":63,"displaystyle":66},[354,6634,6635,6637],{},[48,6636,2450],{},[61,6638,2453],{},[1569,6640,6641],{},[1572,6642,6643],{"scriptlevel":63,"displaystyle":58},[354,6644,6645,6647],{},[48,6646,2450],{},[61,6648,2462],{},[1569,6650,6651],{},[1572,6652,6653],{"scriptlevel":63,"displaystyle":58},[354,6654,6655,6657],{},[48,6656,2450],{},[61,6658,2471],{},[1569,6660,6661],{},[1572,6662,6663],{"scriptlevel":63,"displaystyle":58},[53,6664,6665],{"lspace":2419,"rspace":2419},"⋯",[1566,6667,6668,6678,6696,6706],{},[1569,6669,6670],{},[1572,6671,6672],{"scriptlevel":63,"displaystyle":58},[354,6673,6674,6676],{},[48,6675,2450],{},[61,6677,2513],{},[1569,6679,6680],{},[1572,6681,6682],{"scriptlevel":63,"displaystyle":58},[6624,6683,6684],{"notation":6626},[1572,6685,6686],{"scriptlevel":63,"displaystyle":58},[1572,6687,6688],{"scriptlevel":63,"displaystyle":58},[1572,6689,6690],{"scriptlevel":63,"displaystyle":66},[354,6691,6692,6694],{},[48,6693,2450],{},[61,6695,2522],{},[1569,6697,6698],{},[1572,6699,6700],{"scriptlevel":63,"displaystyle":58},[354,6701,6702,6704],{},[48,6703,2450],{},[61,6705,2531],{},[1569,6707,6708],{},[1572,6709,6710],{"scriptlevel":63,"displaystyle":58},[53,6711,6665],{"lspace":2419,"rspace":2419},[1566,6713,6714,6724,6734,6752],{},[1569,6715,6716],{},[1572,6717,6718],{"scriptlevel":63,"displaystyle":58},[354,6719,6720,6722],{},[48,6721,2450],{},[61,6723,2573],{},[1569,6725,6726],{},[1572,6727,6728],{"scriptlevel":63,"displaystyle":58},[354,6729,6730,6732],{},[48,6731,2450],{},[61,6733,2582],{},[1569,6735,6736],{},[1572,6737,6738],{"scriptlevel":63,"displaystyle":58},[6624,6739,6740],{"notation":6626},[1572,6741,6742],{"scriptlevel":63,"displaystyle":58},[1572,6743,6744],{"scriptlevel":63,"displaystyle":58},[1572,6745,6746],{"scriptlevel":63,"displaystyle":66},[354,6747,6748,6750],{},[48,6749,2450],{},[61,6751,2591],{},[1569,6753,6754],{},[1572,6755,6756],{"scriptlevel":63,"displaystyle":58},[53,6757,6665],{"lspace":2419,"rspace":2419},[1566,6759,6760,6772,6784,6796],{},[1569,6761,6762],{},[1572,6763,6764],{"scriptlevel":63,"displaystyle":58},[45,6765,6766,6768],{},[48,6767,2686],{"mathvariant":363},[2688,6769,6770],{"height":2419,"voffset":2419},[119,6771],{"mathbackground":2692,"width":2419,"height":2693},[1569,6773,6774],{},[1572,6775,6776],{"scriptlevel":63,"displaystyle":58},[45,6777,6778,6780],{},[48,6779,2686],{"mathvariant":363},[2688,6781,6782],{"height":2419,"voffset":2419},[119,6783],{"mathbackground":2692,"width":2419,"height":2693},[1569,6785,6786],{},[1572,6787,6788],{"scriptlevel":63,"displaystyle":58},[45,6789,6790,6792],{},[48,6791,2686],{"mathvariant":363},[2688,6793,6794],{"height":2419,"voffset":2419},[119,6795],{"mathbackground":2692,"width":2419,"height":2693},[1569,6797,6798],{},[1572,6799,6800],{"scriptlevel":63,"displaystyle":58},[53,6801,6802],{"lspace":2419,"rspace":2419},"⋱",[53,6804,387],{"fence":66},[94,6806,6807],{"encoding":96},"\\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}",[28,6809,6811],{"className":6810,"ariaHidden":66},[101],[28,6812,6814,6818],{"className":6813},[105],[28,6815],{"className":6816,"style":6817},[109],"height:6.4333em;vertical-align:-2.9667em;",[28,6819,6821,6871,7697],{"className":6820},[180],[28,6822,6824],{"className":6823},[138],[28,6825,6828],{"className":6826},[1681,6827],"mult",[28,6829,6831,6862],{"className":6830},[427,428],[28,6832,6834,6859],{"className":6833},[432],[28,6835,6838],{"className":6836,"style":6837},[436],"height:3.25em;",[28,6839,6841,6845],{"style":6840},"top:-5.25em;",[28,6842],{"className":6843,"style":6844},[444],"height:8em;",[28,6846,6848],{"style":6847},"width:0.875em;height:6em;",[6849,6850,6855],"svg",{"xmlns":6851,"width":6852,"height":6853,"viewBox":6854},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.875em","6em","0 0 875 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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","",[7752,7753,7754,7779,7786,7792,7798,7803,7814,7832,7860,7878,7906,7918,7936,7943,7948,7959,7984,8007,8020,8026,8031,8042],"code",{"__ignoreMap":7750},[28,7755,7758,7762,7766,7770,7772,7776],{"class":7756,"line":7757},"line",1,[28,7759,7761],{"class":7760},"snl16","def",[28,7763,7765],{"class":7764},"svObZ"," diagonal_construct",[28,7767,7769],{"class":7768},"s95oV","(number_list, digits",[28,7771,55],{"class":7760},[28,7773,7775],{"class":7774},"sDLfK","10",[28,7777,7778],{"class":7768},"):\n",[28,7780,7782],{"class":7756,"line":7781},2,[28,7783,7785],{"class":7784},"sU2Wk","    \"\"\"\n",[28,7787,7789],{"class":7756,"line":7788},3,[28,7790,7791],{"class":7784},"    number_list：字符串列表，每个形如 \"0.d1d2d3...\"\n",[28,7793,7795],{"class":7756,"line":7794},4,[28,7796,7797],{"class":7784},"    返回：通过对角线方法构造出的新数字字符串\n",[28,7799,7801],{"class":7756,"line":7800},5,[28,7802,7785],{"class":7784},[28,7804,7806,7809,7811],{"class":7756,"line":7805},6,[28,7807,7808],{"class":7768},"    new_digits ",[28,7810,55],{"class":7760},[28,7812,7813],{"class":7768}," []\n",[28,7815,7817,7820,7823,7826,7829],{"class":7756,"line":7816},7,[28,7818,7819],{"class":7760},"    for",[28,7821,7822],{"class":7768}," i, num_str ",[28,7824,7825],{"class":7760},"in",[28,7827,7828],{"class":7774}," enumerate",[28,7830,7831],{"class":7768},"(number_list):\n",[28,7833,7835,7838,7840,7843,7846,7849,7851,7854,7857],{"class":7756,"line":7834},8,[28,7836,7837],{"class":7768},"        decimal_part ",[28,7839,55],{"class":7760},[28,7841,7842],{"class":7768}," num_str.split(",[28,7844,7845],{"class":7784},"'.'",[28,7847,7848],{"class":7768},")[",[28,7850,70],{"class":7774},[28,7852,7853],{"class":7768},"].ljust(digits, ",[28,7855,7856],{"class":7784},"'0'",[28,7858,7859],{"class":7768},")\n",[28,7861,7863,7866,7868,7871,7874],{"class":7756,"line":7862},9,[28,7864,7865],{"class":7768},"        d ",[28,7867,55],{"class":7760},[28,7869,7870],{"class":7774}," int",[28,7872,7873],{"class":7768},"(decimal_part[i])       ",[28,7875,7877],{"class":7876},"sAwPA","# 第 i 个数的第 i 位（对角线数字）\n",[28,7879,7881,7884,7886,7889,7891,7894,7897,7900,7903],{"class":7756,"line":7880},10,[28,7882,7883],{"class":7768},"        new_d ",[28,7885,55],{"class":7760},[28,7887,7888],{"class":7768}," (d ",[28,7890,1610],{"class":7760},[28,7892,7893],{"class":7774}," 5",[28,7895,7896],{"class":7768},") ",[28,7898,7899],{"class":7760},"%",[28,7901,7902],{"class":7774}," 10",[28,7904,7905],{"class":7876},"           # 规则：使新数字与原数字不同\n",[28,7907,7909,7912,7915],{"class":7756,"line":7908},11,[28,7910,7911],{"class":7768},"        new_digits.append(",[28,7913,7914],{"class":7774},"str",[28,7916,7917],{"class":7768},"(new_d))\n",[28,7919,7921,7924,7927,7930,7933],{"class":7756,"line":7920},12,[28,7922,7923],{"class":7760},"    return",[28,7925,7926],{"class":7784}," \"0.\"",[28,7928,7929],{"class":7760}," +",[28,7931,7932],{"class":7784}," \"\"",[28,7934,7935],{"class":7768},".join(new_digits)\n",[28,7937,7939],{"class":7756,"line":7938},13,[28,7940,7942],{"emptyLinePlaceholder":7941},true,"\n",[28,7944,7946],{"class":7756,"line":7945},14,[28,7947,7942],{"emptyLinePlaceholder":7941},[28,7949,7951,7954,7956],{"class":7756,"line":7950},15,[28,7952,7953],{"class":7768},"sample_list ",[28,7955,55],{"class":7760},[28,7957,7958],{"class":7768}," [\n",[28,7960,7962,7965,7968,7971,7973,7976,7978,7981],{"class":7756,"line":7961},16,[28,7963,7964],{"class":7784},"    \"0.1234567890\"",[28,7966,7967],{"class":7768},", ",[28,7969,7970],{"class":7784},"\"0.9876543210\"",[28,7972,7967],{"class":7768},[28,7974,7975],{"class":7784},"\"0.5555555555\"",[28,7977,7967],{"class":7768},[28,7979,7980],{"class":7784},"\"0.1111111111\"",[28,7982,7983],{"class":7768},",\n",[28,7985,7987,7990,7992,7995,7997,8000,8002,8005],{"class":7756,"line":7986},17,[28,7988,7989],{"class":7784},"    \"0.3141592653\"",[28,7991,7967],{"class":7768},[28,7993,7994],{"class":7784},"\"0.2718281828\"",[28,7996,7967],{"class":7768},[28,7998,7999],{"class":7784},"\"0.0000000001\"",[28,8001,7967],{"class":7768},[28,8003,8004],{"class":7784},"\"0.9999999998\"",[28,8006,7983],{"class":7768},[28,8008,8010,8013,8015,8018],{"class":7756,"line":8009},18,[28,8011,8012],{"class":7784},"    \"0.4242424242\"",[28,8014,7967],{"class":7768},[28,8016,8017],{"class":7784},"\"0.6180339887\"",[28,8019,7983],{"class":7768},[28,8021,8023],{"class":7756,"line":8022},19,[28,8024,8025],{"class":7768},"]\n",[28,8027,8029],{"class":7756,"line":8028},20,[28,8030,7942],{"emptyLinePlaceholder":7941},[28,8032,8034,8037,8039],{"class":7756,"line":8033},21,[28,8035,8036],{"class":7768},"new_number ",[28,8038,55],{"class":7760},[28,8040,8041],{"class":7768}," diagonal_construct(sample_list)\n",[28,8043,8045,8048,8050,8053],{"class":7756,"line":8044},22,[28,8046,8047],{"class":7774},"print",[28,8049,378],{"class":7768},[28,8051,8052],{"class":7784},"\"对角线构造出的新数：\"",[28,8054,8055],{"class":7768},", new_number)\n",[11,8057,8058],{},"输出：",[7745,8060,8064],{"className":8061,"code":8063,"language":1801},[8062],"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",[7752,8065,8063],{"__ignoreMap":7750},[11,8067,8068,8069,8072,8073,8101,8102,8227],{},"可以清楚地看到，新构造的数 ",[7752,8070,8071],{},"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 ",[28,8074,8076,8089],{"className":8075,"translate":32},[31],[28,8077,8079],{"className":8078},[36],[38,8080,8081],{"xmlns":40},[42,8082,8083,8087],{},[45,8084,8085],{},[48,8086,5020],{},[94,8088,5020],{"encoding":96},[28,8090,8092],{"className":8091,"ariaHidden":66},[101],[28,8093,8095,8098],{"className":8094},[105],[28,8096],{"className":8097,"style":902},[109],[28,8099,5020],{"className":8100,"style":5081},[114,222],"，",[28,8103,8105,8127],{"className":8104,"translate":32},[31],[28,8106,8108],{"className":8107},[36],[38,8109,8110],{"xmlns":40},[42,8111,8112,8124],{},[45,8113,8114,8116,8118],{},[48,8115,4066],{},[53,8117,364],{"mathvariant":363},[354,8119,8120,8122],{},[48,8121,2430],{},[48,8123,5020],{},[94,8125,8126],{"encoding":96},"y \\neq x_k",[28,8128,8130,8181],{"className":8129,"ariaHidden":66},[101],[28,8131,8133,8136,8139,8142,8178],{"className":8132},[105],[28,8134],{"className":8135,"style":292},[109],[28,8137,4066],{"className":8138,"style":2123},[114,222],[28,8140],{"className":8141,"style":120},[119],[28,8143,8145,8172,8175],{"className":8144},[124],[28,8146,8148],{"className":8147},[124],[28,8149,8151],{"className":8150},[114,482],[28,8152,8154],{"className":8153},[486],[28,8155,8157,8160,8169],{"className":8156},[490],[28,8158],{"className":8159,"style":292},[109],[28,8161,8163],{"className":8162},[497],[28,8164,8166],{"className":8165},[114],[28,8167,504],{"className":8168},[124],[28,8170],{"className":8171},[508],[28,8173],{"className":8174},[119,512],[28,8176,55],{"className":8177},[124],[28,8179],{"className":8180,"style":120},[119],[28,8182,8184,8187],{"className":8183},[105],[28,8185],{"className":8186,"style":525},[109],[28,8188,8190,8193],{"className":8189},[114],[28,8191,2430],{"className":8192},[114,222],[28,8194,8196],{"className":8195},[423],[28,8197,8199,8219],{"className":8198},[427,428],[28,8200,8202,8216],{"className":8201},[432],[28,8203,8205],{"className":8204,"style":5069},[436],[28,8206,8207,8210],{"style":440},[28,8208],{"className":8209,"style":445},[444],[28,8211,8213],{"className":8212},[449,450,451,452],[28,8214,5020],{"className":8215,"style":5081},[114,222,452],[28,8217,460],{"className":8218},[459],[28,8220,8222],{"className":8221},[432],[28,8223,8225],{"className":8224,"style":467},[436],[28,8226],{},"——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。",[18,8229,8230],{"id":8230},"康托尔定理",[11,8232,8233],{},"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。",[2132,8235,8236],{},[11,8237,8238,8239,8267,8268,5404,8316,8344,8345,8373,8374,1369],{},"康托尔定理。对任意集合 ",[28,8240,8242,8255],{"className":8241,"translate":32},[31],[28,8243,8245],{"className":8244},[36],[38,8246,8247],{"xmlns":40},[42,8248,8249,8253],{},[45,8250,8251],{},[48,8252,206],{},[94,8254,206],{"encoding":96},[28,8256,8258],{"className":8257,"ariaHidden":66},[101],[28,8259,8261,8264],{"className":8260},[105],[28,8262],{"className":8263,"style":218},[109],[28,8265,206],{"className":8266},[114,222],"，其幂集 ",[28,8269,8271,8293],{"className":8270,"translate":32},[31],[28,8272,8274],{"className":8273},[36],[38,8275,8276],{"xmlns":40},[42,8277,8278,8290],{},[45,8279,8280,8284,8286,8288],{},[48,8281,8283],{"mathvariant":8282},"script","P",[53,8285,378],{"stretchy":58},[48,8287,206],{},[53,8289,387],{"stretchy":58},[94,8291,8292],{"encoding":96},"\\mathcal{P}(A)",[28,8294,8296],{"className":8295,"ariaHidden":66},[101],[28,8297,8299,8302,8307,8310,8313],{"className":8298},[105],[28,8300],{"className":8301,"style":134},[109],[28,8303,8283],{"className":8304,"style":8306},[114,8305],"mathcal","margin-right:0.0822em;",[28,8308,378],{"className":8309},[138],[28,8311,206],{"className":8312},[114,222],[28,8314,387],{"className":8315},[187],[28,8317,8319,8332],{"className":8318,"translate":32},[31],[28,8320,8322],{"className":8321},[36],[38,8323,8324],{"xmlns":40},[42,8325,8326,8330],{},[45,8327,8328],{},[48,8329,206],{},[94,8331,206],{"encoding":96},[28,8333,8335],{"className":8334,"ariaHidden":66},[101],[28,8336,8338,8341],{"className":8337},[105],[28,8339],{"className":8340,"style":218},[109],[28,8342,206],{"className":8343},[114,222]," 的所有子集构成的集合）的势严格大于 ",[28,8346,8348,8361],{"className":8347,"translate":32},[31],[28,8349,8351],{"className":8350},[36],[38,8352,8353],{"xmlns":40},[42,8354,8355,8359],{},[45,8356,8357],{},[48,8358,206],{},[94,8360,206],{"encoding":96},[28,8362,8364],{"className":8363,"ariaHidden":66},[101],[28,8365,8367,8370],{"className":8366},[105],[28,8368],{"className":8369,"style":218},[109],[28,8371,206],{"className":8372},[114,222]," 本身，即 ",[28,8375,8377,8410],{"className":8376,"translate":32},[31],[28,8378,8380],{"className":8379},[36],[38,8381,8382],{"xmlns":40},[42,8383,8384,8407],{},[45,8385,8386,8388,8390,8392,8395,8397,8399,8401,8403,8405],{},[48,8387,1010],{"mathvariant":363},[48,8389,206],{},[48,8391,1010],{"mathvariant":363},[53,8393,8394],{},"\u003C",[48,8396,1010],{"mathvariant":363},[48,8398,8283],{"mathvariant":8282},[53,8400,378],{"stretchy":58},[48,8402,206],{},[53,8404,387],{"stretchy":58},[48,8406,1010],{"mathvariant":363},[94,8408,8409],{"encoding":96},"|A| \u003C |\\mathcal{P}(A)|",[28,8411,8413,8437],{"className":8412,"ariaHidden":66},[101],[28,8414,8416,8419,8422,8425,8428,8431,8434],{"className":8415},[105],[28,8417],{"className":8418,"style":134},[109],[28,8420,1010],{"className":8421},[114],[28,8423,206],{"className":8424},[114,222],[28,8426,1010],{"className":8427},[114],[28,8429],{"className":8430,"style":120},[119],[28,8432,8394],{"className":8433},[124],[28,8435],{"className":8436,"style":120},[119],[28,8438,8440,8443,8446,8449,8452,8455,8458],{"className":8439},[105],[28,8441],{"className":8442,"style":134},[109],[28,8444,1010],{"className":8445},[114],[28,8447,8283],{"className":8448,"style":8306},[114,8305],[28,8450,378],{"className":8451},[138],[28,8453,206],{"className":8454},[114,222],[28,8456,387],{"className":8457},[187],[28,8459,1010],{"className":8460},[114],[11,8462,8463,8464,8492,8493,8536],{},"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 ",[28,8465,8467,8480],{"className":8466,"translate":32},[31],[28,8468,8470],{"className":8469},[36],[38,8471,8472],{"xmlns":40},[42,8473,8474,8478],{},[45,8475,8476],{},[48,8477,206],{},[94,8479,206],{"encoding":96},[28,8481,8483],{"className":8482,"ariaHidden":66},[101],[28,8484,8486,8489],{"className":8485},[105],[28,8487],{"className":8488,"style":218},[109],[28,8490,206],{"className":8491},[114,222]," 多大，",[28,8494,8496,8515],{"className":8495,"translate":32},[31],[28,8497,8499],{"className":8498},[36],[38,8500,8501],{"xmlns":40},[42,8502,8503,8513],{},[45,8504,8505,8507,8509,8511],{},[48,8506,8283],{"mathvariant":8282},[53,8508,378],{"stretchy":58},[48,8510,206],{},[53,8512,387],{"stretchy":58},[94,8514,8292],{"encoding":96},[28,8516,8518],{"className":8517,"ariaHidden":66},[101],[28,8519,8521,8524,8527,8530,8533],{"className":8520},[105],[28,8522],{"className":8523,"style":134},[109],[28,8525,8283],{"className":8526,"style":8306},[114,8305],[28,8528,378],{"className":8529},[138],[28,8531,206],{"className":8532},[114,222],[28,8534,387],{"className":8535},[187]," 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。",[8538,8539,8540],"h3",{"id":8540},"证明",[11,8542,8543,8544,8632,8633,4048],{},"假设存在一个满射 ",[28,8545,8547,8575],{"className":8546,"translate":32},[31],[28,8548,8550],{"className":8549},[36],[38,8551,8552],{"xmlns":40},[42,8553,8554,8572],{},[45,8555,8556,8558,8560,8562,8564,8566,8568,8570],{},[48,8557,269],{},[53,8559,272],{},[48,8561,206],{},[53,8563,277],{},[48,8565,8283],{"mathvariant":8282},[53,8567,378],{"stretchy":58},[48,8569,206],{},[53,8571,387],{"stretchy":58},[94,8573,8574],{"encoding":96},"f: A \\to \\mathcal{P}(A)",[28,8576,8578,8596,8614],{"className":8577,"ariaHidden":66},[101],[28,8579,8581,8584,8587,8590,8593],{"className":8580},[105],[28,8582],{"className":8583,"style":292},[109],[28,8585,269],{"className":8586,"style":296},[114,222],[28,8588],{"className":8589,"style":120},[119],[28,8591,272],{"className":8592},[124],[28,8594],{"className":8595,"style":120},[119],[28,8597,8599,8602,8605,8608,8611],{"className":8598},[105],[28,8600],{"className":8601,"style":218},[109],[28,8603,206],{"className":8604},[114,222],[28,8606],{"className":8607,"style":120},[119],[28,8609,277],{"className":8610},[124],[28,8612],{"className":8613,"style":120},[119],[28,8615,8617,8620,8623,8626,8629],{"className":8616},[105],[28,8618],{"className":8619,"style":134},[109],[28,8621,8283],{"className":8622,"style":8306},[114,8305],[28,8624,378],{"className":8625},[138],[28,8627,206],{"className":8628},[114,222],[28,8630,387],{"className":8631},[187],"（我们将证明这样的满射不存在，由此得到 ",[28,8634,8636,8667],{"className":8635,"translate":32},[31],[28,8637,8639],{"className":8638},[36],[38,8640,8641],{"xmlns":40},[42,8642,8643,8665],{},[45,8644,8645,8647,8649,8651,8653,8655,8657,8659,8661,8663],{},[48,8646,1010],{"mathvariant":363},[48,8648,206],{},[48,8650,1010],{"mathvariant":363},[53,8652,8394],{},[48,8654,1010],{"mathvariant":363},[48,8656,8283],{"mathvariant":8282},[53,8658,378],{"stretchy":58},[48,8660,206],{},[53,8662,387],{"stretchy":58},[48,8664,1010],{"mathvariant":363},[94,8666,8409],{"encoding":96},[28,8668,8670,8694],{"className":8669,"ariaHidden":66},[101],[28,8671,8673,8676,8679,8682,8685,8688,8691],{"className":8672},[105],[28,8674],{"className":8675,"style":134},[109],[28,8677,1010],{"className":8678},[114],[28,8680,206],{"className":8681},[114,222],[28,8683,1010],{"className":8684},[114],[28,8686],{"className":8687,"style":120},[119],[28,8689,8394],{"className":8690},[124],[28,8692],{"className":8693,"style":120},[119],[28,8695,8697,8700,8703,8706,8709,8712,8715],{"className":8696},[105],[28,8698],{"className":8699,"style":134},[109],[28,8701,1010],{"className":8702},[114],[28,8704,8283],{"className":8705,"style":8306},[114,8305],[28,8707,378],{"className":8708},[138],[28,8710,206],{"className":8711},[114,222],[28,8713,387],{"className":8714},[187],[28,8716,1010],{"className":8717},[114],[11,8719,8720,8721,8101,8771,8815,8816,8844],{},"对每个元素 ",[28,8722,8724,8741],{"className":8723,"translate":32},[31],[28,8725,8727],{"className":8726},[36],[38,8728,8729],{"xmlns":40},[42,8730,8731,8739],{},[45,8732,8733,8735,8737],{},[48,8734,358],{},[53,8736,748],{},[48,8738,206],{},[94,8740,806],{"encoding":96},[28,8742,8744,8762],{"className":8743,"ariaHidden":66},[101],[28,8745,8747,8750,8753,8756,8759],{"className":8746},[105],[28,8748],{"className":8749,"style":816},[109],[28,8751,358],{"className":8752},[114,222],[28,8754],{"className":8755,"style":120},[119],[28,8757,748],{"className":8758},[124],[28,8760],{"className":8761,"style":120},[119],[28,8763,8765,8768],{"className":8764},[105],[28,8766],{"className":8767,"style":218},[109],[28,8769,206],{"className":8770},[114,222],[28,8772,8774,8794],{"className":8773,"translate":32},[31],[28,8775,8777],{"className":8776},[36],[38,8778,8779],{"xmlns":40},[42,8780,8781,8791],{},[45,8782,8783,8785,8787,8789],{},[48,8784,269],{},[53,8786,378],{"stretchy":58},[48,8788,358],{},[53,8790,387],{"stretchy":58},[94,8792,8793],{"encoding":96},"f(a)",[28,8795,8797],{"className":8796,"ariaHidden":66},[101],[28,8798,8800,8803,8806,8809,8812],{"className":8799},[105],[28,8801],{"className":8802,"style":134},[109],[28,8804,269],{"className":8805,"style":296},[114,222],[28,8807,378],{"className":8808},[138],[28,8810,358],{"className":8811},[114,222],[28,8813,387],{"className":8814},[187]," 都是 ",[28,8817,8819,8832],{"className":8818,"translate":32},[31],[28,8820,8822],{"className":8821},[36],[38,8823,8824],{"xmlns":40},[42,8825,8826,8830],{},[45,8827,8828],{},[48,8829,206],{},[94,8831,206],{"encoding":96},[28,8833,8835],{"className":8834,"ariaHidden":66},[101],[28,8836,8838,8841],{"className":8837},[105],[28,8839],{"className":8840,"style":218},[109],[28,8842,206],{"className":8843},[114,222]," 的一个子集。现在构造一个特殊的子集：",[28,8846,8848],{"className":8847,"translate":32},[2403],[28,8849,8851,8897],{"className":8850,"translate":32},[31],[28,8852,8854],{"className":8853},[36],[38,8855,8856],{"xmlns":40,"display":2412},[42,8857,8858,8894],{},[45,8859,8860,8863,8865,8867,8869,8871,8873,8875,8877,8879,8882,8884,8886,8888,8890,8892],{},[48,8861,8862],{},"D",[53,8864,55],{},[53,8866,59],{"stretchy":58},[87,8868,89],{},[48,8870,358],{},[53,8872,748],{},[48,8874,206],{},[53,8876,272],{},[48,8878,358],{},[53,8880,8881],{"mathvariant":363},"∉",[48,8883,269],{},[53,8885,378],{"stretchy":58},[48,8887,358],{},[53,8889,387],{"stretchy":58},[87,8891,89],{},[53,8893,92],{"stretchy":58},[94,8895,8896],{"encoding":96},"D = \\{\\, a \\in A : a \\notin f(a) \\,\\}",[28,8898,8900,8919,8943,8961,9014],{"className":8899,"ariaHidden":66},[101],[28,8901,8903,8906,8910,8913,8916],{"className":8902},[105],[28,8904],{"className":8905,"style":218},[109],[28,8907,8862],{"className":8908,"style":8909},[114,222],"margin-right:0.0278em;",[28,8911],{"className":8912,"style":120},[119],[28,8914,55],{"className":8915},[124],[28,8917],{"className":8918,"style":120},[119],[28,8920,8922,8925,8928,8931,8934,8937,8940],{"className":8921},[105],[28,8923],{"className":8924,"style":134},[109],[28,8926,59],{"className":8927},[138],[28,8929],{"className":8930,"style":149},[119],[28,8932,358],{"className":8933},[114,222],[28,8935],{"className":8936,"style":120},[119],[28,8938,748],{"className":8939},[124],[28,8941],{"className":8942,"style":120},[119],[28,8944,8946,8949,8952,8955,8958],{"className":8945},[105],[28,8947],{"className":8948,"style":218},[109],[28,8950,206],{"className":8951},[114,222],[28,8953],{"className":8954,"style":120},[119],[28,8956,272],{"className":8957},[124],[28,8959],{"className":8960,"style":120},[119],[28,8962,8964,8967,8970,8973,9011],{"className":8963},[105],[28,8965],{"className":8966,"style":134},[109],[28,8968,358],{"className":8969},[114,222],[28,8971],{"className":8972,"style":120},[119],[28,8974,8976,8982],{"className":8975},[124],[28,8977,8979],{"className":8978},[114],[28,8980,748],{"className":8981},[124],[28,8983,8985],{"className":8984},[114,482],[28,8986,8988],{"className":8987},[486],[28,8989,8992,8995,9008],{"className":8990},[8991],"llap",[28,8993],{"className":8994,"style":134},[109],[28,8996,8998],{"className":8997},[497],[28,8999,9001,9004],{"className":9000},[114],[28,9002,1580],{"className":9003},[114],[28,9005],{"className":9006,"style":9007},[119],"margin-right:0.0556em;",[28,9009],{"className":9010},[508],[28,9012],{"className":9013,"style":120},[119],[28,9015,9017,9020,9023,9026,9029,9032,9035],{"className":9016},[105],[28,9018],{"className":9019,"style":134},[109],[28,9021,269],{"className":9022,"style":296},[114,222],[28,9024,378],{"className":9025},[138],[28,9027,358],{"className":9028},[114,222],[28,9030,387],{"className":9031},[187],[28,9033],{"className":9034,"style":149},[119],[28,9036,92],{"className":9037},[187],[11,9039,9040],{},"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。",[11,9042,9043,9044,9072,9073,9101,9102,9130,9131,9174,9175,838,9226,1369],{},"由于 ",[28,9045,9047,9060],{"className":9046,"translate":32},[31],[28,9048,9050],{"className":9049},[36],[38,9051,9052],{"xmlns":40},[42,9053,9054,9058],{},[45,9055,9056],{},[48,9057,269],{},[94,9059,269],{"encoding":96},[28,9061,9063],{"className":9062,"ariaHidden":66},[101],[28,9064,9066,9069],{"className":9065},[105],[28,9067],{"className":9068,"style":292},[109],[28,9070,269],{"className":9071,"style":296},[114,222]," 是满射，",[28,9074,9076,9089],{"className":9075,"translate":32},[31],[28,9077,9079],{"className":9078},[36],[38,9080,9081],{"xmlns":40},[42,9082,9083,9087],{},[45,9084,9085],{},[48,9086,8862],{},[94,9088,8862],{"encoding":96},[28,9090,9092],{"className":9091,"ariaHidden":66},[101],[28,9093,9095,9098],{"className":9094},[105],[28,9096],{"className":9097,"style":218},[109],[28,9099,8862],{"className":9100,"style":8909},[114,222],"（作为 ",[28,9103,9105,9118],{"className":9104,"translate":32},[31],[28,9106,9108],{"className":9107},[36],[38,9109,9110],{"xmlns":40},[42,9111,9112,9116],{},[45,9113,9114],{},[48,9115,206],{},[94,9117,206],{"encoding":96},[28,9119,9121],{"className":9120,"ariaHidden":66},[101],[28,9122,9124,9127],{"className":9123},[105],[28,9125],{"className":9126,"style":218},[109],[28,9128,206],{"className":9129},[114,222]," 的子集，因而是 ",[28,9132,9134,9153],{"className":9133,"translate":32},[31],[28,9135,9137],{"className":9136},[36],[38,9138,9139],{"xmlns":40},[42,9140,9141,9151],{},[45,9142,9143,9145,9147,9149],{},[48,9144,8283],{"mathvariant":8282},[53,9146,378],{"stretchy":58},[48,9148,206],{},[53,9150,387],{"stretchy":58},[94,9152,8292],{"encoding":96},[28,9154,9156],{"className":9155,"ariaHidden":66},[101],[28,9157,9159,9162,9165,9168,9171],{"className":9158},[105],[28,9160],{"className":9161,"style":134},[109],[28,9163,8283],{"className":9164,"style":8306},[114,8305],[28,9166,378],{"className":9167},[138],[28,9169,206],{"className":9170},[114,222],[28,9172,387],{"className":9173},[187]," 的元素）必有原像；即存在某个 ",[28,9176,9178,9196],{"className":9177,"translate":32},[31],[28,9179,9181],{"className":9180},[36],[38,9182,9183],{"xmlns":40},[42,9184,9185,9193],{},[45,9186,9187,9189,9191],{},[48,9188,2450],{},[53,9190,748],{},[48,9192,206],{},[94,9194,9195],{"encoding":96},"d \\in A",[28,9197,9199,9217],{"className":9198,"ariaHidden":66},[101],[28,9200,9202,9205,9208,9211,9214],{"className":9201},[105],[28,9203],{"className":9204,"style":763},[109],[28,9206,2450],{"className":9207},[114,222],[28,9209],{"className":9210,"style":120},[119],[28,9212,748],{"className":9213},[124],[28,9215],{"className":9216,"style":120},[119],[28,9218,9220,9223],{"className":9219},[105],[28,9221],{"className":9222,"style":218},[109],[28,9224,206],{"className":9225},[114,222],[28,9227,9229,9253],{"className":9228,"translate":32},[31],[28,9230,9232],{"className":9231},[36],[38,9233,9234],{"xmlns":40},[42,9235,9236,9250],{},[45,9237,9238,9240,9242,9244,9246,9248],{},[48,9239,269],{},[53,9241,378],{"stretchy":58},[48,9243,2450],{},[53,9245,387],{"stretchy":58},[53,9247,55],{},[48,9249,8862],{},[94,9251,9252],{"encoding":96},"f(d) = D",[28,9254,9256,9283],{"className":9255,"ariaHidden":66},[101],[28,9257,9259,9262,9265,9268,9271,9274,9277,9280],{"className":9258},[105],[28,9260],{"className":9261,"style":134},[109],[28,9263,269],{"className":9264,"style":296},[114,222],[28,9266,378],{"className":9267},[138],[28,9269,2450],{"className":9270},[114,222],[28,9272,387],{"className":9273},[187],[28,9275],{"className":9276,"style":120},[119],[28,9278,55],{"className":9279},[124],[28,9281],{"className":9282,"style":120},[119],[28,9284,9286,9289],{"className":9285},[105],[28,9287],{"className":9288,"style":218},[109],[28,9290,8862],{"className":9291,"style":8909},[114,222],[11,9293,9294,9295,9346],{},"现在问：",[28,9296,9298,9316],{"className":9297,"translate":32},[31],[28,9299,9301],{"className":9300},[36],[38,9302,9303],{"xmlns":40},[42,9304,9305,9313],{},[45,9306,9307,9309,9311],{},[48,9308,2450],{},[53,9310,748],{},[48,9312,8862],{},[94,9314,9315],{"encoding":96},"d \\in D",[28,9317,9319,9337],{"className":9318,"ariaHidden":66},[101],[28,9320,9322,9325,9328,9331,9334],{"className":9321},[105],[28,9323],{"className":9324,"style":763},[109],[28,9326,2450],{"className":9327},[114,222],[28,9329],{"className":9330,"style":120},[119],[28,9332,748],{"className":9333},[124],[28,9335],{"className":9336,"style":120},[119],[28,9338,9340,9343],{"className":9339},[105],[28,9341],{"className":9342,"style":218},[109],[28,9344,8862],{"className":9345,"style":8909},[114,222]," 吗？",[335,9348,9349,9793],{},[338,9350,9351,9352,9402,9403,9431,9432,9587,9588,9653,9654,9792],{},"若 ",[28,9353,9355,9372],{"className":9354,"translate":32},[31],[28,9356,9358],{"className":9357},[36],[38,9359,9360],{"xmlns":40},[42,9361,9362,9370],{},[45,9363,9364,9366,9368],{},[48,9365,2450],{},[53,9367,748],{},[48,9369,8862],{},[94,9371,9315],{"encoding":96},[28,9373,9375,9393],{"className":9374,"ariaHidden":66},[101],[28,9376,9378,9381,9384,9387,9390],{"className":9377},[105],[28,9379],{"className":9380,"style":763},[109],[28,9382,2450],{"className":9383},[114,222],[28,9385],{"className":9386,"style":120},[119],[28,9388,748],{"className":9389},[124],[28,9391],{"className":9392,"style":120},[119],[28,9394,9396,9399],{"className":9395},[105],[28,9397],{"className":9398,"style":218},[109],[28,9400,8862],{"className":9401,"style":8909},[114,222],"：根据 ",[28,9404,9406,9419],{"className":9405,"translate":32},[31],[28,9407,9409],{"className":9408},[36],[38,9410,9411],{"xmlns":40},[42,9412,9413,9417],{},[45,9414,9415],{},[48,9416,8862],{},[94,9418,8862],{"encoding":96},[28,9420,9422],{"className":9421,"ariaHidden":66},[101],[28,9423,9425,9428],{"className":9424},[105],[28,9426],{"className":9427,"style":218},[109],[28,9429,8862],{"className":9430,"style":8909},[114,222]," 的定义，",[28,9433,9435,9472],{"className":9434,"translate":32},[31],[28,9436,9438],{"className":9437},[36],[38,9439,9440],{"xmlns":40},[42,9441,9442,9469],{},[45,9443,9444,9446,9448,9450,9452,9455,9457,9459,9461,9463,9465,9467],{},[48,9445,2450],{},[53,9447,748],{},[48,9449,8862],{},[87,9451,5539],{},[53,9453,9454],{},"⟺",[87,9456,5539],{},[48,9458,2450],{},[53,9460,8881],{"mathvariant":363},[48,9462,269],{},[53,9464,378],{"stretchy":58},[48,9466,2450],{},[53,9468,387],{"stretchy":58},[94,9470,9471],{"encoding":96},"d \\in D \\iff d \\notin f(d)",[28,9473,9475,9493,9518,9569],{"className":9474,"ariaHidden":66},[101],[28,9476,9478,9481,9484,9487,9490],{"className":9477},[105],[28,9479],{"className":9480,"style":763},[109],[28,9482,2450],{"className":9483},[114,222],[28,9485],{"className":9486,"style":120},[119],[28,9488,748],{"className":9489},[124],[28,9491],{"className":9492,"style":120},[119],[28,9494,9496,9500,9503,9506,9509,9512,9515],{"className":9495},[105],[28,9497],{"className":9498,"style":9499},[109],"height:0.7073em;vertical-align:-0.024em;",[28,9501,8862],{"className":9502,"style":8909},[114,222],[28,9504],{"className":9505,"style":120},[119],[28,9507],{"className":9508,"style":120},[119],[28,9510,9454],{"className":9511},[124],[28,9513],{"className":9514,"style":120},[119],[28,9516],{"className":9517,"style":120},[119],[28,9519,9521,9524,9527,9530,9566],{"className":9520},[105],[28,9522],{"className":9523,"style":134},[109],[28,9525,2450],{"className":9526},[114,222],[28,9528],{"className":9529,"style":120},[119],[28,9531,9533,9539],{"className":9532},[124],[28,9534,9536],{"className":9535},[114],[28,9537,748],{"className":9538},[124],[28,9540,9542],{"className":9541},[114,482],[28,9543,9545],{"className":9544},[486],[28,9546,9548,9551,9563],{"className":9547},[8991],[28,9549],{"className":9550,"style":134},[109],[28,9552,9554],{"className":9553},[497],[28,9555,9557,9560],{"className":9556},[114],[28,9558,1580],{"className":9559},[114],[28,9561],{"className":9562,"style":9007},[119],[28,9564],{"className":9565},[508],[28,9567],{"className":9568,"style":120},[119],[28,9570,9572,9575,9578,9581,9584],{"className":9571},[105],[28,9573],{"className":9574,"style":134},[109],[28,9576,269],{"className":9577,"style":296},[114,222],[28,9579,378],{"className":9580},[138],[28,9582,2450],{"className":9583},[114,222],[28,9585,387],{"className":9586},[187],"。由于 ",[28,9589,9591,9614],{"className":9590,"translate":32},[31],[28,9592,9594],{"className":9593},[36],[38,9595,9596],{"xmlns":40},[42,9597,9598,9612],{},[45,9599,9600,9602,9604,9606,9608,9610],{},[48,9601,269],{},[53,9603,378],{"stretchy":58},[48,9605,2450],{},[53,9607,387],{"stretchy":58},[53,9609,55],{},[48,9611,8862],{},[94,9613,9252],{"encoding":96},[28,9615,9617,9644],{"className":9616,"ariaHidden":66},[101],[28,9618,9620,9623,9626,9629,9632,9635,9638,9641],{"className":9619},[105],[28,9621],{"className":9622,"style":134},[109],[28,9624,269],{"className":9625,"style":296},[114,222],[28,9627,378],{"className":9628},[138],[28,9630,2450],{"className":9631},[114,222],[28,9633,387],{"className":9634},[187],[28,9636],{"className":9637,"style":120},[119],[28,9639,55],{"className":9640},[124],[28,9642],{"className":9643,"style":120},[119],[28,9645,9647,9650],{"className":9646},[105],[28,9648],{"className":9649,"style":218},[109],[28,9651,8862],{"className":9652,"style":8909},[114,222],"，就有 ",[28,9655,9657,9687],{"className":9656,"translate":32},[31],[28,9658,9660],{"className":9659},[36],[38,9661,9662],{"xmlns":40},[42,9663,9664,9684],{},[45,9665,9666,9668,9670,9672,9674,9676,9678,9680,9682],{},[48,9667,2450],{},[53,9669,748],{},[48,9671,8862],{},[87,9673,5539],{},[53,9675,9454],{},[87,9677,5539],{},[48,9679,2450],{},[53,9681,8881],{"mathvariant":363},[48,9683,8862],{},[94,9685,9686],{"encoding":96},"d \\in D \\iff d \\notin D",[28,9688,9690,9708,9732,9783],{"className":9689,"ariaHidden":66},[101],[28,9691,9693,9696,9699,9702,9705],{"className":9692},[105],[28,9694],{"className":9695,"style":763},[109],[28,9697,2450],{"className":9698},[114,222],[28,9700],{"className":9701,"style":120},[119],[28,9703,748],{"className":9704},[124],[28,9706],{"className":9707,"style":120},[119],[28,9709,9711,9714,9717,9720,9723,9726,9729],{"className":9710},[105],[28,9712],{"className":9713,"style":9499},[109],[28,9715,8862],{"className":9716,"style":8909},[114,222],[28,9718],{"className":9719,"style":120},[119],[28,9721],{"className":9722,"style":120},[119],[28,9724,9454],{"className":9725},[124],[28,9727],{"className":9728,"style":120},[119],[28,9730],{"className":9731,"style":120},[119],[28,9733,9735,9738,9741,9744,9780],{"className":9734},[105],[28,9736],{"className":9737,"style":134},[109],[28,9739,2450],{"className":9740},[114,222],[28,9742],{"className":9743,"style":120},[119],[28,9745,9747,9753],{"className":9746},[124],[28,9748,9750],{"className":9749},[114],[28,9751,748],{"className":9752},[124],[28,9754,9756],{"className":9755},[114,482],[28,9757,9759],{"className":9758},[486],[28,9760,9762,9765,9777],{"className":9761},[8991],[28,9763],{"className":9764,"style":134},[109],[28,9766,9768],{"className":9767},[497],[28,9769,9771,9774],{"className":9770},[114],[28,9772,1580],{"className":9773},[114],[28,9775],{"className":9776,"style":9007},[119],[28,9778],{"className":9779},[508],[28,9781],{"className":9782,"style":120},[119],[28,9784,9786,9789],{"className":9785},[105],[28,9787],{"className":9788,"style":218},[109],[28,9790,8862],{"className":9791,"style":8909},[114,222],"。矛盾！",[338,9794,9351,9795,9879,9880,10080,10081,10169,10170,9792],{},[28,9796,9798,9816],{"className":9797,"translate":32},[31],[28,9799,9801],{"className":9800},[36],[38,9802,9803],{"xmlns":40},[42,9804,9805,9813],{},[45,9806,9807,9809,9811],{},[48,9808,2450],{},[53,9810,8881],{"mathvariant":363},[48,9812,8862],{},[94,9814,9815],{"encoding":96},"d \\notin D",[28,9817,9819,9870],{"className":9818,"ariaHidden":66},[101],[28,9820,9822,9825,9828,9831,9867],{"className":9821},[105],[28,9823],{"className":9824,"style":134},[109],[28,9826,2450],{"className":9827},[114,222],[28,9829],{"className":9830,"style":120},[119],[28,9832,9834,9840],{"className":9833},[124],[28,9835,9837],{"className":9836},[114],[28,9838,748],{"className":9839},[124],[28,9841,9843],{"className":9842},[114,482],[28,9844,9846],{"className":9845},[486],[28,9847,9849,9852,9864],{"className":9848},[8991],[28,9850],{"className":9851,"style":134},[109],[28,9853,9855],{"className":9854},[497],[28,9856,9858,9861],{"className":9857},[114],[28,9859,1580],{"className":9860},[114],[28,9862],{"className":9863,"style":9007},[119],[28,9865],{"className":9866},[508],[28,9868],{"className":9869,"style":120},[119],[28,9871,9873,9876],{"className":9872},[105],[28,9874],{"className":9875,"style":218},[109],[28,9877,8862],{"className":9878,"style":8909},[114,222],"：同理，",[28,9881,9883,9926],{"className":9882,"translate":32},[31],[28,9884,9886],{"className":9885},[36],[38,9887,9888],{"xmlns":40},[42,9889,9890,9923],{},[45,9891,9892,9894,9896,9898,9900,9902,9904,9907,9909,9911,9913,9915,9917,9919,9921],{},[48,9893,2450],{},[53,9895,8881],{"mathvariant":363},[48,9897,8862],{},[87,9899,5539],{},[53,9901,9454],{},[87,9903,5539],{},[48,9905,9906],{"mathvariant":363},"¬",[53,9908,378],{"stretchy":58},[48,9910,2450],{},[53,9912,8881],{"mathvariant":363},[48,9914,269],{},[53,9916,378],{"stretchy":58},[48,9918,2450],{},[53,9920,387],{"stretchy":58},[53,9922,387],{"stretchy":58},[94,9924,9925],{"encoding":96},"d \\notin D \\iff \\neg(d \\notin f(d))",[28,9927,9929,9980,10004,10061],{"className":9928,"ariaHidden":66},[101],[28,9930,9932,9935,9938,9941,9977],{"className":9931},[105],[28,9933],{"className":9934,"style":134},[109],[28,9936,2450],{"className":9937},[114,222],[28,9939],{"className":9940,"style":120},[119],[28,9942,9944,9950],{"className":9943},[124],[28,9945,9947],{"className":9946},[114],[28,9948,748],{"className":9949},[124],[28,9951,9953],{"className":9952},[114,482],[28,9954,9956],{"className":9955},[486],[28,9957,9959,9962,9974],{"className":9958},[8991],[28,9960],{"className":9961,"style":134},[109],[28,9963,9965],{"className":9964},[497],[28,9966,9968,9971],{"className":9967},[114],[28,9969,1580],{"className":9970},[114],[28,9972],{"className":9973,"style":9007},[119],[28,9975],{"className":9976},[508],[28,9978],{"className":9979,"style":120},[119],[28,9981,9983,9986,9989,9992,9995,9998,10001],{"className":9982},[105],[28,9984],{"className":9985,"style":9499},[109],[28,9987,8862],{"className":9988,"style":8909},[114,222],[28,9990],{"className":9991,"style":120},[119],[28,9993],{"className":9994,"style":120},[119],[28,9996,9454],{"className":9997},[124],[28,9999],{"className":10000,"style":120},[119],[28,10002],{"className":10003,"style":120},[119],[28,10005,10007,10010,10013,10016,10019,10022,10058],{"className":10006},[105],[28,10008],{"className":10009,"style":134},[109],[28,10011,9906],{"className":10012},[114],[28,10014,378],{"className":10015},[138],[28,10017,2450],{"className":10018},[114,222],[28,10020],{"className":10021,"style":120},[119],[28,10023,10025,10031],{"className":10024},[124],[28,10026,10028],{"className":10027},[114],[28,10029,748],{"className":10030},[124],[28,10032,10034],{"className":10033},[114,482],[28,10035,10037],{"className":10036},[486],[28,10038,10040,10043,10055],{"className":10039},[8991],[28,10041],{"className":10042,"style":134},[109],[28,10044,10046],{"className":10045},[497],[28,10047,10049,10052],{"className":10048},[114],[28,10050,1580],{"className":10051},[114],[28,10053],{"className":10054,"style":9007},[119],[28,10056],{"className":10057},[508],[28,10059],{"className":10060,"style":120},[119],[28,10062,10064,10067,10070,10073,10076],{"className":10063},[105],[28,10065],{"className":10066,"style":134},[109],[28,10068,269],{"className":10069,"style":296},[114,222],[28,10071,378],{"className":10072},[138],[28,10074,2450],{"className":10075},[114,222],[28,10077,10079],{"className":10078},[187],"))","，即 ",[28,10082,10084,10112],{"className":10083,"translate":32},[31],[28,10085,10087],{"className":10086},[36],[38,10088,10089],{"xmlns":40},[42,10090,10091,10109],{},[45,10092,10093,10095,10097,10099,10101,10103,10105,10107],{},[48,10094,2450],{},[53,10096,748],{},[48,10098,269],{},[53,10100,378],{"stretchy":58},[48,10102,2450],{},[53,10104,387],{"stretchy":58},[53,10106,55],{},[48,10108,8862],{},[94,10110,10111],{"encoding":96},"d \\in f(d) = D",[28,10113,10115,10133,10160],{"className":10114,"ariaHidden":66},[101],[28,10116,10118,10121,10124,10127,10130],{"className":10117},[105],[28,10119],{"className":10120,"style":763},[109],[28,10122,2450],{"className":10123},[114,222],[28,10125],{"className":10126,"style":120},[119],[28,10128,748],{"className":10129},[124],[28,10131],{"className":10132,"style":120},[119],[28,10134,10136,10139,10142,10145,10148,10151,10154,10157],{"className":10135},[105],[28,10137],{"className":10138,"style":134},[109],[28,10140,269],{"className":10141,"style":296},[114,222],[28,10143,378],{"className":10144},[138],[28,10146,2450],{"className":10147},[114,222],[28,10149,387],{"className":10150},[187],[28,10152],{"className":10153,"style":120},[119],[28,10155,55],{"className":10156},[124],[28,10158],{"className":10159,"style":120},[119],[28,10161,10163,10166],{"className":10162},[105],[28,10164],{"className":10165,"style":218},[109],[28,10167,8862],{"className":10168,"style":8909},[114,222],"，从而 ",[28,10171,10173,10190],{"className":10172,"translate":32},[31],[28,10174,10176],{"className":10175},[36],[38,10177,10178],{"xmlns":40},[42,10179,10180,10188],{},[45,10181,10182,10184,10186],{},[48,10183,2450],{},[53,10185,748],{},[48,10187,8862],{},[94,10189,9315],{"encoding":96},[28,10191,10193,10211],{"className":10192,"ariaHidden":66},[101],[28,10194,10196,10199,10202,10205,10208],{"className":10195},[105],[28,10197],{"className":10198,"style":763},[109],[28,10200,2450],{"className":10201},[114,222],[28,10203],{"className":10204,"style":120},[119],[28,10206,748],{"className":10207},[124],[28,10209],{"className":10210,"style":120},[119],[28,10212,10214,10217],{"className":10213},[105],[28,10215],{"className":10216,"style":218},[109],[28,10218,8862],{"className":10219,"style":8909},[114,222],[11,10221,10222,10223,10310,10311,1369],{},"两种情况都导致自相矛盾，说明「存在满射 ",[28,10224,10226,10253],{"className":10225,"translate":32},[31],[28,10227,10229],{"className":10228},[36],[38,10230,10231],{"xmlns":40},[42,10232,10233,10251],{},[45,10234,10235,10237,10239,10241,10243,10245,10247,10249],{},[48,10236,269],{},[53,10238,272],{},[48,10240,206],{},[53,10242,277],{},[48,10244,8283],{"mathvariant":8282},[53,10246,378],{"stretchy":58},[48,10248,206],{},[53,10250,387],{"stretchy":58},[94,10252,8574],{"encoding":96},[28,10254,10256,10274,10292],{"className":10255,"ariaHidden":66},[101],[28,10257,10259,10262,10265,10268,10271],{"className":10258},[105],[28,10260],{"className":10261,"style":292},[109],[28,10263,269],{"className":10264,"style":296},[114,222],[28,10266],{"className":10267,"style":120},[119],[28,10269,272],{"className":10270},[124],[28,10272],{"className":10273,"style":120},[119],[28,10275,10277,10280,10283,10286,10289],{"className":10276},[105],[28,10278],{"className":10279,"style":218},[109],[28,10281,206],{"className":10282},[114,222],[28,10284],{"className":10285,"style":120},[119],[28,10287,277],{"className":10288},[124],[28,10290],{"className":10291,"style":120},[119],[28,10293,10295,10298,10301,10304,10307],{"className":10294},[105],[28,10296],{"className":10297,"style":134},[109],[28,10299,8283],{"className":10300,"style":8306},[114,8305],[28,10302,378],{"className":10303},[138],[28,10305,206],{"className":10306},[114,222],[28,10308,387],{"className":10309},[187],"」这一假设站不住脚。因此 ",[28,10312,10314,10345],{"className":10313,"translate":32},[31],[28,10315,10317],{"className":10316},[36],[38,10318,10319],{"xmlns":40},[42,10320,10321,10343],{},[45,10322,10323,10325,10327,10329,10331,10333,10335,10337,10339,10341],{},[48,10324,1010],{"mathvariant":363},[48,10326,206],{},[48,10328,1010],{"mathvariant":363},[53,10330,8394],{},[48,10332,1010],{"mathvariant":363},[48,10334,8283],{"mathvariant":8282},[53,10336,378],{"stretchy":58},[48,10338,206],{},[53,10340,387],{"stretchy":58},[48,10342,1010],{"mathvariant":363},[94,10344,8409],{"encoding":96},[28,10346,10348,10372],{"className":10347,"ariaHidden":66},[101],[28,10349,10351,10354,10357,10360,10363,10366,10369],{"className":10350},[105],[28,10352],{"className":10353,"style":134},[109],[28,10355,1010],{"className":10356},[114],[28,10358,206],{"className":10359},[114,222],[28,10361,1010],{"className":10362},[114],[28,10364],{"className":10365,"style":120},[119],[28,10367,8394],{"className":10368},[124],[28,10370],{"className":10371,"style":120},[119],[28,10373,10375,10378,10381,10384,10387,10390,10393],{"className":10374},[105],[28,10376],{"className":10377,"style":134},[109],[28,10379,1010],{"className":10380},[114],[28,10382,8283],{"className":10383,"style":8306},[114,8305],[28,10385,378],{"className":10386},[138],[28,10388,206],{"className":10389},[114,222],[28,10391,387],{"className":10392},[187],[28,10394,1010],{"className":10395},[114],[11,10397,10398],{},"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：",[10400,10401,10402,10414],"table",{},[10403,10404,10405],"thead",{},[10406,10407,10408,10412],"tr",{},[10409,10410,10411],"th",{},"实数的不可数性",[10409,10413,8230],{},[10415,10416,10417,10608,10895,11183],"tbody",{},[10406,10418,10419,10518],{},[10420,10421,10422,10423],"td",{},"假设双射 ",[28,10424,10426,10455],{"className":10425,"translate":32},[31],[28,10427,10429],{"className":10428},[36],[38,10430,10431],{"xmlns":40},[42,10432,10433,10453],{},[45,10434,10435,10437,10439,10441,10443,10445,10447,10449,10451],{},[48,10436,269],{},[53,10438,272],{},[48,10440,51],{"mathvariant":50},[53,10442,277],{},[53,10444,378],{"stretchy":58},[61,10446,63],{},[53,10448,67],{"separator":66},[61,10450,70],{},[53,10452,387],{"stretchy":58},[94,10454,2335],{"encoding":96},[28,10456,10458,10476,10494],{"className":10457,"ariaHidden":66},[101],[28,10459,10461,10464,10467,10470,10473],{"className":10460},[105],[28,10462],{"className":10463,"style":292},[109],[28,10465,269],{"className":10466,"style":296},[114,222],[28,10468],{"className":10469,"style":120},[119],[28,10471,272],{"className":10472},[124],[28,10474],{"className":10475,"style":120},[119],[28,10477,10479,10482,10485,10488,10491],{"className":10478},[105],[28,10480],{"className":10481,"style":110},[109],[28,10483,51],{"className":10484},[114,115],[28,10486],{"className":10487,"style":120},[119],[28,10489,277],{"className":10490},[124],[28,10492],{"className":10493,"style":120},[119],[28,10495,10497,10500,10503,10506,10509,10512,10515],{"className":10496},[105],[28,10498],{"className":10499,"style":134},[109],[28,10501,378],{"className":10502},[138],[28,10504,63],{"className":10505},[114],[28,10507,67],{"className":10508},[145],[28,10510],{"className":10511,"style":149},[119],[28,10513,70],{"className":10514},[114],[28,10516,387],{"className":10517},[187],[10420,10519,10520,10521],{},"假设满射 ",[28,10522,10524,10551],{"className":10523,"translate":32},[31],[28,10525,10527],{"className":10526},[36],[38,10528,10529],{"xmlns":40},[42,10530,10531,10549],{},[45,10532,10533,10535,10537,10539,10541,10543,10545,10547],{},[48,10534,269],{},[53,10536,272],{},[48,10538,206],{},[53,10540,277],{},[48,10542,8283],{"mathvariant":8282},[53,10544,378],{"stretchy":58},[48,10546,206],{},[53,10548,387],{"stretchy":58},[94,10550,8574],{"encoding":96},[28,10552,10554,10572,10590],{"className":10553,"ariaHidden":66},[101],[28,10555,10557,10560,10563,10566,10569],{"className":10556},[105],[28,10558],{"className":10559,"style":292},[109],[28,10561,269],{"className":10562,"style":296},[114,222],[28,10564],{"className":10565,"style":120},[119],[28,10567,272],{"className":10568},[124],[28,10570],{"className":10571,"style":120},[119],[28,10573,10575,10578,10581,10584,10587],{"className":10574},[105],[28,10576],{"className":10577,"style":218},[109],[28,10579,206],{"className":10580},[114,222],[28,10582],{"className":10583,"style":120},[119],[28,10585,277],{"className":10586},[124],[28,10588],{"className":10589,"style":120},[119],[28,10591,10593,10596,10599,10602,10605],{"className":10592},[105],[28,10594],{"className":10595,"style":134},[109],[28,10597,8283],{"className":10598,"style":8306},[114,8305],[28,10600,378],{"className":10601},[138],[28,10603,206],{"className":10604},[114,222],[28,10606,387],{"className":10607},[187],[10406,10609,10610,10741],{},[10420,10611,10612,10613,10641,10642,5153,10712,10740],{},"构造对角数 ",[28,10614,10616,10629],{"className":10615,"translate":32},[31],[28,10617,10619],{"className":10618},[36],[38,10620,10621],{"xmlns":40},[42,10622,10623,10627],{},[45,10624,10625],{},[48,10626,4066],{},[94,10628,4066],{"encoding":96},[28,10630,10632],{"className":10631,"ariaHidden":66},[101],[28,10633,10635,10638],{"className":10634},[105],[28,10636],{"className":10637,"style":4110},[109],[28,10639,4066],{"className":10640,"style":2123},[114,222],"，每一位都与 ",[28,10643,10645,10663],{"className":10644,"translate":32},[31],[28,10646,10648],{"className":10647},[36],[38,10649,10650],{"xmlns":40},[42,10651,10652,10660],{},[45,10653,10654],{},[354,10655,10656,10658],{},[48,10657,2430],{},[48,10659,1314],{},[94,10661,10662],{"encoding":96},"x_n",[28,10664,10666],{"className":10665,"ariaHidden":66},[101],[28,10667,10669,10672],{"className":10668},[105],[28,10670],{"className":10671,"style":525},[109],[28,10673,10675,10678],{"className":10674},[114],[28,10676,2430],{"className":10677},[114,222],[28,10679,10681],{"className":10680},[423],[28,10682,10684,10704],{"className":10683},[427,428],[28,10685,10687,10701],{"className":10686},[432],[28,10688,10690],{"className":10689,"style":4421},[436],[28,10691,10692,10695],{"style":440},[28,10693],{"className":10694,"style":445},[444],[28,10696,10698],{"className":10697},[449,450,451,452],[28,10699,1314],{"className":10700},[114,222,452],[28,10702,460],{"className":10703},[459],[28,10705,10707],{"className":10706},[432],[28,10708,10710],{"className":10709,"style":467},[436],[28,10711],{},[28,10713,10715,10728],{"className":10714,"translate":32},[31],[28,10716,10718],{"className":10717},[36],[38,10719,10720],{"xmlns":40},[42,10721,10722,10726],{},[45,10723,10724],{},[48,10725,1314],{},[94,10727,1314],{"encoding":96},[28,10729,10731],{"className":10730,"ariaHidden":66},[101],[28,10732,10734,10737],{"className":10733},[105],[28,10735],{"className":10736,"style":5994},[109],[28,10738,1314],{"className":10739},[114,222]," 位不同",[10420,10742,10743,10744],{},"构造对角集 ",[28,10745,10747,10783],{"className":10746,"translate":32},[31],[28,10748,10750],{"className":10749},[36],[38,10751,10752],{"xmlns":40},[42,10753,10754,10780],{},[45,10755,10756,10758,10760,10762,10764,10766,10768,10770,10772,10774,10776,10778],{},[48,10757,8862],{},[53,10759,55],{},[53,10761,59],{"stretchy":58},[48,10763,358],{},[53,10765,272],{},[48,10767,358],{},[53,10769,8881],{"mathvariant":363},[48,10771,269],{},[53,10773,378],{"stretchy":58},[48,10775,358],{},[53,10777,387],{"stretchy":58},[53,10779,92],{"stretchy":58},[94,10781,10782],{"encoding":96},"D = \\{a : a \\notin f(a)\\}",[28,10784,10786,10804,10825,10876],{"className":10785,"ariaHidden":66},[101],[28,10787,10789,10792,10795,10798,10801],{"className":10788},[105],[28,10790],{"className":10791,"style":218},[109],[28,10793,8862],{"className":10794,"style":8909},[114,222],[28,10796],{"className":10797,"style":120},[119],[28,10799,55],{"className":10800},[124],[28,10802],{"className":10803,"style":120},[119],[28,10805,10807,10810,10813,10816,10819,10822],{"className":10806},[105],[28,10808],{"className":10809,"style":134},[109],[28,10811,59],{"className":10812},[138],[28,10814,358],{"className":10815},[114,222],[28,10817],{"className":10818,"style":120},[119],[28,10820,272],{"className":10821},[124],[28,10823],{"className":10824,"style":120},[119],[28,10826,10828,10831,10834,10837,10873],{"className":10827},[105],[28,10829],{"className":10830,"style":134},[109],[28,10832,358],{"className":10833},[114,222],[28,10835],{"className":10836,"style":120},[119],[28,10838,10840,10846],{"className":10839},[124],[28,10841,10843],{"className":10842},[114],[28,10844,748],{"className":10845},[124],[28,10847,10849],{"className":10848},[114,482],[28,10850,10852],{"className":10851},[486],[28,10853,10855,10858,10870],{"className":10854},[8991],[28,10856],{"className":10857,"style":134},[109],[28,10859,10861],{"className":10860},[497],[28,10862,10864,10867],{"className":10863},[114],[28,10865,1580],{"className":10866},[114],[28,10868],{"className":10869,"style":9007},[119],[28,10871],{"className":10872},[508],[28,10874],{"className":10875,"style":120},[119],[28,10877,10879,10882,10885,10888,10891],{"className":10878},[105],[28,10880],{"className":10881,"style":134},[109],[28,10883,269],{"className":10884,"style":296},[114,222],[28,10886,378],{"className":10887},[138],[28,10889,358],{"className":10890},[114,222],[28,10892,10894],{"className":10893},[187],")}",[10406,10896,10897,11054],{},[10420,10898,10899,10900,10928,10929],{},"对所有 ",[28,10901,10903,10916],{"className":10902,"translate":32},[31],[28,10904,10906],{"className":10905},[36],[38,10907,10908],{"xmlns":40},[42,10909,10910,10914],{},[45,10911,10912],{},[48,10913,1314],{},[94,10915,1314],{"encoding":96},[28,10917,10919],{"className":10918,"ariaHidden":66},[101],[28,10920,10922,10925],{"className":10921},[105],[28,10923],{"className":10924,"style":5994},[109],[28,10926,1314],{"className":10927},[114,222]," 有 ",[28,10930,10932,10954],{"className":10931,"translate":32},[31],[28,10933,10935],{"className":10934},[36],[38,10936,10937],{"xmlns":40},[42,10938,10939,10951],{},[45,10940,10941,10943,10945],{},[48,10942,4066],{},[53,10944,364],{"mathvariant":363},[354,10946,10947,10949],{},[48,10948,2430],{},[48,10950,1314],{},[94,10952,10953],{"encoding":96},"y \\neq x_n",[28,10955,10957,11008],{"className":10956,"ariaHidden":66},[101],[28,10958,10960,10963,10966,10969,11005],{"className":10959},[105],[28,10961],{"className":10962,"style":292},[109],[28,10964,4066],{"className":10965,"style":2123},[114,222],[28,10967],{"className":10968,"style":120},[119],[28,10970,10972,10999,11002],{"className":10971},[124],[28,10973,10975],{"className":10974},[124],[28,10976,10978],{"className":10977},[114,482],[28,10979,10981],{"className":10980},[486],[28,10982,10984,10987,10996],{"className":10983},[490],[28,10985],{"className":10986,"style":292},[109],[28,10988,10990],{"className":10989},[497],[28,10991,10993],{"className":10992},[114],[28,10994,504],{"className":10995},[124],[28,10997],{"className":10998},[508],[28,11000],{"className":11001},[119,512],[28,11003,55],{"className":11004},[124],[28,11006],{"className":11007,"style":120},[119],[28,11009,11011,11014],{"className":11010},[105],[28,11012],{"className":11013,"style":525},[109],[28,11015,11017,11020],{"className":11016},[114],[28,11018,2430],{"className":11019},[114,222],[28,11021,11023],{"className":11022},[423],[28,11024,11026,11046],{"className":11025},[427,428],[28,11027,11029,11043],{"className":11028},[432],[28,11030,11032],{"className":11031,"style":4421},[436],[28,11033,11034,11037],{"style":440},[28,11035],{"className":11036,"style":445},[444],[28,11038,11040],{"className":11039},[449,450,451,452],[28,11041,1314],{"className":11042},[114,222,452],[28,11044,460],{"className":11045},[459],[28,11047,11049],{"className":11048},[432],[28,11050,11052],{"className":11051,"style":467},[436],[28,11053],{},[10420,11055,10899,11056,10928,11084],{},[28,11057,11059,11072],{"className":11058,"translate":32},[31],[28,11060,11062],{"className":11061},[36],[38,11063,11064],{"xmlns":40},[42,11065,11066,11070],{},[45,11067,11068],{},[48,11069,358],{},[94,11071,358],{"encoding":96},[28,11073,11075],{"className":11074,"ariaHidden":66},[101],[28,11076,11078,11081],{"className":11077},[105],[28,11079],{"className":11080,"style":5994},[109],[28,11082,358],{"className":11083},[114,222],[28,11085,11087,11111],{"className":11086,"translate":32},[31],[28,11088,11090],{"className":11089},[36],[38,11091,11092],{"xmlns":40},[42,11093,11094,11108],{},[45,11095,11096,11098,11100,11102,11104,11106],{},[48,11097,8862],{},[53,11099,364],{"mathvariant":363},[48,11101,269],{},[53,11103,378],{"stretchy":58},[48,11105,358],{},[53,11107,387],{"stretchy":58},[94,11109,11110],{"encoding":96},"D \\neq f(a)",[28,11112,11114,11165],{"className":11113,"ariaHidden":66},[101],[28,11115,11117,11120,11123,11126,11162],{"className":11116},[105],[28,11118],{"className":11119,"style":292},[109],[28,11121,8862],{"className":11122,"style":8909},[114,222],[28,11124],{"className":11125,"style":120},[119],[28,11127,11129,11156,11159],{"className":11128},[124],[28,11130,11132],{"className":11131},[124],[28,11133,11135],{"className":11134},[114,482],[28,11136,11138],{"className":11137},[486],[28,11139,11141,11144,11153],{"className":11140},[490],[28,11142],{"className":11143,"style":292},[109],[28,11145,11147],{"className":11146},[497],[28,11148,11150],{"className":11149},[114],[28,11151,504],{"className":11152},[124],[28,11154],{"className":11155},[508],[28,11157],{"className":11158},[119,512],[28,11160,55],{"className":11161},[124],[28,11163],{"className":11164,"style":120},[119],[28,11166,11168,11171,11174,11177,11180],{"className":11167},[105],[28,11169],{"className":11170,"style":134},[109],[28,11172,269],{"className":11173,"style":296},[114,222],[28,11175,378],{"className":11176},[138],[28,11178,358],{"className":11179},[114,222],[28,11181,387],{"className":11182},[187],[10406,11184,11185,11216],{},[10420,11186,11187,11215],{},[28,11188,11190,11203],{"className":11189,"translate":32},[31],[28,11191,11193],{"className":11192},[36],[38,11194,11195],{"xmlns":40},[42,11196,11197,11201],{},[45,11198,11199],{},[48,11200,4066],{},[94,11202,4066],{"encoding":96},[28,11204,11206],{"className":11205,"ariaHidden":66},[101],[28,11207,11209,11212],{"className":11208},[105],[28,11210],{"className":11211,"style":4110},[109],[28,11213,4066],{"className":11214,"style":2123},[114,222]," 无法被列表覆盖；矛盾",[10420,11217,11218,11246],{},[28,11219,11221,11234],{"className":11220,"translate":32},[31],[28,11222,11224],{"className":11223},[36],[38,11225,11226],{"xmlns":40},[42,11227,11228,11232],{},[45,11229,11230],{},[48,11231,8862],{},[94,11233,8862],{"encoding":96},[28,11235,11237],{"className":11236,"ariaHidden":66},[101],[28,11238,11240,11243],{"className":11239},[105],[28,11241],{"className":11242,"style":218},[109],[28,11244,8862],{"className":11245,"style":8909},[114,222]," 无法被映射覆盖；矛盾",[11,11248,11249,11250,11301,11302,11391,11392,11476,11477,11521,11522,11573],{},"事实上，",[28,11251,11253,11274],{"className":11252,"translate":32},[31],[28,11254,11256],{"className":11255},[36],[38,11257,11258],{"xmlns":40},[42,11259,11260,11272],{},[45,11261,11262,11264,11266,11268,11270],{},[53,11263,378],{"stretchy":58},[61,11265,63],{},[53,11267,67],{"separator":66},[61,11269,70],{},[53,11271,387],{"stretchy":58},[94,11273,2161],{"encoding":96},[28,11275,11277],{"className":11276,"ariaHidden":66},[101],[28,11278,11280,11283,11286,11289,11292,11295,11298],{"className":11279},[105],[28,11281],{"className":11282,"style":134},[109],[28,11284,378],{"className":11285},[138],[28,11287,63],{"className":11288},[114],[28,11290,67],{"className":11291},[145],[28,11293],{"className":11294,"style":149},[119],[28,11296,70],{"className":11297},[114],[28,11299,387],{"className":11300},[187]," 中的实数可以与 ",[28,11303,11305,11332],{"className":11304,"translate":32},[31],[28,11306,11308],{"className":11307},[36],[38,11309,11310],{"xmlns":40},[42,11311,11312,11329],{},[45,11313,11314,11316,11318,11320,11322],{},[53,11315,59],{"stretchy":58},[61,11317,63],{},[53,11319,67],{"separator":66},[61,11321,70],{},[11323,11324,11325,11327],"msup",{},[53,11326,92],{"stretchy":58},[48,11328,51],{"mathvariant":50},[94,11330,11331],{"encoding":96},"\\{0,1\\}^{\\mathbb{N}}",[28,11333,11335],{"className":11334,"ariaHidden":66},[101],[28,11336,11338,11342,11345,11348,11351,11354,11357],{"className":11337},[105],[28,11339],{"className":11340,"style":11341},[109],"height:1.0952em;vertical-align:-0.25em;",[28,11343,59],{"className":11344},[138],[28,11346,63],{"className":11347},[114],[28,11349,67],{"className":11350},[145],[28,11352],{"className":11353,"style":149},[119],[28,11355,70],{"className":11356},[114],[28,11358,11360,11363],{"className":11359},[187],[28,11361,92],{"className":11362},[187],[28,11364,11366],{"className":11365},[423],[28,11367,11369],{"className":11368},[427],[28,11370,11372],{"className":11371},[432],[28,11373,11376],{"className":11374,"style":11375},[436],"height:0.8452em;",[28,11377,11379,11382],{"style":11378},"top:-3.063em;margin-right:0.05em;",[28,11380],{"className":11381,"style":445},[444],[28,11383,11385],{"className":11384},[449,450,451,452],[28,11386,11388],{"className":11387},[114,452],[28,11389,51],{"className":11390},[114,115,452],"（由 0 和 1 组成的无穷二进制序列集）近似对应，而 ",[28,11393,11395,11420],{"className":11394,"translate":32},[31],[28,11396,11398],{"className":11397},[36],[38,11399,11400],{"xmlns":40},[42,11401,11402,11418],{},[45,11403,11404,11406,11408,11410,11412],{},[53,11405,59],{"stretchy":58},[61,11407,63],{},[53,11409,67],{"separator":66},[61,11411,70],{},[11323,11413,11414,11416],{},[53,11415,92],{"stretchy":58},[48,11417,51],{"mathvariant":50},[94,11419,11331],{"encoding":96},[28,11421,11423],{"className":11422,"ariaHidden":66},[101],[28,11424,11426,11429,11432,11435,11438,11441,11444],{"className":11425},[105],[28,11427],{"className":11428,"style":11341},[109],[28,11430,59],{"className":11431},[138],[28,11433,63],{"className":11434},[114],[28,11436,67],{"className":11437},[145],[28,11439],{"className":11440,"style":149},[119],[28,11442,70],{"className":11443},[114],[28,11445,11447,11450],{"className":11446},[187],[28,11448,92],{"className":11449},[187],[28,11451,11453],{"className":11452},[423],[28,11454,11456],{"className":11455},[427],[28,11457,11459],{"className":11458},[432],[28,11460,11462],{"className":11461,"style":11375},[436],[28,11463,11464,11467],{"style":11378},[28,11465],{"className":11466,"style":445},[444],[28,11468,11470],{"className":11469},[449,450,451,452],[28,11471,11473],{"className":11472},[114,452],[28,11474,51],{"className":11475},[114,115,452]," 本质上就是 ",[28,11478,11480,11500],{"className":11479,"translate":32},[31],[28,11481,11483],{"className":11482},[36],[38,11484,11485],{"xmlns":40},[42,11486,11487,11497],{},[45,11488,11489,11491,11493,11495],{},[48,11490,8283],{"mathvariant":8282},[53,11492,378],{"stretchy":58},[48,11494,51],{"mathvariant":50},[53,11496,387],{"stretchy":58},[94,11498,11499],{"encoding":96},"\\mathcal{P}(\\mathbb{N})",[28,11501,11503],{"className":11502,"ariaHidden":66},[101],[28,11504,11506,11509,11512,11515,11518],{"className":11505},[105],[28,11507],{"className":11508,"style":134},[109],[28,11510,8283],{"className":11511,"style":8306},[114,8305],[28,11513,378],{"className":11514},[138],[28,11516,51],{"className":11517},[114,115],[28,11519,387],{"className":11520},[187],"（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 ",[28,11523,11525,11543],{"className":11524,"translate":32},[31],[28,11526,11528],{"className":11527},[36],[38,11529,11530],{"xmlns":40},[42,11531,11532,11540],{},[45,11533,11534,11536,11538],{},[48,11535,206],{},[53,11537,55],{},[48,11539,51],{"mathvariant":50},[94,11541,11542],{"encoding":96},"A = \\mathbb{N}",[28,11544,11546,11564],{"className":11545,"ariaHidden":66},[101],[28,11547,11549,11552,11555,11558,11561],{"className":11548},[105],[28,11550],{"className":11551,"style":218},[109],[28,11553,206],{"className":11554},[114,222],[28,11556],{"className":11557,"style":120},[119],[28,11559,55],{"className":11560},[124],[28,11562],{"className":11563,"style":120},[119],[28,11565,11567,11570],{"className":11566},[105],[28,11568],{"className":11569,"style":110},[109],[28,11571,51],{"className":11572},[114,115]," 这一特例下的具体体现。",[11,11575,11576],{},"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：",[7745,11578,11580],{"className":7747,"code":11579,"language":7749,"meta":7750,"style":7750},"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",[7752,11581,11582,11592,11597,11607,11619,11667,11671,11689,11699,11733,11737,11742,11775,11797,11801,11830,11863,11867],{"__ignoreMap":7750},[28,11583,11584,11586,11589],{"class":7756,"line":7757},[28,11585,7761],{"class":7760},[28,11587,11588],{"class":7764}," cantor_diagonal_demo",[28,11590,11591],{"class":7768},"(A):\n",[28,11593,11594],{"class":7756,"line":7781},[28,11595,11596],{"class":7876},"    # 构造一个「任意」的映射 f: A -> P(A) 用于演示\n",[28,11598,11599,11602,11604],{"class":7756,"line":7788},[28,11600,11601],{"class":7768},"    f ",[28,11603,55],{"class":7760},[28,11605,11606],{"class":7768}," {}\n",[28,11608,11609,11611,11614,11616],{"class":7756,"line":7794},[28,11610,7819],{"class":7760},[28,11612,11613],{"class":7768}," i ",[28,11615,7825],{"class":7760},[28,11617,11618],{"class":7768}," A:\n",[28,11620,11621,11624,11626,11629,11632,11635,11638,11640,11643,11646,11649,11651,11654,11656,11659,11662,11665],{"class":7756,"line":7800},[28,11622,11623],{"class":7768},"        f[i] ",[28,11625,55],{"class":7760},[28,11627,11628],{"class":7774}," frozenset",[28,11630,11631],{"class":7768},"(j ",[28,11633,11634],{"class":7760},"for",[28,11636,11637],{"class":7768}," j ",[28,11639,7825],{"class":7760},[28,11641,11642],{"class":7768}," A ",[28,11644,11645],{"class":7760},"if",[28,11647,11648],{"class":7768}," (i ",[28,11650,1610],{"class":7760},[28,11652,11653],{"class":7768}," j) ",[28,11655,7899],{"class":7760},[28,11657,11658],{"class":7774}," 2",[28,11660,11661],{"class":7760}," ==",[28,11663,11664],{"class":7774}," 0",[28,11666,7859],{"class":7768},[28,11668,11669],{"class":7756,"line":7805},[28,11670,7942],{"emptyLinePlaceholder":7941},[28,11672,11673,11676,11678,11681,11683,11686],{"class":7756,"line":7816},[28,11674,11675],{"class":7774},"    print",[28,11677,378],{"class":7768},[28,11679,11680],{"class":7784},"\"集合 A =\"",[28,11682,7967],{"class":7768},[28,11684,11685],{"class":7774},"set",[28,11687,11688],{"class":7768},"(A))\n",[28,11690,11691,11693,11695,11697],{"class":7756,"line":7834},[28,11692,7819],{"class":7760},[28,11694,11613],{"class":7768},[28,11696,7825],{"class":7760},[28,11698,11618],{"class":7768},[28,11700,11701,11704,11706,11708,11711,11713,11715,11717,11720,11723,11726,11728,11731],{"class":7756,"line":7862},[28,11702,11703],{"class":7774},"        print",[28,11705,378],{"class":7768},[28,11707,269],{"class":7760},[28,11709,11710],{"class":7784},"\"  f(",[28,11712,59],{"class":7774},[28,11714,3865],{"class":7768},[28,11716,92],{"class":7774},[28,11718,11719],{"class":7784},") = ",[28,11721,11722],{"class":7774},"{set",[28,11724,11725],{"class":7768},"(f[i])",[28,11727,92],{"class":7774},[28,11729,11730],{"class":7784},"\"",[28,11732,7859],{"class":7768},[28,11734,11735],{"class":7756,"line":7880},[28,11736,7942],{"emptyLinePlaceholder":7941},[28,11738,11739],{"class":7756,"line":7908},[28,11740,11741],{"class":7876},"    # 对角线构造：D = { i in A : i not in f(i) }\n",[28,11743,11744,11747,11749,11751,11754,11756,11758,11760,11762,11764,11766,11769,11772],{"class":7756,"line":7920},[28,11745,11746],{"class":7768},"    D ",[28,11748,55],{"class":7760},[28,11750,11628],{"class":7774},[28,11752,11753],{"class":7768},"(i ",[28,11755,11634],{"class":7760},[28,11757,11613],{"class":7768},[28,11759,7825],{"class":7760},[28,11761,11642],{"class":7768},[28,11763,11645],{"class":7760},[28,11765,11613],{"class":7768},[28,11767,11768],{"class":7760},"not",[28,11770,11771],{"class":7760}," in",[28,11773,11774],{"class":7768}," f[i])\n",[28,11776,11777,11779,11781,11783,11786,11788,11791,11793,11795],{"class":7756,"line":7938},[28,11778,11675],{"class":7774},[28,11780,378],{"class":7768},[28,11782,269],{"class":7760},[28,11784,11785],{"class":7784},"\"对角线构造出的集合 D = ",[28,11787,11722],{"class":7774},[28,11789,11790],{"class":7768},"(D)",[28,11792,92],{"class":7774},[28,11794,11730],{"class":7784},[28,11796,7859],{"class":7768},[28,11798,11799],{"class":7756,"line":7945},[28,11800,7942],{"emptyLinePlaceholder":7941},[28,11802,11803,11806,11808,11811,11813,11815,11817,11819,11821,11824,11827],{"class":7756,"line":7950},[28,11804,11805],{"class":7768},"    matches ",[28,11807,55],{"class":7760},[28,11809,11810],{"class":7768}," [i ",[28,11812,11634],{"class":7760},[28,11814,11613],{"class":7768},[28,11816,7825],{"class":7760},[28,11818,11642],{"class":7768},[28,11820,11645],{"class":7760},[28,11822,11823],{"class":7768}," f[i] ",[28,11825,11826],{"class":7760},"==",[28,11828,11829],{"class":7768}," D]\n",[28,11831,11832,11834,11836,11838,11841,11843,11846,11848,11851,11854,11857,11859,11861],{"class":7756,"line":7961},[28,11833,11675],{"class":7774},[28,11835,378],{"class":7768},[28,11837,269],{"class":7760},[28,11839,11840],{"class":7784},"\"是否存在 i 使得 f(i) = D？ -> ",[28,11842,59],{"class":7774},[28,11844,11845],{"class":7768},"matches ",[28,11847,11645],{"class":7760},[28,11849,11850],{"class":7768}," matches ",[28,11852,11853],{"class":7760},"else",[28,11855,11856],{"class":7784}," '不存在（矛盾已建立）'",[28,11858,92],{"class":7774},[28,11860,11730],{"class":7784},[28,11862,7859],{"class":7768},[28,11864,11865],{"class":7756,"line":7986},[28,11866,7942],{"emptyLinePlaceholder":7941},[28,11868,11869,11872,11874,11876,11878,11880,11882,11884,11886],{"class":7756,"line":8009},[28,11870,11871],{"class":7768},"cantor_diagonal_demo([",[28,11873,63],{"class":7774},[28,11875,7967],{"class":7768},[28,11877,70],{"class":7774},[28,11879,7967],{"class":7768},[28,11881,75],{"class":7774},[28,11883,7967],{"class":7768},[28,11885,80],{"class":7774},[28,11887,11888],{"class":7768},"])\n",[11,11890,8058],{},[7745,11892,11895],{"className":11893,"code":11894,"language":1801},[8062],"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",[7752,11896,11894],{"__ignoreMap":7750},[11,11898,11899,11900,11928,11929,11957,11958,11986],{},"这里构造出的 ",[28,11901,11903,11916],{"className":11902,"translate":32},[31],[28,11904,11906],{"className":11905},[36],[38,11907,11908],{"xmlns":40},[42,11909,11910,11914],{},[45,11911,11912],{},[48,11913,8862],{},[94,11915,8862],{"encoding":96},[28,11917,11919],{"className":11918,"ariaHidden":66},[101],[28,11920,11922,11925],{"className":11921},[105],[28,11923],{"className":11924,"style":218},[109],[28,11926,8862],{"className":11927,"style":8909},[114,222]," 恰好是空集，而这个特定的映射 ",[28,11930,11932,11945],{"className":11931,"translate":32},[31],[28,11933,11935],{"className":11934},[36],[38,11936,11937],{"xmlns":40},[42,11938,11939,11943],{},[45,11940,11941],{},[48,11942,269],{},[94,11944,269],{"encoding":96},[28,11946,11948],{"className":11947,"ariaHidden":66},[101],[28,11949,11951,11954],{"className":11950},[105],[28,11952],{"className":11953,"style":292},[109],[28,11955,269],{"className":11956,"style":296},[114,222]," 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 ",[28,11959,11961,11974],{"className":11960,"translate":32},[31],[28,11962,11964],{"className":11963},[36],[38,11965,11966],{"xmlns":40},[42,11967,11968,11972],{},[45,11969,11970],{},[48,11971,269],{},[94,11973,269],{"encoding":96},[28,11975,11977],{"className":11976,"ariaHidden":66},[101],[28,11978,11980,11983],{"className":11979},[105],[28,11981],{"className":11982,"style":292},[109],[28,11984,269],{"className":11985,"style":296},[114,222]," 都成立的普适论证。",[18,11988,11989],{"id":11989},"停机问题",[11,11991,11992],{},"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。",[11,11994,11995,11996,12025,12026,12054,12055,12083,12084,12112],{},"停机问题。给定一个程序 ",[28,11997,11999,12012],{"className":11998,"translate":32},[31],[28,12000,12002],{"className":12001},[36],[38,12003,12004],{"xmlns":40},[42,12005,12006,12010],{},[45,12007,12008],{},[48,12009,8283],{},[94,12011,8283],{"encoding":96},[28,12013,12015],{"className":12014,"ariaHidden":66},[101],[28,12016,12018,12021],{"className":12017},[105],[28,12019],{"className":12020,"style":218},[109],[28,12022,8283],{"className":12023,"style":12024},[114,222],"margin-right:0.1389em;"," 和一个输入 ",[28,12027,12029,12042],{"className":12028,"translate":32},[31],[28,12030,12032],{"className":12031},[36],[38,12033,12034],{"xmlns":40},[42,12035,12036,12040],{},[45,12037,12038],{},[48,12039,2430],{},[94,12041,2430],{"encoding":96},[28,12043,12045],{"className":12044,"ariaHidden":66},[101],[28,12046,12048,12051],{"className":12047},[105],[28,12049],{"className":12050,"style":5994},[109],[28,12052,2430],{"className":12053},[114,222],"，是否存在一个通用算法，能够判断 ",[28,12056,12058,12071],{"className":12057,"translate":32},[31],[28,12059,12061],{"className":12060},[36],[38,12062,12063],{"xmlns":40},[42,12064,12065,12069],{},[45,12066,12067],{},[48,12068,8283],{},[94,12070,8283],{"encoding":96},[28,12072,12074],{"className":12073,"ariaHidden":66},[101],[28,12075,12077,12080],{"className":12076},[105],[28,12078],{"className":12079,"style":218},[109],[28,12081,8283],{"className":12082,"style":12024},[114,222]," 在输入 ",[28,12085,12087,12100],{"className":12086,"translate":32},[31],[28,12088,12090],{"className":12089},[36],[38,12091,12092],{"xmlns":40},[42,12093,12094,12098],{},[45,12095,12096],{},[48,12097,2430],{},[94,12099,2430],{"encoding":96},[28,12101,12103],{"className":12102,"ariaHidden":66},[101],[28,12104,12106,12109],{"className":12105},[105],[28,12107],{"className":12108,"style":5994},[109],[28,12110,2430],{"className":12111},[114,222]," 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）？",[11,12114,12115],{},"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）。",[11,12117,12118],{},"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：",[28,12120,12122],{"className":12121,"translate":32},[2403],[28,12123,12125,12214],{"className":12124,"translate":32},[31],[28,12126,12128],{"className":12127},[36],[38,12129,12130],{"xmlns":40,"display":2412},[42,12131,12132,12211],{},[45,12133,12134,12137,12139,12141,12143,12145,12147,12149],{},[87,12135,12136],{},"halts",[53,12138,378],{"stretchy":58},[48,12140,8283],{},[53,12142,67],{"separator":66},[48,12144,2430],{},[53,12146,387],{"stretchy":58},[53,12148,55],{},[45,12150,12151,12153],{},[53,12152,59],{"fence":66},[1560,12154,12155,12183],{"rowspacing":1562,"columnalign":1563,"columnspacing":1564},[1566,12156,12157,12164],{},[1569,12158,12159],{},[1572,12160,12161],{"scriptlevel":63,"displaystyle":58},[87,12162,12163],{},"真",[1569,12165,12166],{},[1572,12167,12168],{"scriptlevel":63,"displaystyle":58},[45,12169,12170,12172,12174,12176,12178,12180],{},[87,12171,4346],{},[48,12173,8283],{},[53,12175,378],{"stretchy":58},[48,12177,2430],{},[53,12179,387],{"stretchy":58},[87,12181,12182],{}," 停机",[1566,12184,12185,12192],{},[1569,12186,12187],{},[1572,12188,12189],{"scriptlevel":63,"displaystyle":58},[87,12190,12191],{},"假",[1569,12193,12194],{},[1572,12195,12196],{"scriptlevel":63,"displaystyle":58},[45,12197,12198,12200,12202,12204,12206,12208],{},[87,12199,4346],{},[48,12201,8283],{},[53,12203,378],{"stretchy":58},[48,12205,2430],{},[53,12207,387],{"stretchy":58},[87,12209,12210],{}," 不停机",[94,12212,12213],{"encoding":96},"\\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}",[28,12215,12217,12256],{"className":12216,"ariaHidden":66},[101],[28,12218,12220,12223,12229,12232,12235,12238,12241,12244,12247,12250,12253],{"className":12219},[105],[28,12221],{"className":12222,"style":134},[109],[28,12224,12226],{"className":12225},[114,1801],[28,12227,12136],{"className":12228},[114],[28,12230,378],{"className":12231},[138],[28,12233,8283],{"className":12234,"style":12024},[114,222],[28,12236,67],{"className":12237},[145],[28,12239],{"className":12240,"style":149},[119],[28,12242,2430],{"className":12243},[114,222],[28,12245,387],{"className":12246},[187],[28,12248],{"className":12249,"style":120},[119],[28,12251,55],{"className":12252},[124],[28,12254],{"className":12255,"style":120},[119],[28,12257,12259,12262],{"className":12258},[105],[28,12260],{"className":12261,"style":1669},[109],[28,12263,12265,12271,12432],{"className":12264},[180],[28,12266,12268],{"className":12267,"style":1677},[138,1676],[28,12269,59],{"className":12270},[1681,1682],[28,12272,12274],{"className":12273},[114],[28,12275,12277,12328,12331],{"className":12276},[1560],[28,12278,12280],{"className":12279},[1692],[28,12281,12283,12320],{"className":12282},[427,428],[28,12284,12286,12317],{"className":12285},[432],[28,12287,12289,12303],{"className":12288,"style":1702},[436],[28,12290,12291,12294],{"style":1705},[28,12292],{"className":12293,"style":1709},[444],[28,12295,12297],{"className":12296},[114],[28,12298,12300],{"className":12299},[114,1801],[28,12301,12163],{"className":12302},[114,1809],[28,12304,12305,12308],{"style":1722},[28,12306],{"className":12307,"style":1709},[444],[28,12309,12311],{"className":12310},[114],[28,12312,12314],{"className":12313},[114,1801],[28,12315,12191],{"className":12316},[114,1809],[28,12318,460],{"className":12319},[459],[28,12321,12323],{"className":12322},[432],[28,12324,12326],{"className":12325,"style":1767},[436],[28,12327],{},[28,12329],{"className":12330,"style":1774},[1773],[28,12332,12334],{"className":12333},[1692],[28,12335,12337,12424],{"className":12336},[427,428],[28,12338,12340,12421],{"className":12339},[432],[28,12341,12343,12382],{"className":12342,"style":1702},[436],[28,12344,12345,12348],{"style":1705},[28,12346],{"className":12347,"style":1709},[444],[28,12349,12351,12360,12363,12366,12369,12372],{"className":12350},[114],[28,12352,12354,12357],{"className":12353},[114,1801],[28,12355,4548],{"className":12356},[114,1809],[28,12358,1805],{"className":12359},[114],[28,12361,8283],{"className":12362,"style":12024},[114,222],[28,12364,378],{"className":12365},[138],[28,12367,2430],{"className":12368},[114,222],[28,12370,387],{"className":12371},[187],[28,12373,12375,12378],{"className":12374},[114,1801],[28,12376,1805],{"className":12377},[114],[28,12379,12381],{"className":12380},[114,1809],"停机",[28,12383,12384,12387],{"style":1722},[28,12385],{"className":12386,"style":1709},[444],[28,12388,12390,12399,12402,12405,12408,12411],{"className":12389},[114],[28,12391,12393,12396],{"className":12392},[114,1801],[28,12394,4548],{"className":12395},[114,1809],[28,12397,1805],{"className":12398},[114],[28,12400,8283],{"className":12401,"style":12024},[114,222],[28,12403,378],{"className":12404},[138],[28,12406,2430],{"className":12407},[114,222],[28,12409,387],{"className":12410},[187],[28,12412,12414,12417],{"className":12413},[114,1801],[28,12415,1805],{"className":12416},[114],[28,12418,12420],{"className":12419},[114,1809],"不停机",[28,12422,460],{"className":12423},[459],[28,12425,12427],{"className":12426},[432],[28,12428,12430],{"className":12429,"style":1767},[436],[28,12431],{},[28,12433],{"className":12434},[187,1846],[11,12436,12437],{},"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）。",[11,12439,12440,12441,12469],{},"构造一个对角线程序 ",[28,12442,12444,12457],{"className":12443,"translate":32},[31],[28,12445,12447],{"className":12446},[36],[38,12448,12449],{"xmlns":40},[42,12450,12451,12455],{},[45,12452,12453],{},[48,12454,8862],{},[94,12456,8862],{"encoding":96},[28,12458,12460],{"className":12459,"ariaHidden":66},[101],[28,12461,12463,12466],{"className":12462},[105],[28,12464],{"className":12465,"style":218},[109],[28,12467,8862],{"className":12468,"style":8909},[114,222],"，其行为定义如下：",[28,12471,12473],{"className":12472,"translate":32},[2403],[28,12474,12476,12567],{"className":12475,"translate":32},[31],[28,12477,12479],{"className":12478},[36],[38,12480,12481],{"xmlns":40,"display":2412},[42,12482,12483,12564],{},[45,12484,12485,12487,12489,12491,12493,12495],{},[48,12486,8862],{},[53,12488,378],{"stretchy":58},[48,12490,8283],{},[53,12492,387],{"stretchy":58},[53,12494,55],{},[45,12496,12497,12499],{},[53,12498,59],{"fence":66},[1560,12500,12501,12533],{"rowspacing":1562,"columnalign":1563,"columnspacing":1564},[1566,12502,12503,12510],{},[1569,12504,12505],{},[1572,12506,12507],{"scriptlevel":63,"displaystyle":58},[87,12508,12509],{},"死循环",[1569,12511,12512],{},[1572,12513,12514],{"scriptlevel":63,"displaystyle":58},[45,12515,12516,12519,12521,12523,12525,12527,12529,12531],{},[87,12517,12518],{},"若 halts",[53,12520,378],{"stretchy":58},[48,12522,8283],{},[53,12524,67],{"separator":66},[48,12526,8283],{},[53,12528,387],{"stretchy":58},[53,12530,55],{},[87,12532,12163],{},[1566,12534,12535,12542],{},[1569,12536,12537],{},[1572,12538,12539],{"scriptlevel":63,"displaystyle":58},[87,12540,12541],{},"立即停机",[1569,12543,12544],{},[1572,12545,12546],{"scriptlevel":63,"displaystyle":58},[45,12547,12548,12550,12552,12554,12556,12558,12560,12562],{},[87,12549,12518],{},[53,12551,378],{"stretchy":58},[48,12553,8283],{},[53,12555,67],{"separator":66},[48,12557,8283],{},[53,12559,387],{"stretchy":58},[53,12561,55],{},[87,12563,12191],{},[94,12565,12566],{"encoding":96},"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}",[28,12568,12570,12597],{"className":12569,"ariaHidden":66},[101],[28,12571,12573,12576,12579,12582,12585,12588,12591,12594],{"className":12572},[105],[28,12574],{"className":12575,"style":134},[109],[28,12577,8862],{"className":12578,"style":8909},[114,222],[28,12580,378],{"className":12581},[138],[28,12583,8283],{"className":12584,"style":12024},[114,222],[28,12586,387],{"className":12587},[187],[28,12589],{"className":12590,"style":120},[119],[28,12592,55],{"className":12593},[124],[28,12595],{"className":12596,"style":120},[119],[28,12598,12600,12603],{"className":12599},[105],[28,12601],{"className":12602,"style":1669},[109],[28,12604,12606,12612,12807],{"className":12605},[180],[28,12607,12609],{"className":12608,"style":1677},[138,1676],[28,12610,59],{"className":12611},[1681,1682],[28,12613,12615],{"className":12614},[114],[28,12616,12618,12669,12672],{"className":12617},[1560],[28,12619,12621],{"className":12620},[1692],[28,12622,12624,12661],{"className":12623},[427,428],[28,12625,12627,12658],{"className":12626},[432],[28,12628,12630,12644],{"className":12629,"style":1702},[436],[28,12631,12632,12635],{"style":1705},[28,12633],{"className":12634,"style":1709},[444],[28,12636,12638],{"className":12637},[114],[28,12639,12641],{"className":12640},[114,1801],[28,12642,12509],{"className":12643},[114,1809],[28,12645,12646,12649],{"style":1722},[28,12647],{"className":12648,"style":1709},[444],[28,12650,12652],{"className":12651},[114],[28,12653,12655],{"className":12654},[114,1801],[28,12656,12541],{"className":12657},[114,1809],[28,12659,460],{"className":12660},[459],[28,12662,12664],{"className":12663},[432],[28,12665,12667],{"className":12666,"style":1767},[436],[28,12668],{},[28,12670],{"className":12671,"style":1774},[1773],[28,12673,12675],{"className":12674},[1692],[28,12676,12678,12799],{"className":12677},[427,428],[28,12679,12681,12796],{"className":12680},[432],[28,12682,12684,12740],{"className":12683,"style":1702},[436],[28,12685,12686,12689],{"style":1705},[28,12687],{"className":12688,"style":1709},[444],[28,12690,12692,12701,12707,12710,12713,12716,12719,12722,12725,12728,12731,12734],{"className":12691},[114],[28,12693,12695,12698],{"className":12694},[114,1801],[28,12696,4548],{"className":12697},[114,1809],[28,12699,1805],{"className":12700},[114],[28,12702,12704],{"className":12703},[114,1801],[28,12705,12136],{"className":12706},[114],[28,12708,378],{"className":12709},[138],[28,12711,8283],{"className":12712,"style":12024},[114,222],[28,12714,67],{"className":12715},[145],[28,12717],{"className":12718,"style":149},[119],[28,12720,8283],{"className":12721,"style":12024},[114,222],[28,12723,387],{"className":12724},[187],[28,12726],{"className":12727,"style":120},[119],[28,12729,55],{"className":12730},[124],[28,12732],{"className":12733,"style":120},[119],[28,12735,12737],{"className":12736},[114,1801],[28,12738,12163],{"className":12739},[114,1809],[28,12741,12742,12745],{"style":1722},[28,12743],{"className":12744,"style":1709},[444],[28,12746,12748,12757,12763,12766,12769,12772,12775,12778,12781,12784,12787,12790],{"className":12747},[114],[28,12749,12751,12754],{"className":12750},[114,1801],[28,12752,4548],{"className":12753},[114,1809],[28,12755,1805],{"className":12756},[114],[28,12758,12760],{"className":12759},[114,1801],[28,12761,12136],{"className":12762},[114],[28,12764,378],{"className":12765},[138],[28,12767,8283],{"className":12768,"style":12024},[114,222],[28,12770,67],{"className":12771},[145],[28,12773],{"className":12774,"style":149},[119],[28,12776,8283],{"className":12777,"style":12024},[114,222],[28,12779,387],{"className":12780},[187],[28,12782],{"className":12783,"style":120},[119],[28,12785,55],{"className":12786},[124],[28,12788],{"className":12789,"style":120},[119],[28,12791,12793],{"className":12792},[114,1801],[28,12794,12191],{"className":12795},[114,1809],[28,12797,460],{"className":12798},[459],[28,12800,12802],{"className":12801},[432],[28,12803,12805],{"className":12804,"style":1767},[436],[28,12806],{},[28,12808],{"className":12809},[187,1846],[11,12811,12812,12813,12873,12874,3959,12902,12930],{},"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（",[28,12814,12816,12840],{"className":12815,"translate":32},[31],[28,12817,12819],{"className":12818},[36],[38,12820,12821],{"xmlns":40},[42,12822,12823,12837],{},[45,12824,12825,12827,12829,12831,12833,12835],{},[87,12826,12136],{},[53,12828,378],{"stretchy":58},[48,12830,8283],{},[53,12832,67],{"separator":66},[48,12834,8283],{},[53,12836,387],{"stretchy":58},[94,12838,12839],{"encoding":96},"\\text{halts}(P, P)",[28,12841,12843],{"className":12842,"ariaHidden":66},[101],[28,12844,12846,12849,12855,12858,12861,12864,12867,12870],{"className":12845},[105],[28,12847],{"className":12848,"style":134},[109],[28,12850,12852],{"className":12851},[114,1801],[28,12853,12136],{"className":12854},[114],[28,12856,378],{"className":12857},[138],[28,12859,8283],{"className":12860,"style":12024},[114,222],[28,12862,67],{"className":12863},[145],[28,12865],{"className":12866,"style":149},[119],[28,12868,8283],{"className":12869,"style":12024},[114,222],[28,12871,387],{"className":12872},[187],"）。