[{"data":1,"prerenderedAt":17384},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-07-30-what-agi-needs-may-not-be-a-larger-model-but-a-set-of-theories":3,"i-ci:tag":17378,"i-mingcute:time-line":17382},{"post":4,"nextPost":192,"prevPost":16390},{"id":5,"title":6,"body":7,"description":13,"draft":178,"enableComment":179,"extension":180,"image":172,"important":178,"location":181,"meta":182,"navigation":179,"ogImage":181,"path":183,"seo":184,"stem":185,"summary":186,"tags":187,"time":190,"__hash__":191},"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":9,"toc":171},"minimark",[10,14,17,20,26,29,33,36,41,44,49,52,55,58