这正是对角线论证中「取第 ",[28,12875,12877,12890],{"className":12876,"translate":32},[31],[28,12878,12880],{"className":12879},[36],[38,12881,12882],{"xmlns":40},[42,12883,12884,12888],{},[45,12885,12886],{},[48,12887,1314],{},[94,12889,1314],{"encoding":96},[28,12891,12893],{"className":12892,"ariaHidden":66},[101],[28,12894,12896,12899],{"className":12895},[105],[28,12897],{"className":12898,"style":5994},[109],[28,12900,1314],{"className":12901},[114,222],[28,12903,12905,12918],{"className":12904,"translate":32},[31],[28,12906,12908],{"className":12907},[36],[38,12909,12910],{"xmlns":40},[42,12911,12912,12916],{},[45,12913,12914],{},[48,12915,1314],{},[94,12917,1314],{"encoding":96},[28,12919,12921],{"className":12920,"ariaHidden":66},[101],[28,12922,12924,12927],{"className":12923},[105],[28,12925],{"className":12926,"style":5994},[109],[28,12928,1314],{"className":12929},[114,222]," 位」的直接类比——让一个程序「审视」自身的自指动作。",[11,12932,12933,12934,1369],{},"导出矛盾。现在考察具体的执行 ",[28,12935,12937,12957],{"className":12936,"translate":32},[31],[28,12938,12940],{"className":12939},[36],[38,12941,12942],{"xmlns":40},[42,12943,12944,12954],{},[45,12945,12946,12948,12950,12952],{},[48,12947,8862],{},[53,12949,378],{"stretchy":58},[48,12951,8862],{},[53,12953,387],{"stretchy":58},[94,12955,12956],{"encoding":96},"D(D)",[28,12958,12960],{"className":12959,"ariaHidden":66},[101],[28,12961,12963,12966,12969,12972,12975],{"className":12962},[105],[28,12964],{"className":12965,"style":134},[109],[28,12967,8862],{"className":12968,"style":8909},[114,222],[28,12970,378],{"className":12971},[138],[28,12973,8862],{"className":12974,"style":8909},[114,222],[28,12976,387],{"className":12977},[187],[335,12979,12980,13184],{},[338,12981,9351,12982,13067,13068,13111,13112,9431,13140,13183],{},[28,12983,12985,13013],{"className":12984,"translate":32},[31],[28,12986,12988],{"className":12987},[36],[38,12989,12990],{"xmlns":40},[42,12991,12992,13010],{},[45,12993,12994,12996,12998,13000,13002,13004,13006,13008],{},[87,12995,12136],{},[53,12997,378],{"stretchy":58},[48,12999,8862],{},[53,13001,67],{"separator":66},[48,13003,8862],{},[53,13005,387],{"stretchy":58},[53,13007,55],{},[87,13009,12163],{},[94,13011,13012],{"encoding":96},"\\text{halts}(D, D) = \\text{真}",[28,13014,13016,13055],{"className":13015,"ariaHidden":66},[101],[28,13017,13019,13022,13028,13031,13034,13037,13040,13043,13046,13049,13052],{"className":13018},[105],[28,13020],{"className":13021,"style":134},[109],[28,13023,13025],{"className":13024},[114,1801],[28,13026,12136],{"className":13027},[114],[28,13029,378],{"className":13030},[138],[28,13032,8862],{"className":13033,"style":8909},[114,222],[28,13035,67],{"className":13036},[145],[28,13038],{"className":13039,"style":149},[119],[28,13041,8862],{"className":13042,"style":8909},[114,222],[28,13044,387],{"className":13045},[187],[28,13047],{"className":13048,"style":120},[119],[28,13050,55],{"className":13051},[124],[28,13053],{"className":13054,"style":120},[119],[28,13056,13058,13061],{"className":13057},[105],[28,13059],{"className":13060,"style":218},[109],[28,13062,13064],{"className":13063},[114,1801],[28,13065,12163],{"className":13066},[114,1809],"（即判定「",[28,13069,13071,13090],{"className":13070,"translate":32},[31],[28,13072,13074],{"className":13073},[36],[38,13075,13076],{"xmlns":40},[42,13077,13078,13088],{},[45,13079,13080,13082,13084,13086],{},[48,13081,8862],{},[53,13083,378],{"stretchy":58},[48,13085,8862],{},[53,13087,387],{"stretchy":58},[94,13089,12956],{"encoding":96},[28,13091,13093],{"className":13092,"ariaHidden":66},[101],[28,13094,13096,13099,13102,13105,13108],{"className":13095},[105],[28,13097],{"className":13098,"style":134},[109],[28,13100,8862],{"className":13101,"style":8909},[114,222],[28,13103,378],{"className":13104},[138],[28,13106,8862],{"className":13107,"style":8909},[114,222],[28,13109,387],{"className":13110},[187]," 停机」），那么根据 ",[28,13113,13115,13128],{"className":13114,"translate":32},[31],[28,13116,13118],{"className":13117},[36],[38,13119,13120],{"xmlns":40},[42,13121,13122,13126],{},[45,13123,13124],{},[48,13125,8862],{},[94,13127,8862],{"encoding":96},[28,13129,13131],{"className":13130,"ariaHidden":66},[101],[28,13132,13134,13137],{"className":13133},[105],[28,13135],{"className":13136,"style":218},[109],[28,13138,8862],{"className":13139,"style":8909},[114,222],[28,13141,13143,13162],{"className":13142,"translate":32},[31],[28,13144,13146],{"className":13145},[36],[38,13147,13148],{"xmlns":40},[42,13149,13150,13160],{},[45,13151,13152,13154,13156,13158],{},[48,13153,8862],{},[53,13155,378],{"stretchy":58},[48,13157,8862],{},[53,13159,387],{"stretchy":58},[94,13161,12956],{"encoding":96},[28,13163,13165],{"className":13164,"ariaHidden":66},[101],[28,13166,13168,13171,13174,13177,13180],{"className":13167},[105],[28,13169],{"className":13170,"style":134},[109],[28,13172,8862],{"className":13173,"style":8909},[114,222],[28,13175,378],{"className":13176},[138],[28,13178,8862],{"className":13179,"style":8909},[114,222],[28,13181,387],{"className":13182},[187]," 应该死循环。这与「它停机」矛盾。",[338,13185,9351,13186,13067,13271,13314,13315,9431,13343,13386],{},[28,13187,13189,13217],{"className":13188,"translate":32},[31],[28,13190,13192],{"className":13191},[36],[38,13193,13194],{"xmlns":40},[42,13195,13196,13214],{},[45,13197,13198,13200,13202,13204,13206,13208,13210,13212],{},[87,13199,12136],{},[53,13201,378],{"stretchy":58},[48,13203,8862],{},[53,13205,67],{"separator":66},[48,13207,8862],{},[53,13209,387],{"stretchy":58},[53,13211,55],{},[87,13213,12191],{},[94,13215,13216],{"encoding":96},"\\text{halts}(D, D) = \\text{假}",[28,13218,13220,13259],{"className":13219,"ariaHidden":66},[101],[28,13221,13223,13226,13232,13235,13238,13241,13244,13247,13250,13253,13256],{"className":13222},[105],[28,13224],{"className":13225,"style":134},[109],[28,13227,13229],{"className":13228},[114,1801],[28,13230,12136],{"className":13231},[114],[28,13233,378],{"className":13234},[138],[28,13236,8862],{"className":13237,"style":8909},[114,222],[28,13239,67],{"className":13240},[145],[28,13242],{"className":13243,"style":149},[119],[28,13245,8862],{"className":13246,"style":8909},[114,222],[28,13248,387],{"className":13249},[187],[28,13251],{"className":13252,"style":120},[119],[28,13254,55],{"className":13255},[124],[28,13257],{"className":13258,"style":120},[119],[28,13260,13262,13265],{"className":13261},[105],[28,13263],{"className":13264,"style":218},[109],[28,13266,13268],{"className":13267},[114,1801],[28,13269,12191],{"className":13270},[114,1809],[28,13272,13274,13293],{"className":13273,"translate":32},[31],[28,13275,13277],{"className":13276},[36],[38,13278,13279],{"xmlns":40},[42,13280,13281,13291],{},[45,13282,13283,13285,13287,13289],{},[48,13284,8862],{},[53,13286,378],{"stretchy":58},[48,13288,8862],{},[53,13290,387],{"stretchy":58},[94,13292,12956],{"encoding":96},[28,13294,13296],{"className":13295,"ariaHidden":66},[101],[28,13297,13299,13302,13305,13308,13311],{"className":13298},[105],[28,13300],{"className":13301,"style":134},[109],[28,13303,8862],{"className":13304,"style":8909},[114,222],[28,13306,378],{"className":13307},[138],[28,13309,8862],{"className":13310,"style":8909},[114,222],[28,13312,387],{"className":13313},[187]," 不停机」），那么根据 ",[28,13316,13318,13331],{"className":13317,"translate":32},[31],[28,13319,13321],{"className":13320},[36],[38,13322,13323],{"xmlns":40},[42,13324,13325,13329],{},[45,13326,13327],{},[48,13328,8862],{},[94,13330,8862],{"encoding":96},[28,13332,13334],{"className":13333,"ariaHidden":66},[101],[28,13335,13337,13340],{"className":13336},[105],[28,13338],{"className":13339,"style":218},[109],[28,13341,8862],{"className":13342,"style":8909},[114,222],[28,13344,13346,13365],{"className":13345,"translate":32},[31],[28,13347,13349],{"className":13348},[36],[38,13350,13351],{"xmlns":40},[42,13352,13353,13363],{},[45,13354,13355,13357,13359,13361],{},[48,13356,8862],{},[53,13358,378],{"stretchy":58},[48,13360,8862],{},[53,13362,387],{"stretchy":58},[94,13364,12956],{"encoding":96},[28,13366,13368],{"className":13367,"ariaHidden":66},[101],[28,13369,13371,13374,13377,13380,13383],{"className":13370},[105],[28,13372],{"className":13373,"style":134},[109],[28,13375,8862],{"className":13376,"style":8909},[114,222],[28,13378,378],{"className":13379},[138],[28,13381,8862],{"className":13382,"style":8909},[114,222],[28,13384,387],{"className":13385},[187]," 应该立即停机。这与「它不停机」矛盾。",[11,13388,13389,13390,13422],{},"两种情况都导致自相矛盾。因此，「存在通用停机判定器 ",[28,13391,13393,13407],{"className":13392,"translate":32},[31],[28,13394,13396],{"className":13395},[36],[38,13397,13398],{"xmlns":40},[42,13399,13400,13404],{},[45,13401,13402],{},[87,13403,12136],{},[94,13405,13406],{"encoding":96},"\\text{halts}",[28,13408,13410],{"className":13409,"ariaHidden":66},[101],[28,13411,13413,13416],{"className":13412},[105],[28,13414],{"className":13415,"style":902},[109],[28,13417,13419],{"className":13418},[114,1801],[28,13420,12136],{"className":13421},[114],"」的假设是假的。停机问题不可判定。",[11,13424,13425,13426,13619],{},"若按某种规则把所有程序枚举出来（",[28,13427,13429,13467],{"className":13428,"translate":32},[31],[28,13430,13432],{"className":13431},[36],[38,13433,13434],{"xmlns":40},[42,13435,13436,13464],{},[45,13437,13438,13444,13446,13452,13454,13460,13462],{},[354,13439,13440,13442],{},[48,13441,8283],{},[61,13443,70],{},[53,13445,67],{"separator":66},[354,13447,13448,13450],{},[48,13449,8283],{},[61,13451,75],{},[53,13453,67],{"separator":66},[354,13455,13456,13458],{},[48,13457,8283],{},[61,13459,80],{},[53,13461,67],{"separator":66},[53,13463,85],{},[94,13465,13466],{"encoding":96},"P_1, P_2, P_3, \\dots",[28,13468,13470],{"className":13469,"ariaHidden":66},[101],[28,13471,13473,13477,13518,13521,13524,13564,13567,13570,13610,13613,13616],{"className":13472},[105],[28,13474],{"className":13475,"style":13476},[109],"height:0.8778em;vertical-align:-0.1944em;",[28,13478,13480,13483],{"className":13479},[114],[28,13481,8283],{"className":13482,"style":12024},[114,222],[28,13484,13486],{"className":13485},[423],[28,13487,13489,13510],{"className":13488},[427,428],[28,13490,13492,13507],{"className":13491},[432],[28,13493,13495],{"className":13494,"style":437},[436],[28,13496,13498,13501],{"style":13497},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[28,13499],{"className":13500,"style":445},[444],[28,13502,13504],{"className":13503},[449,450,451,452],[28,13505,70],{"className":13506},[114,452],[28,13508,460],{"className":13509},[459],[28,13511,13513],{"className":13512},[432],[28,13514,13516],{"className":13515,"style":467},[436],[28,13517],{},[28,13519,67],{"className":13520},[145],[28,13522],{"className":13523,"style":149},[119],[28,13525,13527,13530],{"className":13526},[114],[28,13528,8283],{"className":13529,"style":12024},[114,222],[28,13531,13533],{"className":13532},[423],[28,13534,13536,13556],{"className":13535},[427,428],[28,13537,13539,13553],{"className":13538},[432],[28,13540,13542],{"className":13541,"style":437},[436],[28,13543,13544,13547],{"style":13497},[28,13545],{"className":13546,"style":445},[444],[28,13548,13550],{"className":13549},[449,450,451,452],[28,13551,75],{"className":13552},[114,452],[28,13554,460],{"className":13555},[459],[28,13557,13559],{"className":13558},[432],[28,13560,13562],{"className":13561,"style":467},[436],[28,13563],{},[28,13565,67],{"className":13566},[145],[28,13568],{"className":13569,"style":149},[119],[28,13571,13573,13576],{"className":13572},[114],[28,13574,8283],{"className":13575,"style":12024},[114,222],[28,13577,13579],{"className":13578},[423],[28,13580,13582,13602],{"className":13581},[427,428],[28,13583,13585,13599],{"className":13584},[432],[28,13586,13588],{"className":13587,"style":437},[436],[28,13589,13590,13593],{"style":13497},[28,13591],{"className":13592,"style":445},[444],[28,13594,13596],{"className":13595},[449,450,451,452],[28,13597,80],{"className":13598},[114,452],[28,13600,460],{"className":13601},[459],[28,13603,13605],{"className":13604},[432],[28,13606,13608],{"className":13607,"style":467},[436],[28,13609],{},[28,13611,67],{"className":13612},[145],[28,13614],{"className":13615,"style":149},[119],[28,13617,85],{"className":13618},[180],"——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：",[28,13621,13623],{"className":13622,"translate":32},[2403],[28,13624,13626,13791],{"className":13625,"translate":32},[31],[28,13627,13629],{"className":13628},[36],[38,13630,13631],{"xmlns":40,"display":2412},[42,13632,13633,13788],{},[45,13634,13635,13637,13786],{},[53,13636,378],{"fence":66},[1560,13638,13640,13700,13760],{"rowspacing":6615,"columnalign":13639,"columnspacing":1564},"center center center",[1566,13641,13642,13668,13694],{},[1569,13643,13644],{},[1572,13645,13646],{"scriptlevel":63,"displaystyle":58},[45,13647,13648,13650,13652,13658,13660,13666],{},[87,13649,12136],{},[53,13651,378],{"stretchy":58},[354,13653,13654,13656],{},[48,13655,8283],{},[61,13657,70],{},[53,13659,67],{"separator":66},[354,13661,13662,13664],{},[48,13663,8283],{},[61,13665,70],{},[53,13667,387],{"stretchy":58},[1569,13669,13670],{},[1572,13671,13672],{"scriptlevel":63,"displaystyle":58},[45,13673,13674,13676,13678,13684,13686,13692],{},[87,13675,12136],{},[53,13677,378],{"stretchy":58},[354,13679,13680,13682],{},[48,13681,8283],{},[61,13683,70],{},[53,13685,67],{"separator":66},[354,13687,13688,13690],{},[48,13689,8283],{},[61,13691,75],{},[53,13693,387],{"stretchy":58},[1569,13695,13696],{},[1572,13697,13698],{"scriptlevel":63,"displaystyle":58},[53,13699,6665],{"lspace":2419,"rspace":2419},[1566,13701,13702,13728,13754],{},[1569,13703,13704],{},[1572,13705,13706],{"scriptlevel":63,"displaystyle":58},[45,13707,13708,13710,13712,13718,13720,13726],{},[87,13709,12136],{},[53,13711,378],{"stretchy":58},[354,13713,13714,13716],{},[48,13715,8283],{},[61,13717,75],{},[53,13719,67],{"separator":66},[354,13721,13722,13724],{},[48,13723,8283],{},[61,13725,70],{},[53,13727,387],{"stretchy":58},[1569,13729,13730],{},[1572,13731,13732],{"scriptlevel":63,"displaystyle":58},[45,13733,13734,13736,13738,13744,13746,13752],{},[87,13735,12136],{},[53,13737,378],{"stretchy":58},[354,13739,13740,13742],{},[48,13741,8283],{},[61,13743,75],{},[53,13745,67],{"separator":66},[354,13747,13748,13750],{},[48,13749,8283],{},[61,13751,75],{},[53,13753,387],{"stretchy":58},[1569,13755,13756],{},[1572,13757,13758],{"scriptlevel":63,"displaystyle":58},[53,13759,6665],{"lspace":2419,"rspace":2419},[1566,13761,13762,13774,13780],{},[1569,13763,13764],{},[1572,13765,13766],{"scriptlevel":63,"displaystyle":58},[45,13767,13768,13770],{},[48,13769,2686],{"mathvariant":363},[2688,13771,13772],{"height":2419,"voffset":2419},[119,13773],{"mathbackground":2692,"width":2419,"height":2693},[1569,13775,13776],{},[1572,13777,13778],{"scriptlevel":63,"displaystyle":58},[45,13779],{},[1569,13781,13782],{},[1572,13783,13784],{"scriptlevel":63,"displaystyle":58},[53,13785,6802],{"lspace":2419,"rspace":2419},[53,13787,387],{"fence":66},[94,13789,13790],{"encoding":96},"\\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}",[28,13792,13794],{"className":13793,"ariaHidden":66},[101],[28,13795,13797,13801],{"className":13796},[105],[28,13798],{"className":13799,"style":13800},[109],"height:4.26em;vertical-align:-1.88em;",[28,13802,13804,13849,14424],{"className":13803},[180],[28,13805,13807],{"className":13806},[138],[28,13808,13810],{"className":13809},[1681,6827],[28,13811,13813,13840],{"className":13812},[427,428],[28,13814,13816,13837],{"className":13815},[432],[28,13817,13820],{"className":13818,"style":13819},[436],"height:2.35em;",[28,13821,13823,13827],{"style":13822},"top:-4.35em;",[28,13824],{"className":13825,"style":13826},[444],"height:6.2em;",[28,13828,13830],{"style":13829},"width:0.875em;height:4.2em;",[6849,13831,13834],{"xmlns":6851,"width":6852,"height":13832,"viewBox":13833},"4.2em","0 0 875 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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",[28,13838,460],{"className":13839},[459],[28,13841,13843],{"className":13842},[432],[28,13844,13847],{"className":13845,"style":13846},[436],"height:1.85em;",[28,13848],{},[28,13850,13852],{"className":13851},[114],[28,13853,13855,14111,14114,14117,14362,14365,14368],{"className":13854},[1560],[28,13856,13858],{"className":13857},[6880],[28,13859,13861,14102],{"className":13860},[427,428],[28,13862,13864,14099],{"className":13863},[432],[28,13865,13868,13975,14081],{"className":13866,"style":13867},[436],"height:2.38em;",[28,13869,13871,13874],{"style":13870},"top:-5.2275em;",[28,13872],{"className":13873,"style":2963},[444],[28,13875,13877,13883,13886,13926,13929,13932,13972],{"className":13876},[114],[28,13878,13880],{"className":13879},[114,1801],[28,13881,12136],{"className":13882},[114],[28,13884,378],{"className":13885},[138],[28,13887,13889,13892],{"className":13888},[114],[28,13890,8283],{"className":13891,"style":12024},[114,222],[28,13893,13895],{"className":13894},[423],[28,13896,13898,13918],{"className":13897},[427,428],[28,13899,13901,13915],{"className":13900},[432],[28,13902,13904],{"className":13903,"style":437},[436],[28,13905,13906,13909],{"style":13497},[28,13907],{"className":13908,"style":445},[444],[28,13910,13912],{"className":13911},[449,450,451,452],[28,13913,70],{"className":13914},[114,452],[28,13916,460],{"className":13917},[459],[28,13919,13921],{"className":13920},[432],[28,13922,13924],{"className":13923,"style":467},[436],[28,13925],{},[28,13927,67],{"className":13928},[145],[28,13930],{"className":13931,"style":149},[119],[28,13933,13935,13938],{"className":13934},[114],[28,13936,8283],{"className":13937,"style":12024},[114,222],[28,13939,13941],{"className":13940},[423],[28,13942,13944,13964],{"className":13943},[427,428],[28,13945,13947,13961],{"className":13946},[432],[28,13948,13950],{"className":13949,"style":437},[436],[28,13951,13952,13955],{"style":13497},[28,13953],{"className":13954,"style":445},[444],[28,13956,13958],{"className":13957},[449,450,451,452],[28,13959,70],{"className":13960},[114,452],[28,13962,460],{"className":13963},[459],[28,13965,13967],{"className":13966},[432],[28,13968,13970],{"className":13969,"style":467},[436],[28,13971],{},[28,13973,387],{"className":13974},[187],[28,13976,13977,13980],{"style":3382},[28,13978],{"className":13979,"style":2963},[444],[28,13981,13983,13989,13992,14032,14035,14038,14078],{"className":13982},[114],[28,13984,13986],{"className":13985},[114,1801],[28,13987,12136],{"className":13988},[114],[28,13990,378],{"className":13991},[138],[28,13993,13995,13998],{"className":13994},[114],[28,13996,8283],{"className":13997,"style":12024},[114,222],[28,13999,14001],{"className":14000},[423],[28,14002,14004,14024],{"className":14003},[427,428],[28,14005,14007,14021],{"className":14006},[432],[28,14008,14010],{"className":14009,"style":437},[436],[28,14011,14012,14015],{"style":13497},[28,14013],{"className":14014,"style":445},[444],[28,14016,14018],{"className":14017},[449,450,451,452],[28,14019,75],{"className":14020},[114,452],[28,14022,460],{"className":14023},[459],[28,14025,14027],{"className":14026},[432],[28,14028,14030],{"className":14029,"style":467},[436],[28,14031],{},[28,14033,67],{"className":14034},[145],[28,14036],{"className":14037,"style":149},[119],[28,14039,14041,14044],{"className":14040},[114],[28,14042,8283],{"className":14043,"style":12024},[114,222],[28,14045,14047],{"className":14046},[423],[28,14048,14050,14070],{"className":14049},[427,428],[28,14051,14053,14067],{"className":14052},[432],[28,14054,14056],{"className":14055,"style":437},[436],[28,14057,14058,14061],{"style":13497},[28,14059],{"className":14060,"style":445},[444],[28,14062,14064],{"className":14063},[449,450,451,452],[28,14065,70],{"className":14066},[114,452],[28,14068,460],{"className":14069},[459],[28,14071,14073],{"className":14072},[432],[28,14074,14076],{"className":14075,"style":467},[436],[28,14077],{},[28,14079,387],{"className":14080},[187],[28,14082,14084,14087],{"style":14083},"top:-2.1675em;",[28,14085],{"className":14086,"style":2963},[444],[28,14088,14090],{"className":14089},[114],[28,14091,14093,14096],{"className":14092},[114],[28,14094,2686],{"className":14095},[114],[28,14097],{"className":14098,"style":3830},[114,3829],[28,14100,460],{"className":14101},[459],[28,14103,14105],{"className":14104},[432],[28,14106,14109],{"className":14107,"style":14108},[436],"height:1.88em;",[28,14110],{},[28,14112],{"className":14113,"style":7134},[1773],[28,14115],{"className":14116,"style":7134},[1773],[28,14118,14120],{"className":141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,{"className":14429},[1681,6827],[28,14431,14433,14454],{"className":14432},[427,428],[28,14434,14436,14451],{"className":14435},[432],[28,14437,14439],{"className":14438,"style":13819},[436],[28,14440,14441,14444],{"style":13822},[28,14442],{"className":14443,"style":13826},[444],[28,14445,14446],{"style":13829},[6849,14447,14448],{"xmlns":6851,"width":6852,"height":13832,"viewBox":13833},[6856,14449],{"d":14450},"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",[28,14452,460],{"className":14453},[459],[28,14455,14457],{"className":14456},[432],[28,14458,14460],{"className":14459,"style":13846},[436],[28,14461],{},[11,14463,14464,14465,14493,14494,14636],{},"对角线程序 ",[28,14466,14468,14481],{"className":14467,"translate":32},[31],[28,14469,14471],{"className":14470},[36],[38,14472,14473],{"xmlns":40},[42,14474,14475,14479],{},[45,14476,14477],{},[48,14478,8862],{},[94,14480,8862],{"encoding":96},[28,14482,14484],{"className":14483,"ariaHidden":66},[101],[28,14485,14487,14490],{"className":14486},[105],[28,14488],{"className":14489,"style":218},[109],[28,14491,8862],{"className":14492,"style":8909},[114,222]," 的行为恰恰取决于这张表格的主对角线 ",[28,14495,14497,14529],{"className":14496,"translate":32},[31],[28,14498,14500],{"className":14499},[36],[38,14501,14502],{"xmlns":40},[42,14503,14504,14526],{},[45,14505,14506,14508,14510,14516,14518,14524],{},[87,14507,12136],{},[53,14509,378],{"stretchy":58},[354,14511,14512,14514],{},[48,14513,8283],{},[48,14515,3865],{},[53,14517,67],{"separator":66},[354,14519,14520,14522],{},[48,14521,8283],{},[48,14523,3865],{},[53,14525,387],{"stretchy":58},[94,14527,14528],{"encoding":96},"\\text{halts}(P_i, P_i)",[28,14530,14532],{"className":14531,"ariaHidden":66},[101],[28,14533,14535,14538,14544,14547,14587,14590,14593,14633],{"className":14534},[105],[28,14536],{"className":14537,"style":134},[109],[28,14539,14541],{"className":14540},[114,1801],[28,14542,12136],{"className":14543},[114],[28,14545,378],{"className":14546},[138],[28,14548,14550,14553],{"className":14549},[114],[28,14551,8283],{"className":14552,"style":12024},[114,222],[28,14554,14556],{"className":14555},[423],[28,14557,14559,14579],{"className":14558},[427,428],[28,14560,14562,14576],{"className":14561},[432],[28,14563,14565],{"className":14564,"style":3900},[436],[28,14566,14567,14570],{"style":13497},[28,14568],{"className":14569,"style":445},[444],[28,14571,14573],{"className":14572},[449,450,451,452],[28,14574,3865],{"className":14575},[114,222,452],[28,14577,460],{"className":14578},[459],[28,14580,14582],{"className":14581},[432],[28,14583,14585],{"className":14584,"style":467},[436],[28,14586],{},[28,14588,67],{"className":14589},[145],[28,14591],{"className":14592,"style":149},[119],[28,14594,14596,14599],{"className":14595},[114],[28,14597,8283],{"className":14598,"style":12024},[114,222],[28,14600,14602],{"className":14601},[423],[28,14603,14605,14625],{"className":14604},[427,428],[28,14606,14608,14622],{"className":14607},[432],[28,14609,14611],{"className":14610,"style":3900},[436],[28,14612,14613,14616],{"style":13497},[28,14614],{"className":14615,"style":445},[444],[28,14617,14619],{"className":14618},[449,450,451,452],[28,14620,3865],{"className":14621},[114,222,452],[28,14623,460],{"className":14624},[459],[28,14626,14628],{"className":14627},[432],[28,14629,14631],{"className":14630,"style":467},[436],[28,14632],{},[28,14634,387],{"className":14635},[187],"，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。",[11,14638,14639],{},"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：",[7745,14641,14643],{"className":7747,"code":14642,"language":7749,"meta":7750,"style":7750},"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",[7752,14644,14645,14653,14663,14667,14683,14688,14696,14706,14723,14733,14740,14744,14760,14775,14789,14807,14814,14827,14834,14841,14845,14866],{"__ignoreMap":7750},[28,14646,14647,14650],{"class":7756,"line":7757},[28,14648,14649],{"class":7760},"import",[28,14651,14652],{"class":7768}," sys\n",[28,14654,14655,14658,14661],{"class":7756,"line":7781},[28,14656,14657],{"class":7768},"sys.setrecursionlimit(",[28,14659,14660],{"class":7774},"50",[28,14662,7859],{"class":7768},[28,14664,14665],{"class":7756,"line":7788},[28,14666,7942],{"emptyLinePlaceholder":7941},[28,14668,14669,14671,14674,14677,14679,14681],{"class":7756,"line":7794},[28,14670,7761],{"class":7760},[28,14672,14673],{"class":7764}," hypothetical_halts",[28,14675,14676],{"class":7768},"(f, x, fuel",[28,14678,55],{"class":7760},[28,14680,7775],{"class":7774},[28,14682,7778],{"class":7768},[28,14684,14685],{"class":7756,"line":7800},[28,14686,14687],{"class":7784},"    \"\"\"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",[28,14689,14690,14693],{"class":7756,"line":7805},[28,14691,14692],{"class":7760},"    try",[28,14694,14695],{"class":7768},":\n",[28,14697,14698,14701,14703],{"class":7756,"line":7816},[28,14699,14700],{"class":7768},"        result ",[28,14702,55],{"class":7760},[28,14704,14705],{"class":7768}," f(x, fuel)\n",[28,14707,14708,14711,14714,14717,14720],{"class":7756,"line":7834},[28,14709,14710],{"class":7760},"        return",[28,14712,14713],{"class":7768}," result ",[28,14715,14716],{"class":7760},"is",[28,14718,14719],{"class":7760}," not",[28,14721,14722],{"class":7774}," None\n",[28,14724,14725,14728,14731],{"class":7756,"line":7862},[28,14726,14727],{"class":7760},"    except",[28,14729,14730],{"class":7774}," RecursionError",[28,14732,14695],{"class":7768},[28,14734,14735,14737],{"class":7756,"line":7880},[28,14736,14710],{"class":7760},[28,14738,14739],{"class":7774}," False\n",[28,14741,14742],{"class":7756,"line":7908},[28,14743,7942],{"emptyLinePlaceholder":7941},[28,14745,14746,14748,14751,14754,14756,14758],{"class":7756,"line":7920},[28,14747,7761],{"class":7760},[28,14749,14750],{"class":7764}," D",[28,14752,14753],{"class":7768},"(x, fuel",[28,14755,55],{"class":7760},[28,14757,7775],{"class":7774},[28,14759,7778],{"class":7768},[28,14761,14762,14765,14768,14771,14773],{"class":7756,"line":7938},[28,14763,14764],{"class":7760},"    if",[28,14766,14767],{"class":7768}," fuel ",[28,14769,14770],{"class":7760},"\u003C=",[28,14772,11664],{"class":7774},[28,14774,14695],{"class":7768},[28,14776,14777,14780,14782,14784,14787],{"class":7756,"line":7945},[28,14778,14779],{"class":7760},"        raise",[28,14781,14730],{"class":7774},[28,14783,378],{"class":7768},[28,14785,14786],{"class":7784},"\"燃料耗尽，模拟死循环\"",[28,14788,7859],{"class":7768},[28,14790,14791,14794,14796,14799,14802,14805],{"class":7756,"line":7950},[28,14792,14793],{"class":7768},"    will_halt ",[28,14795,55],{"class":7760},[28,14797,14798],{"class":7768}," hypothetical_halts(x, x, fuel ",[28,14800,14801],{"class":7760},"-",[28,14803,14804],{"class":7774}," 1",[28,14806,7859],{"class":7768},[28,14808,14809,14811],{"class":7756,"line":7961},[28,14810,14764],{"class":7760},[28,14812,14813],{"class":7768}," will_halt:\n",[28,14815,14816,14818,14820,14822,14825],{"class":7756,"line":7986},[28,14817,14779],{"class":7760},[28,14819,14730],{"class":7774},[28,14821,378],{"class":7768},[28,14823,14824],{"class":7784},"\"模拟死循环：halts 判定其停机，故 D 刻意不停机\"",[28,14826,7859],{"class":7768},[28,14828,14829,14832],{"class":7756,"line":8009},[28,14830,14831],{"class":7760},"    else",[28,14833,14695],{"class":7768},[28,14835,14836,14838],{"class":7756,"line":8022},[28,14837,14710],{"class":7760},[28,14839,14840],{"class":7784}," \"D 停机\"\n",[28,14842,14843],{"class":7756,"line":8028},[28,14844,7942],{"emptyLinePlaceholder":7941},[28,14846,14847,14850,14852,14855,14859,14861,14864],{"class":7756,"line":8033},[28,14848,14849],{"class":7768},"outcome ",[28,14851,55],{"class":7760},[28,14853,14854],{"class":7768}," D(D, ",[28,14856,14858],{"class":14857},"s9osk","fuel",[28,14860,55],{"class":7760},[28,14862,14863],{"class":7774},"8",[28,14865,7859],{"class":7768},[28,14867,14868,14870,14872,14875],{"class":7756,"line":8044},[28,14869,8047],{"class":7774},[28,14871,378],{"class":7768},[28,14873,14874],{"class":7784},"\"D(D) 的执行结果：\"",[28,14876,14877],{"class":7768},", outcome)\n",[11,14879,8058],{},[7745,14881,14884],{"className":14882,"code":14883,"language":1801},[8062],"=== 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",[7752,14885,14883],{"__ignoreMap":7750},[11,14887,14888,14889,14892,14893,14895],{},"这段代码里的 ",[7752,14890,14891],{},"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 ",[7752,14894,14858],{},"（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）。",[18,14897,14898],{"id":14898},"哥德尔不完备定理",[11,14900,14901],{},"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：",[2132,14903,14904],{},[11,14905,14906],{},"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。",[11,14908,14909],{},"哥德尔的证明大体分为三步：",[14911,14912,14913,14916,15273],"ol",{},[338,14914,14915],{},"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。",[338,14917,14918,14919,14948,14949,15032,15035,15036,15081,15082,15112,15113,15199,15200,15243,15244,15272],{},"构造自指命题。利用编码技巧，构造一个命题 ",[28,14920,14922,14936],{"className":14921,"translate":32},[31],[28,14923,14925],{"className":14924},[36],[38,14926,14927],{"xmlns":40},[42,14928,14929,14934],{},[45,14930,14931],{},[48,14932,14933],{},"G",[94,14935,14933],{"encoding":96},[28,14937,14939],{"className":14938,"ariaHidden":66},[101],[28,14940,14942,14945],{"className":14941},[105],[28,14943],{"className":14944,"style":218},[109],[28,14946,14933],{"className":14947},[114,222],"，其核心内容是：",[28,14950,14952],{"className":14951,"translate":32},[2403],[28,14953,14955,14981],{"className":14954,"translate":32},[31],[28,14956,14958],{"className":14957},[36],[38,14959,14960],{"xmlns":40,"display":2412},[42,14961,14962,14978],{},[45,14963,14964,14966,14968,14970,14973,14975],{},[48,14965,14933],{},[53,14967,272],{},[53,14969,55],{},[87,14971,14972],{},"「命题 ",[48,14974,14933],{},[87,14976,14977],{}," 本身在该系统内不可证明」",[94,14979,14980],{"encoding":96},"G := \\text{「命题 } G \\text{ 本身在该系统内不可证明」}",[28,14982,14984,15003],{"className":14983,"ariaHidden":66},[101],[28,14985,14987,14990,14993,14996,15000],{"className":14986},[105],[28,14988],{"className":14989,"style":218},[109],[28,14991,14933],{"className":14992},[114,222],[28,14994],{"className":14995,"style":120},[119],[28,14997,14999],{"className":14998},[124],":=",[28,15001],{"className":15002,"style":120},[119],[28,15004,15006,15009,15019,15022],{"className":15005},[105],[28,15007],{"className":15008,"style":218},[109],[28,15010,15012,15016],{"className":15011},[114,1801],[28,15013,15015],{"className":15014},[114,1809],"「命题",[28,15017,1805],{"className":15018},[114],[28,15020,14933],{"className":15021},[114,222],[28,15023,15025,15028],{"className":15024},[114,1801],[28,15026,1805],{"className":15027},[114],[28,15029,15031],{"className":15030},[114,1809],"本身在该系统内不可证明」",[15033,15034],"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 ",[28,15037,15039,15060],{"className":15038,"translate":32},[31],[28,15040,15042],{"className":15041},[36],[38,15043,15044],{"xmlns":40},[42,15045,15046,15057],{},[45,15047,15048,15051,15053,15055],{},[48,15049,15050],{},"ϕ",[53,15052,378],{"stretchy":58},[48,15054,2430],{},[53,15056,387],{"stretchy":58},[94,15058,15059],{"encoding":96},"\\phi(x)",[28,15061,15063],{"className":15062,"ariaHidden":66},[101],[28,15064,15066,15069,15072,15075,15078],{"className":15065},[105],[28,15067],{"className":15068,"style":134},[109],[28,15070,15050],{"className":15071},[114,222],[28,15073,378],{"className":15074},[138],[28,15076,2430],{"className":15077},[114,222],[28,15079,387],{"className":15080},[187],"，都能构造一个句子 ",[28,15083,15085,15100],{"className":15084,"translate":32},[31],[28,15086,15088],{"className":15087},[36],[38,15089,15090],{"xmlns":40},[42,15091,15092,15097],{},[45,15093,15094],{},[48,15095,15096],{},"ψ",[94,15098,15099],{"encoding":96},"\\psi",[28,15101,15103],{"className":15102,"ariaHidden":66},[101],[28,15104,15106,15109],{"className":15105},[105],[28,15107],{"className":15108,"style":292},[109],[28,15110,15096],{"className":15111,"style":2123},[114,222],"，使系统证明 ",[28,15114,15116,15151],{"className":15115,"translate":32},[31],[28,15117,15119],{"className":15118},[36],[38,15120,15121],{"xmlns":40},[42,15122,15123,15148],{},[45,15124,15125,15127,15130,15132,15134,15139,15141,15146],{},[48,15126,15096],{},[53,15128,15129],{},"↔",[48,15131,15050],{},[53,15133,378],{"stretchy":58},[53,15135,15136],{},[48,15137,15138],{"mathvariant":363},"⌜",[48,15140,15096],{},[53,15142,15143],{},[48,15144,15145],{"mathvariant":363},"⌝",[53,15147,387],{"stretchy":58},[94,15149,15150],{"encoding":96},"\\psi \\leftrightarrow \\phi(\\ulcorner \\psi \\urcorner)",[28,15152,15154,15172],{"className":15153,"ariaHidden":66},[101],[28,15155,15157,15160,15163,15166,15169],{"className":15156},[105],[28,15158],{"className":15159,"style":292},[109],[28,15161,15096],{"className":15162,"style":2123},[114,222],[28,15164],{"className":15165,"style":120},[119],[28,15167,15129],{"className":15168},[124],[28,15170],{"className":15171,"style":120},[119],[28,15173,15175,15178,15181,15184,15189,15192,15196],{"className":15174},[105],[28,15176],{"className":15177,"style":134},[109],[28,15179,15050],{"className":15180},[114,222],[28,15182,378],{"className":15183},[138],[28,15185,15188],{"className":15186},[138,15187],"amsrm","┌",[28,15190,15096],{"className":15191,"style":2123},[114,222],[28,15193,15195],{"className":15194},[187,15187],"┐",[28,15197,387],{"className":15198},[187],"（其中 ",[28,15201,15203,15225],{"className":15202,"translate":32},[31],[28,15204,15206],{"className":15205},[36],[38,15207,15208],{"xmlns":40},[42,15209,15210,15222],{},[45,15211,15212,15216,15218],{},[53,15213,15214],{},[48,15215,15138],{"mathvariant":363},[48,15217,15096],{},[53,15219,15220],{},[48,15221,15145],{"mathvariant":363},[94,15223,15224],{"encoding":96},"\\ulcorner \\psi \\urcorner",[28,15226,15228],{"className":15227,"ariaHidden":66},[101],[28,15229,15231,15234,15237,15240],{"className":15230},[105],[28,15232],{"className":15233,"style":292},[109],[28,15235,15188],{"className":15236},[138,15187],[28,15238,15096],{"className":15239,"style":2123},[114,222],[28,15241,15195],{"className":15242},[187,15187]," 表示 ",[28,15245,15247,15260],{"className":15246,"translate":32},[31],[28,15248,15250],{"className":15249},[36],[38,15251,15252],{"xmlns":40},[42,15253,15254,15258],{},[45,15255,15256],{},[48,15257,15096],{},[94,15259,15099],{"encoding":96},[28,15261,15263],{"className":15262,"ariaHidden":66},[101],[28,15264,15266,15269],{"className":15265},[105],[28,15267],{"className":15268,"style":292},[109],[28,15270,15096],{"className":15271,"style":2123},[114,222]," 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。",[338,15274,15275,15276],{},"导出矛盾的两难。",[335,15277,15278,15368,15464],{},[338,15279,15280,15281,15309,15310,15338,15339,15367],{},"若系统能证明 ",[28,15282,15284,15297],{"className":15283,"translate":32},[31],[28,15285,15287],{"className":15286},[36],[38,15288,15289],{"xmlns":40},[42,15290,15291,15295],{},[45,15292,15293],{},[48,15294,14933],{},[94,15296,14933],{"encoding":96},[28,15298,15300],{"className":15299,"ariaHidden":66},[101],[28,15301,15303,15306],{"className":15302},[105],[28,15304],{"className":15305,"style":218},[109],[28,15307,14933],{"className":15308},[114,222],"，那么根据 ",[28,15311,15313,15326],{"className":15312,"translate":32},[31],[28,15314,15316],{"className":15315},[36],[38,15317,15318],{"xmlns":40},[42,15319,15320,15324],{},[45,15321,15322],{},[48,15323,14933],{},[94,15325,14933],{"encoding":96},[28,15327,15329],{"className":15328,"ariaHidden":66},[101],[28,15330,15332,15335],{"className":15331},[105],[28,15333],{"className":15334,"style":218},[109],[28,15336,14933],{"className":15337},[114,222]," 的含义（「",[28,15340,15342,15355],{"className":15341,"translate":32},[31],[28,15343,15345],{"className":15344},[36],[38,15346,15347],{"xmlns":40},[42,15348,15349,15353],{},[45,15350,15351],{},[48,15352,14933],{},[94,15354,14933],{"encoding":96},[28,15356,15358],{"className":15357,"ariaHidden":66},[101],[28,15359,15361,15364],{"className":15360},[105],[28,15362],{"className":15363,"style":218},[109],[28,15365,14933],{"className":15366},[114,222]," 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。",[338,15369,15370,15371,15399,15400,15434,15435,15463],{},"若系统能否证 ",[28,15372,15374,15387],{"className":15373,"translate":32},[31],[28,15375,15377],{"className":15376},[36],[38,15378,15379],{"xmlns":40},[42,15380,15381,15385],{},[45,15382,15383],{},[48,15384,14933],{},[94,15386,14933],{"encoding":96},[28,15388,15390],{"className":15389,"ariaHidden":66},[101],[28,15391,15393,15396],{"className":15392},[105],[28,15394],{"className":15395,"style":218},[109],[28,15397,14933],{"className":15398},[114,222],"（即证明 ",[28,15401,15403,15419],{"className":15402,"translate":32},[31],[28,15404,15406],{"className":15405},[36],[38,15407,15408],{"xmlns":40},[42,15409,15410,15416],{},[45,15411,15412,15414],{},[48,15413,9906],{"mathvariant":363},[48,15415,14933],{},[94,15417,15418],{"encoding":96},"\\neg G",[28,15420,15422],{"className":15421,"ariaHidden":66},[101],[28,15423,15425,15428,15431],{"className":15424},[105],[28,15426],{"className":15427,"style":218},[109],[28,15429,9906],{"className":15430},[114],[28,15432,14933],{"className":15433},[114,222],"，相当于证明「",[28,15436,15438,15451],{"className":15437,"translate":32},[31],[28,15439,15441],{"className":15440},[36],[38,15442,15443],{"xmlns":40},[42,15444,15445,15449],{},[45,15446,15447],{},[48,15448,14933],{},[94,15450,14933],{"encoding":96},[28,15452,15454],{"className":15453,"ariaHidden":66},[101],[28,15455,15457,15460],{"className":15456},[105],[28,15458],{"className":15459,"style":218},[109],[28,15461,14933],{"className":15462},[114,222]," 可证明」），那么在合理的补充条件下，这同样会导致矛盾。",[338,15465,15466,15467,15495],{},"因此 ",[28,15468,15470,15483],{"className":15469,"translate":32},[31],[28,15471,15473],{"className":15472},[36],[38,15474,15475],{"xmlns":40},[42,15476,15477,15481],{},[45,15478,15479],{},[48,15480,14933],{},[94,15482,14933],{"encoding":96},[28,15484,15486],{"className":15485,"ariaHidden":66},[101],[28,15487,15489,15492],{"className":15488},[105],[28,15490],{"className":15491,"style":218},[109],[28,15493,14933],{"className":15494},[114,222]," 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」。",[11,15497,15498],{},"与前面两个证明的共通之处：",[10400,15500,15501,15631],{},[10403,15502,15503],{},[10406,15504,15505,15507,15538,15569,15600],{},[10409,15506],{},[10409,15508,15509,15510],{},"对角数 ",[28,15511,15513,15526],{"className":15512,"translate":32},[31],[28,15514,15516],{"className":15515},[36],[38,15517,15518],{"xmlns":40},[42,15519,15520,15524],{},[45,15521,15522],{},[48,15523,4066],{},[94,15525,4066],{"encoding":96},[28,15527,15529],{"className":15528,"ariaHidden":66},[101],[28,15530,15532,15535],{"className":15531},[105],[28,15533],{"className":15534,"style":4110},[109],[28,15536,4066],{"className":15537,"style":2123},[114,222],[10409,15539,15540,15541],{},"对角集 ",[28,15542,15544,15557],{"className":15543,"translate":32},[31],[28,15545,15547],{"className":15546},[36],[38,15548,15549],{"xmlns":40},[42,15550,15551,15555],{},[45,15552,15553],{},[48,15554,8862],{},[94,15556,8862],{"encoding":96},[28,15558,15560],{"className":15559,"ariaHidden":66},[101],[28,15561,15563,15566],{"className":15562},[105],[28,15564],{"className":15565,"style":218},[109],[28,15567,8862],{"className":15568,"style":8909},[114,222],[10409,15570,15571,15572],{},"对角程序 ",[28,15573,15575,15588],{"className":15574,"translate":32},[31],[28,15576,15578],{"className":15577},[36],[38,15579,15580],{"xmlns":40},[42,15581,15582,15586],{},[45,15583,15584],{},[48,15585,8862],{},[94,15587,8862],{"encoding":96},[28,15589,15591],{"className":15590,"ariaHidden":66},[101],[28,15592,15594,15597],{"className":15593},[105],[28,15595],{"className":15596,"style":218},[109],[28,15598,8862],{"className":15599,"style":8909},[114,222],[10409,15601,15602,15603],{},"哥德尔句 ",[28,15604,15606,15619],{"className":15605,"translate":32},[31],[28,15607,15609],{"className":15608},[36],[38,15610,15611],{"xmlns":40},[42,15612,15613,15617],{},[45,15614,15615],{},[48,15616,14933],{},[94,15618,14933],{"encoding":96},[28,15620,15622],{"className":15621,"ariaHidden":66},[101],[28,15623,15625,15628],{"className":15624},[105],[28,15626],{"className":15627,"style":218},[109],[28,15629,14933],{"className":15630},[114,222],[10415,15632,15633,15650,15667],{},[10406,15634,15635,15638,15641,15644,15647],{},[10420,15636,15637],{},"自指的对象",[10420,15639,15640],{},"每一位「避开」自身",[10420,15642,15643],{},"收集「不属于自己」的元素",[10420,15645,15646],{},"用自己的行为否定对自身的判定",[10420,15648,15649],{},"断言「我不可证明」",[10406,15651,15652,15655,15658,15661,15664],{},[10420,15653,15654],{},"依赖的机制",[10420,15656,15657],{},"十进制展开的对角线",[10420,15659,15660],{},"集合属于关系的自指",[10420,15662,15663],{},"程序以自身为输入",[10420,15665,15666],{},"通过哥德尔编号实现的自指",[10406,15668,15669,15672,15703,15734,15737],{},[10420,15670,15671],{},"矛盾的来源",[10420,15673,15674,15675],{},"假定的双射无法覆盖 ",[28,15676,15678,15691],{"className":15677,"translate":32},[31],[28,15679,15681],{"className":15680},[36],[38,15682,15683],{"xmlns":40},[42,15684,15685,15689],{},[45,15686,15687],{},[48,15688,4066],{},[94,15690,4066],{"encoding":96},[28,15692,15694],{"className":15693,"ariaHidden":66},[101],[28,15695,15697,15700],{"className":15696},[105],[28,15698],{"className":15699,"style":4110},[109],[28,15701,4066],{"className":15702,"style":2123},[114,222],[10420,15704,15705,15706],{},"假定的满射无法覆盖 ",[28,15707,15709,15722],{"className":15708,"translate":32},[31],[28,15710,15712],{"className":15711},[36],[38,15713,15714],{"xmlns":40},[42,15715,15716,15720],{},[45,15717,15718],{},[48,15719,8862],{},[94,15721,8862],{"encoding":96},[28,15723,15725],{"className":15724,"ariaHidden":66},[101],[28,15726,15728,15731],{"className":15727},[105],[28,15729],{"className":15730,"style":218},[109],[28,15732,8862],{"className":15733,"style":8909},[114,222],[10420,15735,15736],{},"假定的判定器导致悖论",[10420,15738,15739],{},"假定的可证明性导致悖论",[11,15741,15742],{},"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。",[18,15744,15745],{"id":15745},"延伸",[11,15747,15748],{},"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：",[335,15750,15751,15754,15815,15818],{},[338,15752,15753],{},"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。",[338,15755,15756,15757,15814],{},"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 ",[28,15758,15760,15780],{"className":15759,"translate":32},[31],[28,15761,15763],{"className":15762},[36],[38,15764,15765],{"xmlns":40},[42,15766,15767,15777],{},[45,15768,15769,15771,15774],{},[48,15770,8283],{},[53,15772,15773],{},"⊊",[87,15775,15776],{},"EXP",[94,15778,15779],{"encoding":96},"P \\subsetneq \\text{EXP}",[28,15781,15783,15802],{"className":15782,"ariaHidden":66},[101],[28,15784,15786,15790,15793,15796,15799],{"className":15785},[105],[28,15787],{"className":15788,"style":15789},[109],"height:0.8193em;vertical-align:-0.136em;",[28,15791,8283],{"className":15792,"style":12024},[114,222],[28,15794],{"className":15795,"style":120},[119],[28,15797,15773],{"className":15798},[124,15187],[28,15800],{"className":15801,"style":120},[119],[28,15803,15805,15808],{"className":15804},[105],[28,15806],{"className":15807,"style":218},[109],[28,15809,15811],{"className":15810},[114,1801],[28,15812,15776],{"className":15813},[114]," 等谱系关系的基础）。",[338,15816,15817],{},"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。",[338,15819,15820,15821,15961,15962,16111],{},"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，",[28,15822,15824,15855],{"className":15823,"translate":32},[31],[28,15825,15827],{"className":15826},[36],[38,15828,15829],{"xmlns":40},[42,15830,15831,15852],{},[45,15832,15833,15836,15838,15840,15842,15844,15846,15848,15850],{},[48,15834,15835],{},"R",[53,15837,55],{},[53,15839,59],{"stretchy":58},[48,15841,2430],{},[53,15843,272],{},[48,15845,2430],{},[53,15847,8881],{"mathvariant":363},[48,15849,2430],{},[53,15851,92],{"stretchy":58},[94,15853,15854],{"encoding":96},"R = \\{x : x \\notin x\\}",[28,15856,15858,15877,15898,15949],{"className":15857,"ariaHidden":66},[101],[28,15859,15861,15864,15868,15871,15874],{"className":15860},[105],[28,15862],{"className":15863,"style":218},[109],[28,15865,15835],{"className":15866,"style":15867},[114,222],"margin-right:0.0077em;",[28,15869],{"className":15870,"style":120},[119],[28,15872,55],{"className":15873},[124],[28,15875],{"className":15876,"style":120},[119],[28,15878,15880,15883,15886,15889,15892,15895],{"className":15879},[105],[28,15881],{"className":15882,"style":134},[109],[28,15884,59],{"className":15885},[138],[28,15887,2430],{"className":15888},[114,222],[28,15890],{"className":15891,"style":120},[119],[28,15893,272],{"className":15894},[124],[28,15896],{"className":15897,"style":120},[119],[28,15899,15901,15904,15907,15910,15946],{"className":15900},[105],[28,15902],{"className":15903,"style":134},[109],[28,15905,2430],{"className":15906},[114,222],[28,15908],{"className":15909,"style":120},[119],[28,15911,15913,15919],{"className":15912},[124],[28,15914,15916],{"className":15915},[114],[28,15917,748],{"className":15918},[124],[28,15920,15922],{"className":15921},[114,482],[28,15923,15925],{"className":15924},[486],[28,15926,15928,15931,15943],{"className":15927},[8991],[28,15929],{"className":15930,"style":134},[109],[28,15932,15934],{"className":15933},[497],[28,15935,15937,15940],{"className":15936},[114],[28,15938,1580],{"className":15939},[114],[28,15941],{"className":15942,"style":9007},[119],[28,15944],{"className":15945},[508],[28,15947],{"className":15948,"style":120},[119],[28,15950,15952,15955,15958],{"className":15951},[105],[28,15953],{"className":15954,"style":134},[109],[28,15956,2430],{"className":15957},[114,222],[28,15959,92],{"className":15960},[187],"，与康托尔定理证明中的集合 ",[28,15963,15965,16000],{"className":15964,"translate":32},[31],[28,15966,15968],{"className":15967},[36],[38,15969,15970],{"xmlns":40},[42,15971,15972,15998],{},[45,15973,15974,15976,15978,15980,15982,15984,15986,15988,15990,15992,15994,15996],{},[48,15975,8862],{},[53,15977,55],{},[53,15979,59],{"stretchy":58},[48,15981,358],{},[53,15983,272],{},[48,15985,358],{},[53,15987,8881],{"mathvariant":363},[48,15989,269],{},[53,15991,378],{"stretchy":58},[48,15993,358],{},[53,15995,387],{"stretchy":58},[53,15997,92],{"stretchy":58},[94,15999,10782],{"encoding":96},[28,16001,16003,16021,16042,16093],{"className":16002,"ariaHidden":66},[101],[28,16004,16006,16009,16012,16015,16018],{"className":16005},[105],[28,16007],{"className":16008,"style":218},[109],[28,16010,8862],{"className":16011,"style":8909},[114,222],[28,16013],{"className":16014,"style":120},[119],[28,16016,55],{"className":16017},[124],[28,16019],{"className":16020,"style":120},[119],[28,16022,16024,16027,16030,16033,16036,16039],{"className":16023},[105],[28,16025],{"className":16026,"style":134},[109],[28,16028,59],{"className":16029},[138],[28,16031,358],{"className":16032},[114,222],[28,16034],{"className":16035,"style":120},[119],[28,16037,272],{"className":16038},[124],[28,16040],{"className":16041,"style":120},[119],[28,16043,16045,16048,16051,16054,16090],{"className":16044},[105],[28,16046],{"className":16047,"style":134},[109],[28,16049,358],{"className":16050},[114,222],[28,16052],{"className":16053,"style":120},[119],[28,16055,16057,16063],{"className":16056},[124],[28,16058,16060],{"className":16059},[114],[28,16061,748],{"className":16062},[124],[28,16064,16066],{"className":16065},[114,482],[28,16067,16069],{"className":16068},[486],[28,16070,16072,16075,16087],{"className":16071},[8991],[28,16073],{"className":16074,"style":134},[109],[28,16076,16078],{"className":16077},[497],[28,16079,16081,16084],{"className":16080},[114],[28,16082,1580],{"className":16083},[114],[28,16085],{"className":16086,"style":9007},[119],[28,16088],{"className":16089},[508],[28,16091],{"className":16092,"style":120},[119],[28,16094,16096,16099,16102,16105,16108],{"className":16095},[105],[28,16097],{"className":16098,"style":134},[109],[28,16100,269],{"className":16101,"style":296},[114,222],[28,16103,378],{"className":16104},[138],[28,16106,358],{"className":16107},[114,222],[28,16109,10894],{"className":16110},[187]," 具有完全相同的结构——两者都是「自指 + 自我排除」的产物。",[11,16113,16114],{},"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：",[14911,16116,16117,16120,16123,16184],{},[338,16118,16119],{},"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。",[338,16121,16122],{},"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）。",[338,16124,16125,16126,16154,16155,16183],{},"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 ",[28,16127,16129,16142],{"className":16128,"translate":32},[31],[28,16130,16132],{"className":16131},[36],[38,16133,16134],{"xmlns":40},[42,16135,16136,16140],{},[45,16137,16138],{},[48,16139,1314],{},[94,16141,1314],{"encoding":96},[28,16143,16145],{"className":16144,"ariaHidden":66},[101],[28,16146,16148,16151],{"className":16147},[105],[28,16149],{"className":16150,"style":5994},[109],[28,16152,1314],{"className":16153},[114,222]," 个条目在第 ",[28,16156,16158,16171],{"className":16157,"translate":32},[31],[28,16159,16161],{"className":16160},[36],[38,16162,16163],{"xmlns":40},[42,16164,16165,16169],{},[45,16166,16167],{},[48,16168,1314],{},[94,16170,1314],{"encoding":96},[28,16172,16174],{"className":16173,"ariaHidden":66},[101],[28,16175,16177,16180],{"className":16176},[105],[28,16178],{"className":16179,"style":5994},[109],[28,16181,1314],{"className":16182},[114,222]," 个位置不同」这一模式）。",[338,16185,16186],{},"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。",[16188,16189,16190],"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":7750,"searchDepth":7781,"depth":7781,"links":16192},[16193,16194,16195,16196,16199,16200,16201],{"id":20,"depth":7781,"text":20},{"id":2130,"depth":7781,"text":2130},{"id":7740,"depth":7781,"text":7740},{"id":8230,"depth":7781,"text":8230,"children":16197},[16198],{"id":8540,"depth":7788,"text":8540},{"id":11989,"depth":7781,"text":11989},{"id":14898,"depth":7781,"text":14898},{"id":15745,"depth":7781,"text":15745},false,"md",null,{},"\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en",{"title":6,"description":13},"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。",[16211,16212],"mathematics","computation","2026-07-31T00:00:00+08:00","ZcA-tfQ1zVCr2HL5QHuWWDVkSCGS-RbN98cAuxLrnw0",{"id":16216,"title":16217,"body":16218,"description":20900,"draft":16202,"enableComment":7941,"extension":16203,"image":7750,"important":16202,"location":16204,"meta":20901,"navigation":7941,"ogImage":16204,"path":20902,"seo":20903,"stem":20904,"summary":20905,"tags":20906,"time":20909,"__hash__":20910},"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":16219,"toc":20890},[16220,16230,16233,16242,16245,16248,16251,16254,16540,16887,16989,16993,16996,17136,17429,17432,17435,18034,18671,18674,18677,18680,18695,18698,18701,18704,18707,19294,19297,19370,19373,19376,19379,19818,20021,20226,20485,20558,20561,20564,20567,20570,20610,20675,20758,20761,20764,20767,20770,20773,20776,20787,20790,20793,20796,20799,20802,20805,20808,20881,20884,20887],[11,16221,16222,16223,16229],{},"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 ",[358,16224,16228],{"href":16225,"rel":16226},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2405.14860",[16227],"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。",[11,16231,16232],{},"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？",[11,16234,16235,16236,16241],{},"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 ",[358,16237,16240],{"href":16238,"rel":16239},"https:\u002F\u002Farxiv.org\u002Fpdf\u002F2505.18235",[16227],"《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。",[18,16243,16244],{"id":16244},"线性表征假说",[11,16246,16247],{},"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」。",[11,16249,16250],{},"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。",[11,16252,16253],{},"因此，该领域提出了 LRH 的推广版本——多维线性表征假说：",[28,16255,16257],{"className":16256,"translate":32},[2403],[28,16258,16260,16330],{"className":16259,"translate":32},[31],[28,16261,16263],{"className":16262},[36],[38,16264,16265],{"xmlns":40,"display":2412},[42,16266,16267,16327],{},[45,16268,16269,16272,16274,16276,16278,16280,16301,16308,16310,16312,16314,16321,16323,16325],{},[48,16270,16271],{"mathvariant":363},"Ψ",[53,16273,378],{"stretchy":58},[48,16275,2430],{},[53,16277,387],{"stretchy":58},[53,16279,55],{},[16281,16282,16283,16286],"munder",{},[53,16284,16285],{},"∑",[45,16287,16288,16290,16292,16295,16297,16299],{},[48,16289,269],{},[53,16291,748],{},[48,16293,16294],{},"F",[53,16296,378],{"stretchy":58},[48,16298,2430],{},[53,16300,387],{"stretchy":58},[354,16302,16303,16306],{},[48,16304,16305],{},"ρ",[48,16307,269],{},[53,16309,378],{"stretchy":58},[48,16311,2430],{},[53,16313,387],{"stretchy":58},[354,16315,16316,16319],{},[48,16317,16318],{},"v",[48,16320,269],{},[53,16322,378],{"stretchy":58},[48,16324,2430],{},[53,16326,387],{"stretchy":58},[94,16328,16329],{"encoding":96},"\\Psi(x) = \\sum_{f \\in F(x)} \\rho_f(x) v_f(x)",[28,16331,16333,16360],{"className":16332,"ariaHidden":66},[101],[28,16334,16336,16339,16342,16345,16348,16351,16354,16357],{"className":16335},[105],[28,16337],{"className":16338,"style":134},[109],[28,16340,16271],{"className":16341},[114],[28,16343,378],{"className":16344},[138],[28,16346,2430],{"className":16347},[114,222],[28,16349,387],{"className":16350},[187],[28,16352],{"className":16353,"style":120},[119],[28,16355,55],{"className":16356},[124],[28,16358],{"className":16359,"style":120},[119],[28,16361,16363,16367,16438,16441,16481,16484,16487,16490,16531,16534,16537],{"className":16362},[105],[28,16364],{"className":16365,"style":16366},[109],"height:2.566em;vertical-align:-1.516em;",[28,16368,16372],{"className":16369},[16370,16371],"mop","op-limits",[28,16373,16375,16429],{"className":16374},[427,428],[28,16376,16378,16426],{"className":16377},[432],[28,16379,16382,16413],{"className":16380,"style":16381},[436],"height:1.05em;",[28,16383,16385,16389],{"style":16384},"top:-1.809em;margin-left:0em;",[28,16386],{"className":16387,"style":16388},[444],"height:3.05em;",[28,16390,16392],{"className":16391},[449,450,451,452],[28,16393,16395,16398,16401,16404,16407,16410],{"className":16394},[114,452],[28,16396,269],{"className":16397,"style":296},[114,222,452],[28,16399,748],{"className":16400},[124,452],[28,16402,16294],{"className":16403,"style":12024},[114,222,452],[28,16405,378],{"className":16406},[138,452],[28,16408,2430],{"className":16409},[114,222,452],[28,16411,387],{"className":16412},[187,452],[28,16414,16416,16419],{"style":16415},"top:-3.05em;",[28,16417],{"className":16418,"style":16388},[444],[28,16420,16421],{},[28,16422,16285],{"className":16423},[16370,16424,16425],"op-symbol","large-op",[28,16427,460],{"className":16428},[459],[28,16430,16432],{"className":16431},[432],[28,16433,16436],{"className":16434,"style":16435},[436],"height:1.516em;",[28,16437],{},[28,16439],{"className":16440,"style":149},[119],[28,16442,16444,16447],{"className":16443},[114],[28,16445,16305],{"className":16446},[114,222],[28,16448,16450],{"className":16449},[423],[28,16451,16453,16473],{"className":16452},[427,428],[28,16454,16456,16470],{"className":16455},[432],[28,16457,16459],{"className":16458,"style":5069},[436],[28,16460,16461,16464],{"style":440},[28,16462],{"className":16463,"style":445},[444],[28,16465,16467],{"className":16466},[449,450,451,452],[28,16468,269],{"className":16469,"style":296},[114,222,452],[28,16471,460],{"className":16472},[459],[28,16474,16476],{"className":16475},[432],[28,16477,16479],{"className":16478,"style":3926},[436],[28,16480],{},[28,16482,378],{"className":16483},[138],[28,16485,2430],{"className":16486},[114,222],[28,16488,387],{"className":16489},[187],[28,16491,16493,16496],{"className":16492},[114],[28,16494,16318],{"className":16495,"style":2123},[114,222],[28,16497,16499],{"className":16498},[423],[28,16500,16502,16523],{"className":16501},[427,428],[28,16503,16505,16520],{"className":16504},[432],[28,16506,16508],{"className":16507,"style":5069},[436],[28,16509,16511,16514],{"style":16510},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[28,16512],{"className":16513,"style":445},[444],[28,16515,16517],{"className":16516},[449,450,451,452],[28,16518,269],{"className":16519,"style":296},[114,222,452],[28,16521,460],{"className":16522},[459],[28,16524,16526],{"className":16525},[432],[28,16527,16529],{"className":16528,"style":3926},[436],[28,16530],{},[28,16532,378],{"className":16533},[138],[28,16535,2430],{"className":16536},[114,222],[28,16538,387],{"className":16539},[187],[11,16541,3844,16542,16628,16629,16702,16703,16731,16732,16816,16817,16886],{},[28,16543,16545,16569],{"className":16544,"translate":32},[31],[28,16546,16548],{"className":16547},[36],[38,16549,16550],{"xmlns":40},[42,16551,16552,16566],{},[45,16553,16554,16560,16562,16564],{},[354,16555,16556,16558],{},[48,16557,16318],{},[48,16559,269],{},[53,16561,378],{"stretchy":58},[48,16563,2430],{},[53,16565,387],{"stretchy":58},[94,16567,16568],{"encoding":96},"v_f(x)",[28,16570,16572],{"className":16571,"ariaHidden":66},[101],[28,16573,16575,16579,16619,16622,16625],{"className":16574},[105],[28,16576],{"className":16577,"style":16578},[109],"height:1.0361em;vertical-align:-0.2861em;",[28,16580,16582,16585],{"className":16581},[114],[28,16583,16318],{"className":16584,"style":2123},[114,222],[28,16586,16588],{"className":16587},[423],[28,16589,16591,16611],{"className":16590},[427,428],[28,16592,16594,16608],{"className":16593},[432],[28,16595,16597],{"className":16596,"style":5069},[436],[28,16598,16599,16602],{"style":16510},[28,16600],{"className":16601,"style":445},[444],[28,16603,16605],{"className":16604},[449,450,451,452],[28,16606,269],{"className":16607,"style":296},[114,222,452],[28,16609,460],{"className":16610},[459],[28,16612,16614],{"className":16613},[432],[28,16615,16617],{"className":16616,"style":3926},[436],[28,16618],{},[28,16620,378],{"className":16621},[138],[28,16623,2430],{"className":16624},[114,222],[28,16626,387],{"className":16627},[187]," 不再是固定方向，而是可以在某个子空间 ",[28,16630,16632,16651],{"className":16631,"translate":32},[31],[28,16633,16635],{"className":16634},[36],[38,16636,16637],{"xmlns":40},[42,16638,16639,16648],{},[45,16640,16641],{},[354,16642,16643,16646],{},[48,16644,16645],{},"V",[48,16647,269],{},[94,16649,16650],{"encoding":96},"V_f",[28,16652,16654],{"className":16653,"ariaHidden":66},[101],[28,16655,16657,16661],{"className":16656},[105],[28,16658],{"className":16659,"style":16660},[109],"height:0.9694em;vertical-align:-0.2861em;",[28,16662,16664,16667],{"className":16663},[114],[28,16665,16645],{"className":16666,"style":1741},[114,222],[28,16668,16670],{"className":16669},[423],[28,16671,16673,16694],{"className":16672},[427,428],[28,16674,16676,16691],{"className":16675},[432],[28,16677,16679],{"className":16678,"style":5069},[436],[28,16680,16682,16685],{"style":16681},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[28,16683],{"className":16684,"style":445},[444],[28,16686,16688],{"className":16687},[449,450,451,452],[28,16689,269],{"className":16690,"style":296},[114,222,452],[28,16692,460],{"className":16693},[459],[28,16695,16697],{"className":16696},[432],[28,16698,16700],{"className":16699,"style":3926},[436],[28,16701],{}," 内随输入 ",[28,16704,16706,16719],{"className":16705,"translate":32},[31],[28,16707,16709],{"className":16708},[36],[38,16710,16711],{"xmlns":40},[42,16712,16713,16717],{},[45,16714,16715],{},[48,16716,2430],{},[94,16718,2430],{"encoding":96},[28,16720,16722],{"className":16721,"ariaHidden":66},[101],[28,16723,16725,16728],{"className":16724},[105],[28,16726],{"className":16727,"style":5994},[109],[28,16729,2430],{"className":16730},[114,222]," 连续变化。标准的 LRH 只是 ",[28,16733,16735,16758],{"className":16734,"translate":32},[31],[28,16736,16738],{"className":16737},[36],[38,16739,16740],{"xmlns":40},[42,16741,16742,16756],{},[45,16743,16744,16750,16752,16754],{},[354,16745,16746,16748],{},[48,16747,16318],{},[48,16749,269],{},[53,16751,378],{"stretchy":58},[48,16753,2430],{},[53,16755,387],{"stretchy":58},[94,16757,16568],{"encoding":96},[28,16759,16761],{"className":16760,"ariaHidden":66},[101],[28,16762,16764,16767,16807,16810,16813],{"className":16763},[105],[28,16765],{"className":16766,"style":16578},[109],[28,16768,16770,16773],{"className":16769},[114],[28,16771,16318],{"className":16772,"style":2123},[114,222],[28,16774,16776],{"className":16775},[423],[28,16777,16779,16799],{"className":16778},[427,428],[28,16780,16782,16796],{"className":16781},[432],[28,16783,16785],{"className":16784,"style":5069},[436],[28,16786,16787,16790],{"style":16510},[28,16788],{"className":16789,"style":445},[444],[28,16791,16793],{"className":16792},[449,450,451,452],[28,16794,269],{"className":16795,"style":296},[114,222,452],[28,16797,460],{"className":16798},[459],[28,16800,16802],{"className":16801},[432],[28,16803,16805],{"className":16804,"style":3926},[436],[28,16806],{},[28,16808,378],{"className":16809},[138],[28,16811,2430],{"className":16812},[114,222],[28,16814,387],{"className":16815},[187]," 恒定且 ",[28,16818,16820,16837],{"className":16819,"translate":32},[31],[28,16821,16823],{"className":16822},[36],[38,16824,16825],{"xmlns":40},[42,16826,16827,16835],{},[45,16828,16829],{},[354,16830,16831,16833],{},[48,16832,16645],{},[48,16834,269],{},[94,16836,16650],{"encoding":96},[28,16838,16840],{"className":16839,"ariaHidden":66},[101],[28,16841,16843,16846],{"className":16842},[105],[28,16844],{"className":16845,"style":16660},[109],[28,16847,16849,16852],{"className":16848},[114],[28,16850,16645],{"className":16851,"style":1741},[114,222],[28,16853,16855],{"className":16854},[423],[28,16856,16858,16878],{"className":16857},[427,428],[28,16859,16861,16875],{"className":16860},[432],[28,16862,16864],{"className":16863,"style":5069},[436],[28,16865,16866,16869],{"style":16681},[28,16867],{"className":16868,"style":445},[444],[28,16870,16872],{"className":16871},[449,450,451,452],[28,16873,269],{"className":16874,"style":296},[114,222,452],[28,16876,460],{"className":16877},[459],[28,16879,16881],{"className":16880},[432],[28,16882,16884],{"className":16883,"style":3926},[436],[28,16885],{}," 一维的特例。",[11,16888,16889,16890,16918,16919,16988],{},"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 ",[28,16891,16893,16906],{"className":16892,"translate":32},[31],[28,16894,16896],{"className":16895},[36],[38,16897,16898],{"xmlns":40},[42,16899,16900,16904],{},[45,16901,16902],{},[48,16903,269],{},[94,16905,269],{"encoding":96},[28,16907,16909],{"className":16908,"ariaHidden":66},[101],[28,16910,16912,16915],{"className":16911},[105],[28,16913],{"className":16914,"style":292},[109],[28,16916,269],{"className":16917,"style":296},[114,222]," 在子空间 ",[28,16920,16922,16939],{"className":16921,"translate":32},[31],[28,16923,16925],{"className":16924},[36],[38,16926,16927],{"xmlns":40},[42,16928,16929,16937],{},[45,16930,16931],{},[354,16932,16933,16935],{},[48,16934,16645],{},[48,16936,269],{},[94,16938,16650],{"encoding":96},[28,16940,16942],{"className":16941,"ariaHidden":66},[101],[28,16943,16945,16948],{"className":16944},[105],[28,16946],{"className":16947,"style":16660},[109],[28,16949,16951,16954],{"className":16950},[114],[28,16952,16645],{"className":16953,"style":1741},[114,222],[28,16955,16957],{"className":16956},[423],[28,16958,16960,16980],{"className":16959},[427,428],[28,16961,16963,16977],{"className":16962},[432],[28,16964,16966],{"className":16965,"style":5069},[436],[28,16967,16968,16971],{"style":16681},[28,16969],{"className":16970,"style":445},[444],[28,16972,16974],{"className":16973},[449,450,451,452],[28,16975,269],{"className":16976,"style":296},[114,222,452],[28,16978,460],{"className":16979},[459],[28,16981,16983],{"className":16982},[432],[28,16984,16986],{"className":16985,"style":3926},[436],[28,16987],{}," 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？",[18,16990,16992],{"id":16991},"特征即度量空间","「特征」即度量空间",[11,16994,16995],{},"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？",[11,16997,16998,16999,17135],{},"作者给出的答案出人意料地简洁：特征是一个度量空间 ",[28,17000,17002,17032],{"className":17001,"translate":32},[31],[28,17003,17005],{"className":17004},[36],[38,17006,17007],{"xmlns":40},[42,17008,17009,17029],{},[45,17010,17011,17013,17019,17021,17027],{},[53,17012,378],{"stretchy":58},[354,17014,17015,17017],{},[48,17016,1387],{},[48,17018,269],{},[53,17020,67],{"separator":66},[354,17022,17023,17025],{},[48,17024,2450],{},[48,17026,269],{},[53,17028,387],{"stretchy":58},[94,17030,17031],{"encoding":96},"(Z_f, d_f)",[28,17033,17035],{"className":17034,"ariaHidden":66},[101],[28,17036,17038,17041,17044,17086,17089,17092,17132],{"className":17037},[105],[28,17039],{"className":17040,"style":16578},[109],[28,17042,378],{"className":17043},[138],[28,17045,17047,17051],{"className":17046},[114],[28,17048,1387],{"className":17049,"style":17050},[114,222],"margin-right:0.0715em;",[28,17052,17054],{"className":17053},[423],[28,17055,17057,17078],{"className":17056},[427,428],[28,17058,17060,17075],{"className":17059},[432],[28,17061,17063],{"className":17062,"style":5069},[436],[28,17064,17066,17069],{"style":17065},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[28,17067],{"className":17068,"style":445},[444],[28,17070,17072],{"className":17071},[449,450,451,452],[28,17073,269],{"className":17074,"style":296},[114,222,452],[28,17076,460],{"className":17077},[459],[28,17079,17081],{"className":17080},[432],[28,17082,17084],{"className":17083,"style":3926},[436],[28,17085],{},[28,17087,67],{"className":17088},[145],[28,17090],{"className":17091,"style":149},[119],[28,17093,17095,17098],{"className":17094},[114],[28,17096,2450],{"className":17097},[114,222],[28,17099,17101],{"className":17100},[423],[28,17102,17104,17124],{"className":17103},[427,428],[28,17105,17107,17121],{"className":17106},[432],[28,17108,17110],{"className":17109,"style":5069},[436],[28,17111,17112,17115],{"style":440},[28,17113],{"className":17114,"style":445},[444],[28,17116,17118],{"className":17117},[449,450,451,452],[28,17119,269],{"className":17120,"style":296},[114,222,452],[28,17122,460],{"className":17123},[459],[28,17125,17127],{"className":17126},[432],[28,17128,17130],{"className":17129,"style":3926},[436],[28,17131],{},[28,17133,387],{"className":17134},[187],"——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：",[335,17137,17138,17212,17356],{},[338,17139,17140,17141,17211],{},"原子特征（是否有猫）：",[28,17142,17144,17162],{"className":17143,"translate":32},[31],[28,17145,17147],{"className":17146},[36],[38,17148,17149],{"xmlns":40},[42,17150,17151,17159],{},[45,17152,17153],{},[354,17154,17155,17157],{},[48,17156,1387],{},[48,17158,269],{},[94,17160,17161],{"encoding":96},"Z_f",[28,17163,17165],{"className":17164,"ariaHidden":66},[101],[28,17166,17168,17171],{"className":17167},[105],[28,17169],{"className":17170,"style":16660},[109],[28,17172,17174,17177],{"className":17173},[114],[28,17175,1387],{"className":17176,"style":17050},[114,222],[28,17178,17180],{"className":17179},[423],[28,17181,17183,17203],{"className":17182},[427,428],[28,17184,17186,17200],{"className":17185},[432],[28,17187,17189],{"className":17188,"style":5069},[436],[28,17190,17191,17194],{"style":17065},[28,17192],{"className":17193,"style":445},[444],[28,17195,17197],{"className":17196},[449,450,451,452],[28,17198,269],{"className":17199,"style":296},[114,222,452],[28,17201,460],{"className":17202},[459],[28,17204,17206],{"className":17205},[432],[28,17207,17209],{"className":17208,"style":3926},[436],[28,17210],{}," 是单元素集合；",[338,17213,17214,17215,17284,17285,17355],{},"层次特征（动物的分类学树）：",[28,17216,17218,17235],{"className":17217,"translate":32},[31],[28,17219,17221],{"className":17220},[36],[38,17222,17223],{"xmlns":40},[42,17224,17225,17233],{},[45,17226,17227],{},[354,17228,17229,17231],{},[48,17230,1387],{},[48,17232,269],{},[94,17234,17161],{"encoding":96},[28,17236,17238],{"className":17237,"ariaHidden":66},[101],[28,17239,17241,17244],{"className":17240},[105],[28,17242],{"className":17243,"style":16660},[109],[28,17245,17247,17250],{"className":17246},[114],[28,17248,1387],{"className":17249,"style":17050},[114,222],[28,17251,17253],{"className":17252},[423],[28,17254,17256,17276],{"className":17255},[427,428],[28,17257,17259,17273],{"className":17258},[432],[28,17260,17262],{"className":17261,"style":5069},[436],[28,17263,17264,17267],{"style":17065},[28,17265],{"className":17266,"style":445},[444],[28,17268,17270],{"className":17269},[449,450,451,452],[28,17271,269],{"className":17272,"style":296},[114,222,452],[28,17274,460],{"className":17275},[459],[28,17277,17279],{"className":17278},[432],[28,17280,17282],{"className":17281,"style":3926},[436],[28,17283],{}," 是离散集合，",[28,17286,17288,17306],{"className":17287,"translate":32},[31],[28,17289,17291],{"className":17290},[36],[38,17292,17293],{"xmlns":40},[42,17294,17295,17303],{},[45,17296,17297],{},[354,17298,17299,17301],{},[48,17300,2450],{},[48,17302,269],{},[94,17304,17305],{"encoding":96},"d_f",[28,17307,17309],{"className":17308,"ariaHidden":66},[101],[28,17310,17312,17315],{"className":17311},[105],[28,17313],{"className":17314,"style":3881},[109],[28,17316,17318,17321],{"className":17317},[114],[28,17319,2450],{"className":17320},[114,222],[28,17322,17324],{"className":17323},[423],[28,17325,17327,17347],{"className":17326},[427,428],[28,17328,17330,17344],{"className":17329},[432],[28,17331,17333],{"className":17332,"style":5069},[436],[28,17334,17335,17338],{"style":440},[28,17336],{"className":17337,"style":445},[444],[28,17339,17341],{"className":17340},[449,450,451,452],[28,17342,269],{"className":17343,"style":296},[114,222,452],[28,17345,460],{"className":17346},[459],[28,17348,17350],{"className":17349},[432],[28,17351,17353],{"className":17352,"style":3926},[436],[28,17354],{}," 是其上的树距离；",[338,17357,17358,17359,17428],{},"连续特征（颜色色相、一年中的第几天、日历年）：",[28,17360,17362,17379],{"className":17361,"translate":32},[31],[28,17363,17365],{"className":17364},[36],[38,17366,17367],{"xmlns":40},[42,17368,17369,17377],{},[45,17370,17371],{},[354,17372,17373,17375],{},[48,17374,1387],{},[48,17376,269],{},[94,17378,17161],{"encoding":96},[28,17380,17382],{"className":17381,"ariaHidden":66},[101],[28,17383,17385,17388],{"className":17384},[105],[28,17386],{"className":17387,"style":16660},[109],[28,17389,17391,17394],{"className":17390},[114],[28,17392,1387],{"className":17393,"style":17050},[114,222],[28,17395,17397],{"className":17396},[423],[28,17398,17400,17420],{"className":17399},[427,428],[28,17401,17403,17417],{"className":17402},[432],[28,17404,17406],{"className":17405,"style":5069},[436],[28,17407,17408,17411],{"style":17065},[28,17409],{"className":17410,"style":445},[444],[28,17412,17414],{"className":17413},[449,450,451,452],[28,17415,269],{"className":17416,"style":296},[114,222,452],[28,17418,460],{"className":17419},[459],[28,17421,17423],{"className":17422},[432],[28,17424,17426],{"className":17425,"style":3926},[436],[28,17427],{}," 可以是区间、圆或更高维的欧几里得空间。",[11,17430,17431],{},"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。",[11,17433,17434],{},"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：",[2132,17436,17437],{},[11,17438,17439,17440,17528,17529,17613,17614,17817,17818,1369],{},"假说 1（连续对应假说）。特征值 ",[28,17441,17443,17468],{"className":17442,"translate":32},[31],[28,17444,17446],{"className":17445},[36],[38,17447,17448],{"xmlns":40},[42,17449,17450,17465],{},[45,17451,17452,17459,17461,17463],{},[354,17453,17454,17457],{},[48,17455,17456],{},"z",[48,17458,269],{},[53,17460,378],{"stretchy":58},[48,17462,2430],{},[53,17464,387],{"stretchy":58},[94,17466,17467],{"encoding":96},"z_f(x)",[28,17469,17471],{"className":17470,"ariaHidden":66},[101],[28,17472,17474,17477,17519,17522,17525],{"className":17473},[105],[28,17475],{"className":17476,"style":16578},[109],[28,17478,17480,17484],{"className":17479},[114],[28,17481,17456],{"className":17482,"style":17483},[114,222],"margin-right:0.044em;",[28,17485,17487],{"className":17486},[423],[28,17488,17490,17511],{"className":17489},[427,428],[28,17491,17493,17508],{"className":17492},[432],[28,17494,17496],{"className":17495,"style":5069},[436],[28,17497,17499,17502],{"style":17498},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[28,17500],{"className":17501,"style":445},[444],[28,17503,17505],{"className":17504},[449,450,451,452],[28,17506,269],{"className":17507,"style":296},[114,222,452],[28,17509,460],{"className":17510},[459],[28,17512,17514],{"className":17513},[432],[28,17515,17517],{"className":17516,"style":3926},[436],[28,17518],{},[28,17520,378],{"className":17521},[138],[28,17523,2430],{"className":17524},[114,222],[28,17526,387],{"className":17527},[187]," 与表示方向 ",[28,17530,17532,17555],{"className":17531,"translate":32},[31],[28,17533,17535],{"className":17534},[36],[38,17536,17537],{"xmlns":40},[42,17538,17539,17553],{},[45,17540,17541,17547,17549,17551],{},[354,17542,17543,17545],{},[48,17544,16318],{},[48,17546,269],{},[53,17548,378],{"stretchy":58},[48,17550,2430],{},[53,17552,387],{"stretchy":58},[94,17554,16568],{"encoding":96},[28,17556,17558],{"className":17557,"ariaHidden":66},[101],[28,17559,17561,17564,17604,17607,17610],{"className":17560},[105],[28,17562],{"className":17563,"style":16578},[109],[28,17565,17567,17570],{"className":17566},[114],[28,17568,16318],{"className":17569,"style":2123},[114,222],[28,17571,17573],{"className":17572},[423],[28,17574,17576,17596],{"className":17575},[427,428],[28,17577,17579,17593],{"className":17578},[432],[28,17580,17582],{"className":17581,"style":5069},[436],[28,17583,17584,17587],{"style":16510},[28,17585],{"className":17586,"style":445},[444],[28,17588,17590],{"className":17589},[449,450,451,452],[28,17591,269],{"className":17592,"style":296},[114,222,452],[28,17594,460],{"className":17595},[459],[28,17597,17599],{"className":17598},[432],[28,17600,17602],{"className":17601,"style":3926},[436],[28,17603],{},[28,17605,378],{"className":17606},[138],[28,17608,2430],{"className":17609},[114,222],[28,17611,387],{"className":17612},[187]," 之间存在连续、可逆的一一对应；即存在连续映射 ",[28,17615,17617,17658],{"className":17616,"translate":32},[31],[28,17618,17620],{"className":17619},[36],[38,17621,17622],{"xmlns":40},[42,17623,17624,17655],{},[45,17625,17626,17632,17634,17640,17642],{},[354,17627,17628,17630],{},[48,17629,15050],{},[48,17631,269],{},[53,17633,272],{},[354,17635,17636,17638],{},[48,17637,1387],{},[48,17639,269],{},[53,17641,277],{},[11323,17643,17644,17647],{},[48,17645,17646],{},"S",[45,17648,17649,17651,17653],{},[48,17650,8862],{},[53,17652,1398],{},[61,17654,70],{},[94,17656,17657],{"encoding":96},"\\phi_f: Z_f \\to S^{D-1}",[28,17659,17661,17716,17771],{"className":17660,"ariaHidden":66},[101],[28,17662,17664,17667,17707,17710,17713],{"className":17663},[105],[28,17665],{"className":17666,"style":3881},[109],[28,17668,17670,17673],{"className":17669},[114],[28,17671,15050],{"className":17672},[114,222],[28,17674,17676],{"className":17675},[423],[28,17677,17679,17699],{"className":17678},[427,428],[28,17680,17682,17696],{"className":17681},[432],[28,17683,17685],{"className":17684,"style":5069},[436],[28,17686,17687,17690],{"style":440},[28,17688],{"className":17689,"style":445},[444],[28,17691,17693],{"className":17692},[449,450,451,452],[28,17694,269],{"className":17695,"style":296},[114,222,452],[28,17697,460],{"className":17698},[459],[28,17700,17702],{"className":17701},[432],[28,17703,17705],{"className":17704,"style":3926},[436],[28,17706],{},[28,17708],{"className":17709,"style":120},[119],[28,17711,272],{"className":17712},[124],[28,17714],{"className":17715,"style":120},[119],[28,17717,17719,17722,17762,17765,17768],{"className":17718},[105],[28,17720],{"className":17721,"style":16660},[109],[28,17723,17725,17728],{"className":17724},[114],[28,17726,1387],{"className":17727,"style":17050},[114,222],[28,17729,17731],{"className":17730},[423],[28,17732,17734,17754],{"className":17733},[427,428],[28,17735,17737,17751],{"className":17736},[432],[28,17738,17740],{"className":17739,"style":5069},[436],[28,17741,17742,17745],{"style":17065},[28,17743],{"className":17744,"style":445},[444],[28,17746,17748],{"className":17747},[449,450,451,452],[28,17749,269],{"className":17750,"style":296},[114,222,452],[28,17752,460],{"className":17753},[459],[28,17755,17757],{"className":17756},[432],[28,17758,17760],{"className":17759,"style":3926},[436],[28,17761],{},[28,17763],{"className":17764,"style":120},[119],[28,17766,277],{"className":17767},[124],[28,17769],{"className":17770,"style":120},[119],[28,17772,17774,17778],{"className":17773},[105],[28,17775],{"className":17776,"style":17777},[109],"height:0.8413em;",[28,17779,17781,17785],{"className":17780},[114],[28,17782,17646],{"className":17783,"style":17784},[114,222],"margin-right:0.0576em;",[28,17786,17788],{"className":17787},[423],[28,17789,17791],{"className":17790},[427],[28,17792,17794],{"className":17793},[432],[28,17795,17797],{"className":17796,"style":17777},[436],[28,17798,17799,17802],{"style":11378},[28,17800],{"className":17801,"style":445},[444],[28,17803,17805],{"className":17804},[449,450,451,452],[28,17806,17808,17811,17814],{"className":17807},[114,452],[28,17809,8862],{"className":17810,"style":8909},[114,222,452],[28,17812,1398],{"className":17813},[1745,452],[28,17815,70],{"className":17816},[114,452],"（映到单位超球面），使得 ",[28,17819,17821,17869],{"className":17820,"translate":32},[31],[28,17822,17824],{"className":17823},[36],[38,17825,17826],{"xmlns":40},[42,17827,17828,17866],{},[45,17829,17830,17836,17838,17840,17842,17844,17850,17852,17858,17860,17862,17864],{},[354,17831,17832,17834],{},[48,17833,16318],{},[48,17835,269],{},[53,17837,378],{"stretchy":58},[48,17839,2430],{},[53,17841,387],{"stretchy":58},[53,17843,55],{},[354,17845,17846,17848],{},[48,17847,15050],{},[48,17849,269],{},[53,17851,378],{"stretchy":58},[354,17853,17854,17856],{},[48,17855,17456],{},[48,17857,269],{},[53,17859,378],{"stretchy":58},[48,17861,2430],{},[53,17863,387],{"stretchy":58},[53,17865,387],{"stretchy":58},[94,17867,17868],{"encoding":96},"v_f(x) = \\phi_f(z_f(x))",[28,17870,17872,17936],{"className":17871,"ariaHidden":66},[101],[28,17873,17875,17878,17918,17921,17924,17927,17930,17933],{"className":17874},[105],[28,17876],{"className":17877,"style":16578},[109],[28,17879,17881,17884],{"className":17880},[114],[28,17882,16318],{"className":17883,"style":2123},[114,222],[28,17885,17887],{"className":17886},[423],[28,17888,17890,17910],{"className":17889},[427,428],[28,17891,17893,17907],{"className":17892},[432],[28,17894,17896],{"className":17895,"style":5069},[436],[28,17897,17898,17901],{"style":16510},[28,17899],{"className":17900,"style":445},[444],[28,17902,17904],{"className":17903},[449,450,451,452],[28,17905,269],{"className":17906,"style":296},[114,222,452],[28,17908,460],{"className":17909},[459],[28,17911,17913],{"className":17912},[432],[28,17914,17916],{"className":17915,"style":3926},[436],[28,17917],{},[28,17919,378],{"className":17920},[138],[28,17922,2430],{"className":17923},[114,222],[28,17925,387],{"className":17926},[187],[28,17928],{"className":17929,"style":120},[119],[28,17931,55],{"className":17932},[124],[28,17934],{"className":17935,"style":120},[119],[28,17937,17939,17942,17982,17985,18025,18028,18031],{"className":17938},[105],[28,17940],{"className":17941,"style":16578},[109],[28,17943,17945,17948],{"className":17944},[114],[28,17946,15050],{"className":17947},[114,222],[28,17949,17951],{"className":17950},[423],[28,17952,17954,17974],{"className":17953},[427,428],[28,17955,17957,17971],{"className":17956},[432],[28,17958,17960],{"className":17959,"style":5069},[436],[28,17961,17962,17965],{"style":440},[28,17963],{"className":17964,"style":445},[444],[28,17966,17968],{"className":17967},[449,450,451,452],[28,17969,269],{"className":17970,"style":296},[114,222,452],[28,17972,460],{"className":17973},[459],[28,17975,17977],{"className":17976},[432],[28,17978,17980],{"className":17979,"style":3926},[436],[28,17981],{},[28,17983,378],{"className":17984},[138],[28,17986,17988,17991],{"className":17987},[114],[28,17989,17456],{"className":17990,"style":17483},[114,222],[28,17992,17994],{"className":17993},[423],[28,17995,17997,18017],{"className":17996},[427,428],[28,17998,18000,18014],{"className":17999},[432],[28,18001,18003],{"className":18002,"style":5069},[436],[28,18004,18005,18008],{"style":17498},[28,18006],{"className":18007,"style":445},[444],[28,18009,18011],{"className":18010},[449,450,451,452],[28,18012,269],{"className":18013,"style":296},[114,222,452],[28,18015,460],{"className":18016},[459],[28,18018,18020],{"className":18019},[432],[28,18021,18023],{"className":18022,"style":3926},[436],[28,18024],{},[28,18026,378],{"className":18027},[138],[28,18029,2430],{"className":18030},[114,222],[28,18032,10079],{"className":18033},[187],[11,18035,18036,18037,18106,18107,18177,18178,5404,18251,18320,18321,18390,18391,18460,18461,18530,18531,18600,18601,18670],{},"结合 ",[28,18038,18040,18057],{"className":18039,"translate":32},[31],[28,18041,18043],{"className":18042},[36],[38,18044,18045],{"xmlns":40},[42,18046,18047,18055],{},[45,18048,18049],{},[354,18050,18051,18053],{},[48,18052,1387],{},[48,18054,269],{},[94,18056,17161],{"encoding":96},[28,18058,18060],{"className":18059,"ariaHidden":66},[101],[28,18061,18063,18066],{"className":18062},[105],[28,18064],{"className":18065,"style":16660},[109],[28,18067,18069,18072],{"className":18068},[114],[28,18070,1387],{"className":18071,"style":17050},[114,222],[28,18073,18075],{"className":18074},[423],[28,18076,18078,18098],{"className":18077},[427,428],[28,18079,18081,18095],{"className":18080},[432],[28,18082,18084],{"className":18083,"style":5069},[436],[28,18085,18086,18089],{"style":17065},[28,18087],{"className":18088,"style":445},[444],[28,18090,18092],{"className":18091},[449,450,451,452],[28,18093,269],{"className":18094,"style":296},[114,222,452],[28,18096,460],{"className":18097},[459],[28,18099,18101],{"className":18100},[432],[28,18102,18104],{"className":18103,"style":3926},[436],[28,18105],{}," 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）：",[28,18108,18110,18128],{"className":18109,"translate":32},[31],[28,18111,18113],{"className":18112},[36],[38,18114,18115],{"xmlns":40},[42,18116,18117,18125],{},[45,18118,18119],{},[354,18120,18121,18123],{},[48,18122,15050],{},[48,18124,269],{},[94,18126,18127],{"encoding":96},"\\phi_f",[28,18129,18131],{"className":18130,"ariaHidden":66},[101],[28,18132,18134,18137],{"className":18133},[105],[28,18135],{"className":18136,"style":3881},[109],[28,18138,18140,18143],{"className":18139},[114],[28,18141,15050],{"className":18142},[114,222],[28,18144,18146],{"className":18145},[423],[28,18147,18149,18169],{"className":18148},[427,428],[28,18150,18152,18166],{"className":18151},[432],[28,18153,18155],{"className":18154,"style":5069},[436],[28,18156,18157,18160],{"style":440},[28,18158],{"className":18159,"style":445},[444],[28,18161,18163],{"className":18162},[449,450,451,452],[28,18164,269],{"className":18165,"style":296},[114,222,452],[28,18167,460],{"className":18168},[459],[28,18170,18172],{"className":18171},[432],[28,18173,18175],{"className":18174,"style":3926},[436],[28,18176],{}," 是一个同胚。换言之，表征流形 ",[28,18179,18181,18200],{"className":18180,"translate":32},[31],[28,18182,18184],{"className":18183},[36],[38,18185,18186],{"xmlns":40},[42,18187,18188,18197],{},[45,18189,18190],{},[354,18191,18192,18195],{},[48,18193,18194],{},"M",[48,18196,269],{},[94,18198,18199],{"encoding":96},"M_f",[28,18201,18203],{"className":18202,"ariaHidden":66},[101],[28,18204,18206,18209],{"className":18205},[105],[28,18207],{"className":18208,"style":16660},[109],[28,18210,18212,18216],{"className":18211},[114],[28,18213,18194],{"className":18214,"style":18215},[114,222],"margin-right:0.109em;",[28,18217,18219],{"className":18218},[423],[28,18220,18222,18243],{"className":18221},[427,428],[28,18223,18225,18240],{"className":18224},[432],[28,18226,18228],{"className":18227,"style":5069},[436],[28,18229,18231,18234],{"style":18230},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[28,18232],{"className":18233,"style":445},[444],[28,18235,18237],{"className":18236},[449,450,451,452],[28,18238,269],{"className":18239,"style":296},[114,222,452],[28,18241,460],{"className":18242},[459],[28,18244,18246],{"className":18245},[432],[28,18247,18249],{"className":18248,"style":3926},[436],[28,18250],{},[28,18252,18254,18271],{"className":18253,"translate":32},[31],[28,18255,18257],{"className":18256},[36],[38,18258,18259],{"xmlns":40},[42,18260,18261,18269],{},[45,18262,18263],{},[354,18264,18265,18267],{},[48,18266,15050],{},[48,18268,269],{},[94,18270,18127],{"encoding":96},[28,18272,18274],{"className":18273,"ariaHidden":66},[101],[28,18275,18277,18280],{"className":18276},[105],[28,18278],{"className":18279,"style":3881},[109],[28,18281,18283,18286],{"className":18282},[114],[28,18284,15050],{"className":18285},[114,222],[28,18287,18289],{"className":18288},[423],[28,18290,18292,18312],{"className":18291},[427,428],[28,18293,18295,18309],{"className":18294},[432],[28,18296,18298],{"className":18297,"style":5069},[436],[28,18299,18300,18303],{"style":440},[28,18301],{"className":18302,"style":445},[444],[28,18304,18306],{"className":18305},[449,450,451,452],[28,18307,269],{"className":18308,"style":296},[114,222,452],[28,18310,460],{"className":18311},[459],[28,18313,18315],{"className":18314},[432],[28,18316,18318],{"className":18317,"style":3926},[436],[28,18319],{}," 的像）与原始特征空间 ",[28,18322,18324,18341],{"className":18323,"translate":32},[31],[28,18325,18327],{"className":18326},[36],[38,18328,18329],{"xmlns":40},[42,18330,18331,18339],{},[45,18332,18333],{},[354,18334,18335,18337],{},[48,18336,1387],{},[48,18338,269],{},[94,18340,17161],{"encoding":96},[28,18342,18344],{"className":18343,"ariaHidden":66},[101],[28,18345,18347,18350],{"className":18346},[105],[28,18348],{"className":18349,"style":16660},[109],[28,18351,18353,18356],{"className":18352},[114],[28,18354,1387],{"className":18355,"style":17050},[114,222],[28,18357,18359],{"className":18358},[423],[28,18360,18362,18382],{"className":18361},[427,428],[28,18363,18365,18379],{"className":18364},[432],[28,18366,18368],{"className":18367,"style":5069},[436],[28,18369,18370,18373],{"style":17065},[28,18371],{"className":18372,"style":445},[444],[28,18374,18376],{"className":18375},[449,450,451,452],[28,18377,269],{"className":18378,"style":296},[114,222,452],[28,18380,460],{"className":18381},[459],[28,18383,18385],{"className":18384},[432],[28,18386,18388],{"className":18387,"style":3926},[436],[28,18389],{}," 在拓扑上「形状相同」：若 ",[28,18392,18394,18411],{"className":18393,"translate":32},[31],[28,18395,18397],{"className":18396},[36],[38,18398,18399],{"xmlns":40},[42,18400,18401,18409],{},[45,18402,18403],{},[354,18404,18405,18407],{},[48,18406,1387],{},[48,18408,269],{},[94,18410,17161],{"encoding":96},[28,18412,18414],{"className":18413,"ariaHidden":66},[101],[28,18415,18417,18420],{"className":18416},[105],[28,18418],{"className":18419,"style":16660},[109],[28,18421,18423,18426],{"className":18422},[114],[28,18424,1387],{"className":18425,"style":17050},[114,222],[28,18427,18429],{"className":18428},[423],[28,18430,18432,18452],{"className":18431},[427,428],[28,18433,18435,18449],{"className":18434},[432],[28,18436,18438],{"className":18437,"style":5069},[436],[28,18439,18440,18443],{"style":17065},[28,18441],{"className":18442,"style":445},[444],[28,18444,18446],{"className":18445},[449,450,451,452],[28,18447,269],{"className":18448,"style":296},[114,222,452],[28,18450,460],{"className":18451},[459],[28,18453,18455],{"className":18454},[432],[28,18456,18458],{"className":18457,"style":3926},[436],[28,18459],{}," 是区间，",[28,18462,18464,18481],{"className":18463,"translate":32},[31],[28,18465,18467],{"className":18466},[36],[38,18468,18469],{"xmlns":40},[42,18470,18471,18479],{},[45,18472,18473],{},[354,18474,18475,18477],{},[48,18476,18194],{},[48,18478,269],{},[94,18480,18199],{"encoding":96},[28,18482,18484],{"className":18483,"ariaHidden":66},[101],[28,18485,18487,18490],{"className":18486},[105],[28,18488],{"className":18489,"style":16660},[109],[28,18491,18493,18496],{"className":18492},[114],[28,18494,18194],{"className":18495,"style":18215},[114,222],[28,18497,18499],{"className":18498},[423],[28,18500,18502,18522],{"className":18501},[427,428],[28,18503,18505,18519],{"className":18504},[432],[28,18506,18508],{"className":18507,"style":5069},[436],[28,18509,18510,18513],{"style":18230},[28,18511],{"className":18512,"style":445},[444],[28,18514,18516],{"className":18515},[449,450,451,452],[28,18517,269],{"className":18518,"style":296},[114,222,452],[28,18520,460],{"className":18521},[459],[28,18523,18525],{"className":18524},[432],[28,18526,18528],{"className":18527,"style":3926},[436],[28,18529],{}," 就是一条曲线；若 ",[28,18532,18534,18551],{"className":18533,"translate":32},[31],[28,18535,18537],{"className":18536},[36],[38,18538,18539],{"xmlns":40},[42,18540,18541,18549],{},[45,18542,18543],{},[354,18544,18545,18547],{},[48,18546,1387],{},[48,18548,269],{},[94,18550,17161],{"encoding":96},[28,18552,18554],{"className":18553,"ariaHidden":66},[101],[28,18555,18557,18560],{"className":18556},[105],[28,18558],{"className":18559,"style":16660},[109],[28,18561,18563,18566],{"className":18562},[114],[28,18564,1387],{"className":18565,"style":17050},[114,222],[28,18567,18569],{"className":18568},[423],[28,18570,18572,18592],{"className":18571},[427,428],[28,18573,18575,18589],{"className":18574},[432],[28,18576,18578],{"className":18577,"style":5069},[436],[28,18579,18580,18583],{"style":17065},[28,18581],{"className":18582,"style":445},[444],[28,18584,18586],{"className":18585},[449,450,451,452],[28,18587,269],{"className":18588,"style":296},[114,222,452],[28,18590,460],{"className":18591},[459],[28,18593,18595],{"className":18594},[432],[28,18596,18598],{"className":18597,"style":3926},[436],[28,18599],{}," 是圆，",[28,18602,18604,18621],{"className":18603,"translate":32},[31],[28,18605,18607],{"className":18606},[36],[38,18608,18609],{"xmlns":40},[42,18610,18611,18619],{},[45,18612,18613],{},[354,18614,18615,18617],{},[48,18616,18194],{},[48,18618,269],{},[94,18620,18199],{"encoding":96},[28,18622,18624],{"className":18623,"ariaHidden":66},[101],[28,18625,18627,18630],{"className":18626},[105],[28,18628],{"className":18629,"style":16660},[109],[28,18631,18633,18636],{"className":18632},[114],[28,18634,18194],{"className":18635,"style":18215},[114,222],[28,18637,18639],{"className":18638},[423],[28,18640,18642,18662],{"className":18641},[427,428],[28,18643,18645,18659],{"className":18644},[432],[28,18646,18648],{"className":18647,"style":5069},[436],[28,18649,18650,18653],{"style":18230},[28,18651],{"className":18652,"style":445},[444],[28,18654,18656],{"className":18655},[449,450,451,452],[28,18657,269],{"className":18658,"style":296},[114,222,452],[28,18660,460],{"className":18661},[459],[28,18663,18665],{"className":18664},[432],[28,18666,18668],{"className":18667,"style":3926},[436],[28,18669],{}," 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。",[11,18672,18673],{},"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。",[18,18675,18676],{"id":18676},"实证验证",[11,18678,18679],{},"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：",[14911,18681,18682,18689,18692],{},[338,18683,18684,18685,18688],{},"颜色：用 OpenAI 的 ",[7752,18686,18687],{},"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；",[338,18690,18691],{},"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；",[338,18693,18694],{},"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。",[11,18696,18697],{},"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。",[11,18699,18700],{},"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。",[18,18702,18703],{"id":18703},"计算表达能力角度",[11,18705,18706],{},"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？",[11,18708,18709,18710,18738,18739,18814,18815,18939,18940,18968,18969,19029,19030,19186,19187,19231,19232,19293],{},"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」 ",[28,18711,18713,18726],{"className":18712,"translate":32},[31],[28,18714,18716],{"className":18715},[36],[38,18717,18718],{"xmlns":40},[42,18719,18720,18724],{},[45,18721,18722],{},[48,18723,17456],{},[94,18725,17456],{"encoding":96},[28,18727,18729],{"className":18728,"ariaHidden":66},[101],[28,18730,18732,18735],{"className":18731},[105],[28,18733],{"className":18734,"style":5994},[109],[28,18736,17456],{"className":18737,"style":17483},[114,222]," 本身（即让恒等函数 ",[28,18740,18742,18770],{"className":18741,"translate":32},[31],[28,18743,18745],{"className":18744},[36],[38,18746,18747],{"xmlns":40},[42,18748,18749,18767],{},[45,18750,18751,18757,18759,18761,18763,18765],{},[45,18752,18753,18755],{},[48,18754,3865],{"mathvariant":363},[48,18756,2450],{"mathvariant":363},[53,18758,378],{"stretchy":58},[48,18760,17456],{},[53,18762,387],{"stretchy":58},[53,18764,55],{},[48,18766,17456],{},[94,18768,18769],{"encoding":96},"\\mathrm{id}(z) = z",[28,18771,18773,18805],{"className":18772,"ariaHidden":66},[101],[28,18774,18776,18779,18787,18790,18793,18796,18799,18802],{"className":18775},[105],[28,18777],{"className":18778,"style":134},[109],[28,18780,18782],{"className":18781},[114],[28,18783,18786],{"className":18784},[114,18785],"mathrm","id",[28,18788,378],{"className":18789},[138],[28,18791,17456],{"className":18792,"style":17483},[114,222],[28,18794,387],{"className":18795},[187],[28,18797],{"className":18798,"style":120},[119],[28,18800,55],{"className":18801},[124],[28,18803],{"className":18804,"style":120},[119],[28,18806,18808,18811],{"className":18807},[105],[28,18809],{"className":18810,"style":5994},[109],[28,18812,17456],{"className":18813,"style":17483},[114,222]," 能通过一次线性运算算出），那么两个正交方向 ",[28,18816,18818,18844],{"className":18817,"translate":32},[31],[28,18819,18821],{"className":18820},[36],[38,18822,18823],{"xmlns":40},[42,18824,18825,18841],{},[45,18826,18827,18833,18835],{},[354,18828,18829,18831],{},[48,18830,16318],{},[61,18832,63],{},[53,18834,67],{"separator":66},[354,18836,18837,18839],{},[48,18838,16318],{},[61,18840,70],{},[94,18842,18843],{"encoding":96},"v_0, v_1",[28,18845,18847],{"className":18846,"ariaHidden":66},[101],[28,18848,18850,18853,18893,18896,18899],{"className":18849},[105],[28,18851],{"className":18852,"style":4110},[109],[28,18854,18856,18859],{"className":18855},[114],[28,18857,16318],{"className":18858,"style":2123},[114,222],[28,18860,18862],{"className":18861},[423],[28,18863,18865,18885],{"className":18864},[427,428],[28,18866,18868,18882],{"className":18867},[432],[28,18869,18871],{"className":18870,"style":437},[436],[28,18872,18873,18876],{"style":16510},[28,18874],{"className":18875,"style":445},[444],[28,18877,18879],{"className":18878},[449,450,451,452],[28,18880,63],{"className":18881},[114,452],[28,18883,460],{"className":18884},[459],[28,18886,18888],{"className":18887},[432],[28,18889,18891],{"className":18890,"style":467},[436],[28,18892],{},[28,18894,67],{"className":18895},[145],[28,18897],{"className":18898,"style":149},[119],[28,18900,18902,18905],{"className":18901},[114],[28,18903,16318],{"className":18904,"style":2123},[114,222],[28,18906,18908],{"className":18907},[423],[28,18909,18911,18931],{"className":18910},[427,428],[28,18912,18914,18928],{"className":18913},[432],[28,18915,18917],{"className":18916,"style":437},[436],[28,18918,18919,18922],{"style":16510},[28,18920],{"className":18921,"style":445},[444],[28,18923,18925],{"className":18924},[449,450,451,452],[28,18926,70],{"className":18927},[114,452],[28,18929,460],{"className":18930},[459],[28,18932,18934],{"className":18933},[432],[28,18935,18937],{"className":18936,"style":467},[436],[28,18938],{}," 就足够了。但如果还希望让 ",[28,18941,18943,18956],{"className":18942,"translate":32},[31],[28,18944,18946],{"className":18945},[36],[38,18947,18948],{"xmlns":40},[42,18949,18950,18954],{},[45,18951,18952],{},[48,18953,17456],{},[94,18955,17456],{"encoding":96},[28,18957,18959],{"className":18958,"ariaHidden":66},[101],[28,18960,18962,18965],{"className":18961},[105],[28,18963],{"className":18964,"style":5994},[109],[28,18966,17456],{"className":18967,"style":17483},[114,222]," 的高阶多项式（如 ",[28,18970,18972,18990],{"className":18971,"translate":32},[31],[28,18973,18975],{"className":18974},[36],[38,18976,18977],{"xmlns":40},[42,18978,18979,18987],{},[45,18980,18981],{},[11323,18982,18983,18985],{},[48,18984,17456],{},[61,18986,75],{},[94,18988,18989],{"encoding":96},"z^2",[28,18991,18993],{"className":18992,"ariaHidden":66},[101],[28,18994,18996,19000],{"className":18995},[105],[28,18997],{"className":18998,"style":18999},[109],"height:0.8141em;",[28,19001,19003,19006],{"className":19002},[114],[28,19004,17456],{"className":19005,"style":17483},[114,222],[28,19007,19009],{"className":19008},[423],[28,19010,19012],{"className":19011},[427],[28,19013,19015],{"className":19014},[432],[28,19016,19018],{"className":19017,"style":18999},[436],[28,19019,19020,19023],{"style":11378},[28,19021],{"className":19022,"style":445},[444],[28,19024,19026],{"className":19025},[449,450,451,452],[28,19027,75],{"className":19028},[114,452],"）也能被线性读出，就需要更多的正交方向 ",[28,19031,19033,19069],{"className":19032,"translate":32},[31],[28,19034,19036],{"className":19035},[36],[38,19037,19038],{"xmlns":40},[42,19039,19040,19066],{},[45,19041,19042,19048,19050,19052,19054],{},[354,19043,19044,19046],{},[48,19045,16318],{},[61,19047,63],{},[53,19049,67],{"separator":66},[53,19051,85],{},[53,19053,67],{"separator":66},[354,19055,19056,19058],{},[48,19057,16318],{},[45,19059,19060,19062,19064],{},[48,19061,11],{},[53,19063,1610],{},[61,19065,70],{},[94,19067,19068],{"encoding":96},"v_0, \\dots, v_{p+1}",[28,19070,19072],{"className":19071,"ariaHidden":66},[101],[28,19073,19075,19079,19119,19122,19125,19128,19131,19134,19137],{"className":19074},[105],[28,19076],{"className":19077,"style":19078},[109],"height:0.7167em;vertical-align:-0.2861em;",[28,19080,19082,19085],{"className":19081},[114],[28,19083,16318],{"className":19084,"style":2123},[114,222],[28,19086,19088],{"className":19087},[423],[28,19089,19091,19111],{"className":19090},[427,428],[28,19092,19094,19108],{"className":19093},[432],[28,19095,19097],{"className":19096,"style":437},[436],[28,19098,19099,19102],{"style":16510},[28,19100],{"className":19101,"style":445},[444],[28,19103,19105],{"className":19104},[449,450,451,452],[28,19106,63],{"className":19107},[114,452],[28,19109,460],{"className":19110},[459],[28,19112,19114],{"className":19113},[432],[28,19115,19117],{"className":19116,"style":467},[436],[28,19118],{},[28,19120,67],{"className":19121},[145],[28,19123],{"className":19124,"style":149},[119],[28,19126,85],{"className":19127},[180],[28,19129],{"className":19130,"style":149},[119],[28,19132,67],{"className":19133},[145],[28,19135],{"className":19136,"style":149},[119],[28,19138,19140,19143],{"className":19139},[114],[28,19141,16318],{"className":19142,"style":2123},[114,222],[28,19144,19146],{"className":19145},[423],[28,19147,19149,19178],{"className":19148},[427,428],[28,19150,19152,19175],{"className":19151},[432],[28,19153,19155],{"className":19154,"style":437},[436],[28,19156,19157,19160],{"style":16510},[28,19158],{"className":19159,"style":445},[444],[28,19161,19163],{"className":19162},[449,450,451,452],[28,19164,19166,19169,19172],{"className":19165},[114,452],[28,19167,11],{"className":19168},[114,222,452],[28,19170,1610],{"className":19171},[1745,452],[28,19173,70],{"className":19174},[114,452],[28,19176,460],{"className":19177},[459],[28,19179,19181],{"className":19180},[432],[28,19182,19184],{"className":19183,"style":3926},[436],[28,19185],{},"，相应地 ",[28,19188,19190,19210],{"className":19189,"translate":32},[31],[28,19191,19193],{"className":19192},[36],[38,19194,19195],{"xmlns":40},[42,19196,19197,19207],{},[45,19198,19199,19201,19203,19205],{},[48,19200,15050],{},[53,19202,378],{"stretchy":58},[48,19204,17456],{},[53,19206,387],{"stretchy":58},[94,19208,19209],{"encoding":96},"\\phi(z)",[28,19211,19213],{"className":19212,"ariaHidden":66},[101],[28,19214,19216,19219,19222,19225,19228],{"className":19215},[105],[28,19217],{"className":19218,"style":134},[109],[28,19220,15050],{"className":19221},[114,222],[28,19223,378],{"className":19224},[138],[28,19226,17456],{"className":19227,"style":17483},[114,222],[28,19229,387],{"className":19230},[187]," 描出的路径就必须在一个 ",[28,19233,19235,19257],{"className":19234,"translate":32},[31],[28,19236,19238],{"className":19237},[36],[38,19239,19240],{"xmlns":40},[42,19241,19242,19254],{},[45,19243,19244,19246,19248,19250,19252],{},[53,19245,378],{"stretchy":58},[48,19247,11],{},[53,19249,1610],{},[61,19251,75],{},[53,19253,387],{"stretchy":58},[94,19255,19256],{"encoding":96},"(p+2)",[28,19258,19260,19281],{"className":19259,"ariaHidden":66},[101],[28,19261,19263,19266,19269,19272,19275,19278],{"className":19262},[105],[28,19264],{"className":19265,"style":134},[109],[28,19267,378],{"className":19268},[138],[28,19270,11],{"className":19271},[114,222],[28,19273],{"className":19274,"style":1741},[119],[28,19276,1610],{"className":19277},[1745],[28,19279],{"className":19280,"style":1741},[119],[28,19282,19284,19287,19290],{"className":19283},[105],[28,19285],{"className":19286,"style":134},[109],[28,19288,75],{"className":19289},[114],[28,19291,387],{"className":19292},[187]," 维子空间中弯曲。",[11,19295,19296],{},"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。",[11,19298,19299,19300,19369],{},"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，",[28,19301,19303,19320],{"className":19302,"translate":32},[31],[28,19304,19306],{"className":19305},[36],[38,19307,19308],{"xmlns":40},[42,19309,19310,19318],{},[45,19311,19312],{},[354,19313,19314,19316],{},[48,19315,15050],{},[48,19317,269],{},[94,19319,18127],{"encoding":96},[28,19321,19323],{"className":19322,"ariaHidden":66},[101],[28,19324,19326,19329],{"className":19325},[105],[28,19327],{"className":19328,"style":3881},[109],[28,19330,19332,19335],{"className":19331},[114],[28,19333,15050],{"className":19334},[114,222],[28,19336,19338],{"className":19337},[423],[28,19339,19341,19361],{"className":19340},[427,428],[28,19342,19344,19358],{"className":19343},[432],[28,19345,19347],{"className":19346,"style":5069},[436],[28,19348,19349,19352],{"style":440},[28,19350],{"className":19351,"style":445},[444],[28,19353,19355],{"className":19354},[449,450,451,452],[28,19356,269],{"className":19357,"style":296},[114,222,452],[28,19359,460],{"className":19360},[459],[28,19362,19364],{"className":19363},[432],[28,19365,19367],{"className":19366,"style":3926},[436],[28,19368],{},"——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。",[18,19371,19372],{"id":19372},"余弦相似度",[11,19374,19375],{},"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。",[11,19377,19378],{},"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：",[28,19380,19382],{"className":19381,"translate":32},[2403],[28,19383,19385,19487],{"className":19384,"translate":32},[31],[28,19386,19388],{"className":19387},[36],[38,19389,19390],{"xmlns":40,"display":2412},[42,19391,19392,19484],{},[45,19393,19394,19412,19414,19420,19422,19424,19426,19428,19434,19436,19443,19445,19447,19449,19456,19458,19464,19466,19468,19470,19476,19482],{},[45,19395,19396,19399,19402,19405,19407,19409],{},[48,19397,19398],{"mathvariant":363},"C",[48,19400,19401],{"mathvariant":363},"o",[48,19403,19404],{"mathvariant":363},"s",[48,19406,17646],{"mathvariant":363},[48,19408,3865],{"mathvariant":363},[48,19410,19411],{"mathvariant":363},"m",[53,19413,378],{"stretchy":58},[354,19415,19416,19418],{},[48,19417,15050],{},[48,19419,269],{},[53,19421,378],{"stretchy":58},[48,19423,17456],{},[53,19425,387],{"stretchy":58},[53,19427,67],{"separator":66},[354,19429,19430,19432],{},[48,19431,15050],{},[48,19433,269],{},[53,19435,378],{"stretchy":58},[11323,19437,19438,19440],{},[48,19439,17456],{},[53,19441,19442],{"mathvariant":363,"lspace":2419,"rspace":2419},"′",[53,19444,387],{"stretchy":58},[53,19446,387],{"stretchy":58},[53,19448,55],{},[354,19450,19451,19454],{},[48,19452,19453],{},"g",[48,19455,269],{},[53,19457,378],{"stretchy":58},[354,19459,19460,19462],{},[48,19461,2450],{},[48,19463,269],{},[53,19465,378],{"stretchy":58},[48,19467,17456],{},[53,19469,67],{"separator":66},[11323,19471,19472,19474],{},[48,19473,17456],{},[53,19475,19442],{"mathvariant":363,"lspace":2419,"rspace":2419},[11323,19477,19478,19480],{},[53,19479,387],{"stretchy":58},[61,19481,75],{},[53,19483,387],{"stretchy":58},[94,19485,19486],{"encoding":96},"\\mathrm{CosSim}(\\phi_f(z), \\phi_f(z')) = g_f(d_f(z, z')^2)",[28,19488,19490,19651],{"className":19489,"ariaHidden":66},[101],[28,19491,19493,19497,19504,19507,19547,19550,19553,19556,19559,19562,19602,19605,19639,19642,19645,19648],{"className":19492},[105],[28,19494],{"className":19495,"style":19496},[109],"height:1.088em;vertical-align:-0.2861em;",[28,19498,19500],{"className":19499},[114],[28,19501,19503],{"className":19502},[114,18785],"CosSim",[28,19505,378],{"className":19506},[138],[28,19508,19510,19513],{"className":19509},[114],[28,19511,15050],{"className":19512},[114,222],[28,19514,19516],{"className":19515},[423],[28,19517,19519,19539],{"className":19518},[427,428],[28,19520,19522,19536],{"className":19521},[432],[28,19523,19525],{"className":19524,"style":5069},[436],[28,19526,19527,19530],{"style":440},[28,19528],{"className":19529,"style":445},[444],[28,19531,19533],{"className":19532},[449,450,451,452],[28,19534,269],{"className":19535,"style":296},[114,222,452],[28,19537,460],{"className":19538},[459],[28,19540,19542],{"className":19541},[432],[28,19543,19545],{"className":19544,"style":3926},[436],[28,19546],{},[28,19548,378],{"className":19549},[138],[28,19551,17456],{"className":19552,"style":17483},[114,222],[28,19554,387],{"className":19555},[187],[28,19557,67],{"className":19558},[145],[28,19560],{"className":19561,"style":149},[119],[28,19563,19565,19568],{"className":19564},[114],[28,19566,15050],{"className":19567},[114,222],[28,19569,19571],{"className":19570},[423],[28,19572,19574,19594],{"className":19573},[427,428],[28,19575,19577,19591],{"className":19576},[432],[28,19578,19580],{"className":19579,"style":5069},[436],[28,19581,19582,19585],{"style":440},[28,19583],{"className":19584,"style":445},[444],[28,19586,19588],{"className":19587},[449,450,451,452],[28,19589,269],{"className":19590,"style":296},[114,222,452],[28,19592,460],{"className":19593},[459],[28,19595,19597],{"className":19596},[432],[28,19598,19600],{"className":19599,"style":3926},[436],[28,19601],{},[28,19603,378],{"className":19604},[138],[28,19606,19608,19611],{"className":19607},[114],[28,19609,17456],{"className":19610,"style":17483},[114,222],[28,19612,19614],{"className":19613},[423],[28,19615,19617],{"className":19616},[427],[28,19618,19620],{"className":19619},[432],[28,19621,19624],{"className":19622,"style":19623},[436],"height:0.8019em;",[28,19625,19627,19630],{"style":19626},"top:-3.113em;margin-right:0.05em;",[28,19628],{"className":19629,"style":445},[444],[28,19631,19633],{"className":19632},[449,450,451,452],[28,19634,19636],{"className":19635},[114,452],[28,19637,19442],{"className":19638},[114,452],[28,19640,10079],{"className":19641},[187],[28,19643],{"className":19644,"style":120},[119],[28,19646,55],{"className":19647},[124],[28,19649],{"className":19650,"style":120},[119],[28,19652,19654,19658,19698,19701,19741,19744,19747,19750,19753,19785,19815],{"className":19653},[105],[28,19655],{"className":19656,"style":19657},[109],"height:1.1502em;vertical-align:-0.2861em;",[28,19659,19661,19664],{"className":19660},[114],[28,19662,19453],{"className":19663,"style":2123},[114,222],[28,19665,19667],{"className":19666},[423],[28,19668,19670,19690],{"className":19669},[427,428],[28,19671,19673,19687],{"className":19672},[432],[28,19674,19676],{"className":19675,"style":5069},[436],[28,19677,19678,19681],{"style":16510},[28,19679],{"className":19680,"style":445},[444],[28,19682,19684],{"className":19683},[449,450,451,452],[28,19685,269],{"className":19686,"style":296},[114,222,452],[28,19688,460],{"className":19689},[459],[28,19691,19693],{"className":19692},[432],[28,19694,19696],{"className":19695,"style":3926},[436],[28,19697],{},[28,19699,378],{"className":19700},[138],[28,19702,19704,19707],{"className":19703},[114],[28,19705,2450],{"className":19706},[114,222],[28,19708,19710],{"className":19709},[423],[28,19711,19713,19733],{"className":19712},[427,428],[28,19714,19716,19730],{"className":19715},[432],[28,19717,19719],{"className":19718,"style":5069},[436],[28,19720,19721,19724],{"style":440},[28,19722],{"className":19723,"style":445},[444],[28,19725,19727],{"className":19726},[449,450,451,452],[28,19728,269],{"className":19729,"style":296},[114,222,452],[28,19731,460],{"className":19732},[459],[28,19734,19736],{"className":19735},[432],[28,19737,19739],{"className":19738,"style":3926},[436],[28,19740],{},[28,19742,378],{"className":19743},[138],[28,19745,17456],{"className":19746,"style":17483},[114,222],[28,19748,67],{"className":19749},[145],[28,19751],{"className":19752,"style":149},[119],[28,19754,19756,19759],{"className":19755},[114],[28,19757,17456],{"className":19758,"style":17483},[114,222],[28,19760,19762],{"className":19761},[423],[28,19763,19765],{"className":19764},[427],[28,19766,19768],{"className":19767},[432],[28,19769,19771],{"className":19770,"style":19623},[436],[28,19772,19773,19776],{"style":19626},[28,19774],{"className":19775,"style":445},[444],[28,19777,19779],{"className":19778},[449,450,451,452],[28,19780,19782],{"className":19781},[114,452],[28,19783,19442],{"className":19784},[114,452],[28,19786,19788,19791],{"className":19787},[187],[28,19789,387],{"className":19790},[187],[28,19792,19794],{"className":19793},[423],[28,19795,19797],{"className":19796},[427],[28,19798,19800],{"className":19799},[432],[28,19801,19804],{"className":19802,"style":19803},[436],"height:0.8641em;",[28,19805,19806,19809],{"style":19626},[28,19807],{"className":19808,"style":445},[444],[28,19810,19812],{"className":19811},[449,450,451,452],[28,19813,75],{"className":19814},[114,452],[28,19816,387],{"className":19817},[187],[11,19819,19820,19821,19891,19892,20020],{},"对 ",[28,19822,19824,19842],{"className":19823,"translate":32},[31],[28,19825,19827],{"className":19826},[36],[38,19828,19829],{"xmlns":40},[42,19830,19831,19839],{},[45,19832,19833],{},[354,19834,19835,19837],{},[48,19836,19453],{},[48,19838,269],{},[94,19840,19841],{"encoding":96},"g_f",[28,19843,19845],{"className":19844,"ariaHidden":66},[101],[28,19846,19848,19851],{"className":19847},[105],[28,19849],{"className":19850,"style":19078},[109],[28,19852,19854,19857],{"className":19853},[114],[28,19855,19453],{"className":19856,"style":2123},[114,222],[28,19858,19860],{"className":19859},[423],[28,19861,19863,19883],{"className":19862},[427,428],[28,19864,19866,19880],{"className":19865},[432],[28,19867,19869],{"className":19868,"style":5069},[436],[28,19870,19871,19874],{"style":16510},[28,19872],{"className":19873,"style":445},[444],[28,19875,19877],{"className":19876},[449,450,451,452],[28,19878,269],{"className":19879,"style":296},[114,222,452],[28,19881,460],{"className":19882},[459],[28,19884,19886],{"className":19885},[432],[28,19887,19889],{"className":19888,"style":3926},[436],[28,19890],{}," 的唯一要求是在 0 附近二次可微且满足 ",[28,19893,19895,19926],{"className":19894,"translate":32},[31],[28,19896,19898],{"className":19897},[36],[38,19899,19900],{"xmlns":40},[42,19901,19902,19923],{},[45,19903,19904,19913,19915,19917,19919,19921],{},[19905,19906,19907,19909,19911],"msubsup",{},[48,19908,19453],{},[48,19910,269],{},[53,19912,19442],{"mathvariant":363,"lspace":2419,"rspace":2419},[53,19914,378],{"stretchy":58},[61,19916,63],{},[53,19918,387],{"stretchy":58},[53,19920,8394],{},[61,19922,63],{},[94,19924,19925],{"encoding":96},"g_f'(0) \u003C 0",[28,19927,19929,20011],{"className":19928,"ariaHidden":66},[101],[28,19930,19932,19936,19993,19996,19999,20002,20005,20008],{"className":19931},[105],[28,19933],{"className":19934,"style":19935},[109],"height:1.1711em;vertical-align:-0.4192em;",[28,19937,19939,19942],{"className":19938},[114],[28,19940,19453],{"className":19941,"style":2123},[114,222],[28,19943,19945],{"className":19944},[423],[28,19946,19948,19984],{"className":19947},[427,428],[28,19949,19951,19981],{"className":19950},[432],[28,19952,19955,19967],{"className":19953,"style":19954},[436],"height:0.7519em;",[28,19956,19958,19961],{"style":19957},"top:-2.4169em;margin-left:-0.0359em;margin-right:0.05em;",[28,19959],{"className":19960,"style":445},[444],[28,19962,19964],{"className":19963},[449,450,451,452],[28,19965,269],{"className":19966,"style":296},[114,222,452],[28,19968,19969,19972],{"style":11378},[28,19970],{"className":19971,"style":445},[444],[28,19973,19975],{"className":19974},[449,450,451,452],[28,19976,19978],{"className":19977},[114,452],[28,19979,19442],{"className":19980},[114,452],[28,19982,460],{"className":19983},[459],[28,19985,19987],{"className":19986},[432],[28,19988,19991],{"className":19989,"style":19990},[436],"height:0.4192em;",[28,19992],{},[28,19994,378],{"className":19995},[138],[28,19997,63],{"className":19998},[114],[28,20000,387],{"className":20001},[187],[28,20003],{"className":20004,"style":120},[119],[28,20006,8394],{"className":20007},[124],[28,20009],{"className":20010,"style":120},[119],[28,20012,20014,20017],{"className":20013},[105],[28,20015],{"className":20016,"style":1362},[109],[28,20018,63],{"className":20019},[114],"；除此之外不再施加任何限制。",[11,20022,20023,20024,20054,20055,20124,20125,20155,20156,20225],{},"在假说 1 与假说 2 同时成立的前提下，作者证明了论文的核心结果定理 1：设 ",[28,20025,20027,20042],{"className":20026,"translate":32},[31],[28,20028,20030],{"className":20029},[36],[38,20031,20032],{"xmlns":40},[42,20033,20034,20039],{},[45,20035,20036],{},[48,20037,20038],{},"η",[94,20040,20041],{"encoding":96},"\\eta",[28,20043,20045],{"className":20044,"ariaHidden":66},[101],[28,20046,20048,20051],{"className":20047},[105],[28,20049],{"className":20050,"style":4110},[109],[28,20052,20038],{"className":20053,"style":2123},[114,222]," 是特征空间 ",[28,20056,20058,20075],{"className":20057,"translate":32},[31],[28,20059,20061],{"className":20060},[36],[38,20062,20063],{"xmlns":40},[42,20064,20065,20073],{},[45,20066,20067],{},[354,20068,20069,20071],{},[48,20070,1387],{},[48,20072,269],{},[94,20074,17161],{"encoding":96},[28,20076,20078],{"className":20077,"ariaHidden":66},[101],[28,20079,20081,20084],{"className":20080},[105],[28,20082],{"className":20083,"style":16660},[109],[28,20085,20087,20090],{"className":20086},[114],[28,20088,1387],{"className":20089,"style":17050},[114,222],[28,20091,20093],{"className":20092},[423],[28,20094,20096,20116],{"className":20095},[427,428],[28,20097,20099,20113],{"className":20098},[432],[28,20100,20102],{"className":20101,"style":5069},[436],[28,20103,20104,20107],{"style":17065},[28,20105],{"className":20106,"style":445},[444],[28,20108,20110],{"className":20109},[449,450,451,452],[28,20111,269],{"className":20112,"style":296},[114,222,452],[28,20114,460],{"className":20115},[459],[28,20117,20119],{"className":20118},[432],[28,20120,20122],{"className":20121,"style":3926},[436],[28,20123],{}," 中的一条有限长路径，",[28,20126,20128,20143],{"className":20127,"translate":32},[31],[28,20129,20131],{"className":20130},[36],[38,20132,20133],{"xmlns":40},[42,20134,20135,20140],{},[45,20136,20137],{},[48,20138,20139],{},"γ",[94,20141,20142],{"encoding":96},"\\gamma",[28,20144,20146],{"className":20145,"ariaHidden":66},[101],[28,20147,20149,20152],{"className":20148},[105],[28,20150],{"className":20151,"style":4110},[109],[28,20153,20139],{"className":20154,"style":9007},[114,222]," 是它在流形 ",[28,20157,20159,20176],{"className":20158,"translate":32},[31],[28,20160,20162],{"className":20161},[36],[38,20163,20164],{"xmlns":40},[42,20165,20166,20174],{},[45,20167,20168],{},[354,20169,20170,20172],{},[48,20171,18194],{},[48,20173,269],{},[94,20175,18199],{"encoding":96},[28,20177,20179],{"className":20178,"ariaHidden":66},[101],[28,20180,20182,20185],{"className":20181},[105],[28,20183],{"className":20184,"style":16660},[109],[28,20186,20188,20191],{"className":20187},[114],[28,20189,18194],{"className":20190,"style":18215},[114,222],[28,20192,20194],{"className":20193},[423],[28,20195,20197,20217],{"className":20196},[427,428],[28,20198,20200,20214],{"className":20199},[432],[28,20201,20203],{"className":20202,"style":5069},[436],[28,20204,20205,20208],{"style":18230},[28,20206],{"className":20207,"style":445},[444],[28,20209,20211],{"className":20210},[449,450,451,452],[28,20212,269],{"className":20213,"style":296},[114,222,452],[28,20215,460],{"className":20216},[459],[28,20218,20220],{"className":20219},[432],[28,20221,20223],{"className":20222,"style":3926},[436],[28,20224],{}," 上的对应路径，则",[28,20227,20229],{"className":20228,"translate":32},[2403],[28,20230,20232,20289],{"className":20231,"translate":32},[31],[28,20233,20235],{"className":20234},[36],[38,20236,20237],{"xmlns":40,"display":2412},[42,20238,20239,20286],{},[45,20240,20241,20244,20246,20248,20250,20252,20275,20278,20280,20282,20284],{},[48,20242,20243],{},"L",[53,20245,378],{"stretchy":58},[48,20247,20139],{},[53,20249,387],{"stretchy":58},[53,20251,55],{},[20253,20254,20255],"msqrt",{},[45,20256,20257,20259,20261,20269,20271,20273],{},[53,20258,1398],{},[61,20260,75],{},[19905,20262,20263,20265,20267],{},[48,20264,19453],{},[48,20266,269],{},[53,20268,19442],{"mathvariant":363,"lspace":2419,"rspace":2419},[53,20270,378],{"stretchy":58},[61,20272,63],{},[53,20274,387],{"stretchy":58},[53,20276,20277],{},"⋅",[48,20279,20243],{},[53,20281,378],{"stretchy":58},[48,20283,20038],{},[53,20285,387],{"stretchy":58},[94,20287,20288],{"encoding":96},"L(\\gamma) = \\sqrt{-2g_f'(0)} \\cdot L(\\eta)",[28,20290,20292,20319,20467],{"className":20291,"ariaHidden":66},[101],[28,20293,20295,20298,20301,20304,20307,20310,20313,20316],{"className":20294},[105],[28,20296],{"className":20297,"style":134},[109],[28,20299,20243],{"className":20300},[114,222],[28,20302,378],{"className":20303},[138],[28,20305,20139],{"className":20306,"style":9007},[114,222],[28,20308,387],{"className":20309},[187],[28,20311],{"className":20312,"style":120},[119],[28,20314,55],{"className":20315},[124],[28,20317],{"className":20318,"style":120},[119],[28,20320,20322,20326,20458,20461,20464],{"className":20321},[105],[28,20323],{"className":20324,"style":20325},[109],"height:1.84em;vertical-align:-0.6498em;",[28,20327,20330],{"className":20328},[114,20329],"sqrt",[28,20331,20333,20449],{"className":20332},[427,428],[28,20334,20336,20446],{"className":20335},[432],[28,20337,20340,20426],{"className":20338,"style":20339},[436],"height:1.1902em;",[28,20341,20345,20349],{"className":20342,"style":20344},[20343],"svg-align","top:-3.8em;",[28,20346],{"className":20347,"style":20348},[444],"height:3.8em;",[28,20350,20353,20356,20359,20417,20420,20423],{"className":20351,"style":20352},[114],"padding-left:1em;",[28,20354,1398],{"className":20355},[114],[28,20357,75],{"className":20358},[114],[28,20360,20362,20365],{"className":20361},[114],[28,20363,19453],{"className":20364,"style":2123},[114,222],[28,20366,20368],{"className":20367},[423],[28,20369,20371,20408],{"className":20370},[427,428],[28,20372,20374,20405],{"className":20373},[432],[28,20375,20378,20390],{"className":20376,"style":20377},[436],"height:0.7337em;",[28,20379,20381,20384],{"style":20380},"top:-2.3987em;margin-left:-0.0359em;margin-right:0.05em;",[28,20382],{"className":20383,"style":445},[444],[28,20385,20387],{"className":20386},[449,450,451,452],[28,20388,269],{"className":20389,"style":296},[114,222,452],[28,20391,20393,20396],{"style":20392},"top:-3.0448em;margin-right:0.05em;",[28,20394],{"className":20395,"style":445},[444],[28,20397,20399],{"className":20398},[449,450,451,452],[28,20400,20402],{"className":20401},[114,452],[28,20403,19442],{"className":20404},[114,452],[28,20406,460],{"className":20407},[459],[28,20409,20411],{"className":20410},[432],[28,20412,20415],{"className":20413,"style":20414},[436],"height:0.4374em;",[28,20416],{},[28,20418,378],{"className":20419},[138],[28,20421,63],{"className":20422},[114],[28,20424,387],{"className":20425},[187],[28,20427,20429,20432],{"style":20428},"top:-3.1502em;",[28,20430],{"className":20431,"style":20348},[444],[28,20433,20437],{"className":20434,"style":20436},[20435],"hide-tail","min-width:1.02em;height:1.88em;",[6849,20438,20443],{"xmlns":6851,"width":20439,"height":20440,"viewBox":20441,"preserveAspectRatio":20442},"400em","1.88em","0 0 400000 1944","xMinYMin slice",[6856,20444],{"d":20445},"M983 90\nl0 -0\nc4,-6.7,10,-10,18,-10 H400000v40\nH1013.1s-83.4,268,-264.1,840c-180.7,572,-277,876.3,-289,913c-4.7,4.7,-12.7,7,-24,7\ns-12,0,-12,0c-1.3,-3.3,-3.7,-11.7,-7,-25c-35.3,-125.3,-106.7,-373.3,-214,-744\nc-10,12,-21,25,-33,39s-32,39,-32,39c-6,-5.3,-15,-14,-27,-26s25,-30,25,-30\nc26.7,-32.7,52,-63,76,-91s52,-60,52,-60s208,722,208,722\nc56,-175.3,126.3,-397.3,211,-666c84.7,-268.7,153.8,-488.2,207.5,-658.5\nc53.7,-170.3,84.5,-266.8,92.5,-289.5z\nM1001 80h400000v40h-400000z",[28,20447,460],{"className":20448},[459],[28,20450,20452],{"className":20451},[432],[28,20453,20456],{"className":20454,"style":20455},[436],"height:0.6498em;",[28,20457],{},[28,20459],{"className":20460,"style":1741},[119],[28,20462,20277],{"className":20463},[1745],[28,20465],{"className":20466,"style":1741},[119],[28,20468,20470,20473,20476,20479,20482],{"className":20469},[105],[28,20471],{"className":20472,"style":134},[109],[28,20474,20243],{"className":20475},[114,222],[28,20477,378],{"className":20478},[138],[28,20480,20038],{"className":20481,"style":2123},[114,222],[28,20483,387],{"className":20484},[187],[11,20486,20487,20488,20557],{},"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 ",[28,20489,20491,20508],{"className":20490,"translate":32},[31],[28,20492,20494],{"className":20493},[36],[38,20495,20496],{"xmlns":40},[42,20497,20498,20506],{},[45,20499,20500],{},[354,20501,20502,20504],{},[48,20503,19453],{},[48,20505,269],{},[94,20507,19841],{"encoding":96},[28,20509,20511],{"className":20510,"ariaHidden":66},[101],[28,20512,20514,20517],{"className":20513},[105],[28,20515],{"className":20516,"style":19078},[109],[28,20518,20520,20523],{"className":20519},[114],[28,20521,19453],{"className":20522,"style":2123},[114,222],[28,20524,20526],{"className":20525},[423],[28,20527,20529,20549],{"className":20528},[427,428],[28,20530,20532,20546],{"className":20531},[432],[28,20533,20535],{"className":20534,"style":5069},[436],[28,20536,20537,20540],{"style":16510},[28,20538],{"className":20539,"style":445},[444],[28,20541,20543],{"className":20542},[449,450,451,452],[28,20544,269],{"className":20545,"style":296},[114,222,452],[28,20547,460],{"className":20548},[459],[28,20550,20552],{"className":20551},[432],[28,20553,20555],{"className":20554,"style":3926},[436],[28,20556],{}," 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。",[11,20559,20560],{},"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。",[11,20562,20563],{},"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）。",[18,20565,20566],{"id":20566},"同胚容易等距困难",[11,20568,20569],{},"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：",[14911,20571,20572,20607],{},[338,20573,20574,20575,20606],{},"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 ",[28,20576,20578,20593],{"className":20577,"translate":32},[31],[28,20579,20581],{"className":20580},[36],[38,20582,20583],{"xmlns":40},[42,20584,20585,20590],{},[45,20586,20587],{},[48,20588,20589],{},"ξ",[94,20591,20592],{"encoding":96},"\\xi",[28,20594,20596],{"className":20595,"ariaHidden":66},[101],[28,20597,20599,20602],{"className":20598},[105],[28,20600],{"className":20601,"style":292},[109],[28,20603,20589],{"className":20604,"style":20605},[114,222],"margin-right:0.046em;"," 量化整体的函数依赖程度；",[338,20608,20609],{},"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。",[11,20611,20612,20613,20674],{},"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 ",[28,20614,20616,20638],{"className":20615,"translate":32},[31],[28,20617,20619],{"className":20618},[36],[38,20620,20621],{"xmlns":40},[42,20622,20623,20635],{},[45,20624,20625,20627,20629,20631,20633],{},[48,20626,1010],{"mathvariant":363},[48,20628,2430],{},[53,20630,1398],{},[48,20632,4066],{},[48,20634,1010],{"mathvariant":363},[94,20636,20637],{"encoding":96},"|x-y|",[28,20639,20641,20662],{"className":20640,"ariaHidden":66},[101],[28,20642,20644,20647,20650,20653,20656,20659],{"className":20643},[105],[28,20645],{"className":20646,"style":134},[109],[28,20648,1010],{"className":20649},[114],[28,20651,2430],{"className":20652},[114,222],[28,20654],{"className":20655,"style":1741},[119],[28,20657,1398],{"className":20658},[1745],[28,20660],{"className":20661,"style":1741},[119],[28,20663,20665,20668,20671],{"className":20664},[105],[28,20666],{"className":20667,"style":134},[109],[28,20669,4066],{"className":20670,"style":2123},[114,222],[28,20672,1010],{"className":20673},[114],"（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。",[11,20676,20677,20678,20757],{},"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 ",[28,20679,20681,20711],{"className":20680,"translate":32},[31],[28,20682,20684],{"className":20683},[36],[38,20685,20686],{"xmlns":40},[42,20687,20688,20708],{},[45,20689,20690,20693,20696,20698,20701,20703,20706],{},[48,20691,20692],{},"log",[53,20694,20695],{},"⁡",[53,20697,378],{"stretchy":58},[61,20699,20700],{},"2019",[53,20702,1398],{},[87,20704,20705],{},"year",[53,20707,387],{"stretchy":58},[94,20709,20710],{"encoding":96},"\\log(2019 - \\text{year})",[28,20712,20714,20742],{"className":20713,"ariaHidden":66},[101],[28,20715,20717,20720,20727,20730,20733,20736,20739],{"className":20716},[105],[28,20718],{"className":20719,"style":134},[109],[28,20721,20723,20724],{"className":20722},[16370],"lo",[28,20725,19453],{"style":20726},"margin-right:0.0139em;",[28,20728,378],{"className":20729},[138],[28,20731,20700],{"className":20732},[114],[28,20734],{"className":20735,"style":1741},[119],[28,20737,1398],{"className":20738},[1745],[28,20740],{"className":20741,"style":1741},[119],[28,20743,20745,20748,20754],{"className":20744},[105],[28,20746],{"className":20747,"style":134},[109],[28,20749,20751],{"className":20750},[114,1801],[28,20752,20705],{"className":20753},[114],[28,20755,387],{"className":20756},[187]," 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）。",[11,20759,20760],{},"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。",[18,20762,20763],{"id":20763},"柏拉图假说",[11,20765,20766],{},"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。",[11,20768,20769],{},"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。",[11,20771,20772],{},"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？",[11,20774,20775],{},"把两篇论文放在一起，可以读出相当连贯的一条逻辑链：",[14911,20777,20778,20781,20784],{},[338,20779,20780],{},"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？",[338,20782,20783],{},"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。",[338,20785,20786],{},"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。",[11,20788,20789],{},"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。",[11,20791,20792],{},"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」。",[18,20794,20795],{"id":20795},"结语",[11,20797,20798],{},"读完这篇论文，我想从几个角度给出我的评价。",[11,20800,20801],{},"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。",[11,20803,20804],{},"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。",[11,20806,20807],{},"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。",[11,20809,20810,20811,20880],{},"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 ",[28,20812,20814,20831],{"className":20813,"translate":32},[31],[28,20815,20817],{"className":20816},[36],[38,20818,20819],{"xmlns":40},[42,20820,20821,20829],{},[45,20822,20823],{},[354,20824,20825,20827],{},[48,20826,15050],{},[48,20828,269],{},[94,20830,18127],{"encoding":96},[28,20832,20834],{"className":20833,"ariaHidden":66},[101],[28,20835,20837,20840],{"className":20836},[105],[28,20838],{"className":20839,"style":3881},[109],[28,20841,20843,20846],{"className":20842},[114],[28,20844,15050],{"className":20845},[114,222],[28,20847,20849],{"className":20848},[423],[28,20850,20852,20872],{"className":20851},[427,428],[28,20853,20855,20869],{"className":20854},[432],[28,20856,20858],{"className":20857,"style":5069},[436],[28,20859,20860,20863],{"style":440},[28,20861],{"className":20862,"style":445},[444],[28,20864,20866],{"className":20865},[449,450,451,452],[28,20867,269],{"className":20868,"style":296},[114,222,452],[28,20870,460],{"className":20871},[459],[28,20873,20875],{"className":20874},[432],[28,20876,20878],{"className":20877,"style":3926},[436],[28,20879],{},"（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。",[11,20882,20883],{},"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。",[11,20885,20886],{},"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。",[11,20888,20889],{},"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":7750,"searchDepth":7781,"depth":7781,"links":20891},[20892,20893,20894,20895,20896,20897,20898,20899],{"id":16244,"depth":7781,"text":16244},{"id":16991,"depth":7781,"text":16992},{"id":18676,"depth":7781,"text":18676},{"id":18703,"depth":7781,"text":18703},{"id":19372,"depth":7781,"text":19372},{"id":20566,"depth":7781,"text":20566},{"id":20763,"depth":7781,"text":20763},{"id":20795,"depth":7781,"text":20795},"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。",{},"\u002Fblog\u002F2026\u002F2026-08-04-the-representations-in-llm",{"title":16217,"description":20900},"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。",[20907,20908],"machine-learning","ai","2026-08-04T00:00:00+08:00","7muZrPVdpjj3Hr2Jo2ta-VJnPzeZr9TL8KvVJunp_nQ",{"id":20912,"title":20913,"body":20914,"description":20918,"draft":16202,"enableComment":7941,"extension":16203,"image":7750,"important":16202,"location":16204,"meta":21075,"navigation":7941,"ogImage":16204,"path":21076,"seo":21077,"stem":21078,"summary":21079,"tags":21080,"time":21082,"__hash__":21083},"blog\u002Fblog\u002F2026\u002F2026-07-30-what-agi-needs-may-not-be-a-larger-model-but-a-set-of-theories.md","在开始 AGI 之前，先定义它",{"type":8,"value":20915,"toc":21070},[20916,20919,20922,20925,20930,20933,20936,20939,20944,20947,20952,20955,20958,20961,20964,20978,20981,20984,20987,20990,20993,20996,20999,21002,21005,21013,21016,21021,21024,21027,21030,21033,21036,21041,21044,21047,21050,21055,21058,21061,21064,21067],[11,20917,20918],{},"最近，我看了梁文锋关于 AGI 的演讲，他指出人工智能的终极目标是通用人工智能（AGI）——一种能以类人方式解决各种问题的一般智能。",[11,20920,20921],{},"然而，我越来越觉得，当下 AI 领域围绕 AGI 的讨论仍停留在经验层面。",[11,20923,20924],{},"我们讨论模型规模、参数量、强化学习、智能体和多模态，却很少有人触及一个更根本的问题：",[2132,20926,20927],{},[11,20928,20929],{},"什么才是 AGI？",[11,20931,20932],{},"直到今天，仍然没有一个被广泛接受的正式定义。大多数讨论都在描述现象，而非把握本质。",[18,20934,20935],{"id":20935},"理论框架",[11,20937,20938],{},"历史上真正改变世界的，从来不是某台机器，而是它背后的理论。为了研究机械计算，图灵提出了图灵机模型。随后，丘奇 - 图灵论题指出：",[2132,20940,20941],{},[11,20942,20943],{},"任何能用有效程序计算的函数，都能被图灵机计算。",[11,20945,20946],{},"几乎所有现代计算机都是这个模型的工程实现：一个有限状态控制器加上外部存储。更重要的是，图灵进一步证明了停机问题不可判定。这意味着，人类第一次意识到：",[2132,20948,20949],{},[11,20950,20951],{},"并非所有问题都是可计算的。",[11,20953,20954],{},"换言之，计算机第一次拥有了明确定义的理论上限。这一发现的意义不亚于物理学中的热力学第二定律。",[18,20956,20957],{"id":20957},"理论的局限",[11,20959,20960],{},"今天的 AGI，本质上正处于相当于计算机发明之前的那个阶段。",[11,20962,20963],{},"每个人都在造越来越大的机器，但下面这些问题仍然没有答案：",[335,20965,20966,20969,20972,20975],{},[338,20967,20968],{},"什么是智能？",[338,20970,20971],{},"智能能被形式化吗？",[338,20973,20974],{},"一般智能的必要条件是什么？",[338,20976,20977],{},"智能有理论极限吗？",[11,20979,20980],{},"如果未来有人能提出一套 AGI 公理，类似于图灵机之于计算理论，其意义或许远远超过再训练一个更大的模型。",[11,20982,20983],{},"只有这样的理论建立起来之后，我们才能真正讨论：什么能做到，什么永远做不到。",[18,20985,20986],{"id":20986},"计算主义的内在边界",[11,20988,20989],{},"当今几乎所有主流 AI 方法，核心都建立在计算主义之上。",[11,20991,20992],{},"也就是说：智能就是计算。",[11,20994,20995],{},"如果这个前提成立，那么任何 AI 系统最终都属于可计算系统这一类。然而可计算性理论告诉我们，存在大量的不可判定问题（例如停机问题）。",[11,20997,20998],{},"于是，一个值得深思的问题浮现出来：如果 AGI 被定义为一种能够解决所有问题的智能，那么它必然要直面不可判定问题。",[11,21000,21001],{},"不可判定问题，按照丘奇 - 图灵框架，本质上是不可能被任何可计算系统解决的。",[11,21003,21004],{},"因此，如果采纳这种「全知全能」式的 AGI 定义，它与可计算性理论之间就出现了明显的张力。当然，这并不意味着 AGI 必然不可能；而是说明，「一般智能」到底意味着什么，需要更精确的定义。",[11,21006,21007,21012],{},[358,21008,21011],{"href":21009,"rel":21010},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fcs\u002F0004001",[16227],"AIXI 早已揭示了这一矛盾","。事实上，这个问题很早就出现了。Marcus Hutter 的 AIXI 是一般智能最著名的数学模型之一。",[11,21014,21015],{},"AIXI 把智能定义为：",[2132,21017,21018],{},[11,21019,21020],{},"在所有可计算环境中，最大化长期期望回报的最优智能体。",[11,21022,21023],{},"它拥有严谨的数学定义，因此常被视为 AGI 的理论上限。但与此同时，AIXI 本身却是不可计算的。原因在于，它依赖所罗门诺夫归纳，需要对所有可能的程序求和——这个操作天然涉及不可计算的对象，因而无法在真实的物理世界中实现。",[11,21025,21026],{},"换言之，我们已经有了一种「完美智能」的形式化定义，却证明了它无法被计算。",[11,21028,21029],{},"这不是工程上的困难，而是理论上的局限。",[11,21031,21032],{},"我认为有一个公理是缺失的。目前，包括 DeepMind 在内的许多机构都在探索元学习、自适应智能体、世界模型等统一框架。在某种意义上，这些努力都可以被理解为对 AGI「公理系统」的搜寻。",[11,21034,21035],{},"然而，我认为有一个关键公理可能是缺失的：",[2132,21037,21038],{},[11,21039,21040],{},"智能体必须与环境保持连续的因果耦合，而不是作为一个超然的、观察者式的求解器运转。",[11,21042,21043],{},"真正的智能不是对固定数据集的一次性推理，而是在与环境、行动、反馈与修正的持续互动中不断塑造自身。",[11,21045,21046],{},"没有这个因果闭环，即使再大的语言模型，也像是静态世界里运转的模板匹配系统，而不是在实质意义上发展智能。",[11,21048,21049],{},"也许，我们真正需要追寻的并不是一个万能 AI。我越来越倾向于认为，真正值得研究的问题不是如何造出一个「全能」的 AI，而是：",[2132,21051,21052],{},[11,21053,21054],{},"智能是否存在类似于热力学第二定律或香农极限那样的根本性约束原理？",[11,21056,21057],{},"如果未来有人能证明，任何位于有限物理时空中的系统都不可能实现绝对的一般智能，",[11,21059,21060],{},"那么这样工作的意义，或许不亚于图灵建立计算理论，也不亚于牛顿和爱因斯坦为物理学奠定的基础框架。",[11,21062,21063],{},"它不会宣告 AI 的失败。",[11,21065,21066],{},"恰恰相反，它将揭示：真正值得追求的并不是无限的智能，而是一种在理论极限内逐步逼近最优的有界智能。",[11,21068,21069],{},"也许，站在未来回望，人们记住的今天将不是 AGI 在 2026 年出现的那一年，而是我们开始重新思考这个根本问题的那一年：「智能，到底是什么？」",{"title":7750,"searchDepth":7781,"depth":7781,"links":21071},[21072,21073,21074],{"id":20935,"depth":7781,"text":20935},{"id":20957,"depth":7781,"text":20957},{"id":20986,"depth":7781,"text":20986},{},"\u002Fblog\u002F2026\u002F2026-07-30-what-agi-needs-may-not-be-a-larger-model-but-a-set-of-theories",{"title":20913,"description":20918},"blog\u002F2026\u002F2026-07-30-what-agi-needs-may-not-be-a-larger-model-but-a-set-of-theories","AGI 研究仍停留在经验层面，缺乏基础性的理论框架。计算主义与不可判定问题之间存在核心张力，AIXI 正是一个例证。作者提出，与环境保持连续的因果耦合可能是缺失的公理，并呼吁寻找能约束任何物理可实现智能的根本性约束原理。",[20908,21081],"thoughts","2026-07-30T00:00:00+08:00","6b3H4Ls4HkUBe5xve0AixDdwujabBMiU5faXjBSLD4U",{"left":21085,"top":21085,"width":21086,"height":21086,"rotate":21085,"vFlip":16202,"hFlip":16202,"body":21087},0,24,"\u003Cg fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\">\u003Cpath d=\"m4.748 7.645l-.331 3.644c-.05.54-.074.811-.03 1.07a2 2 0 0 0 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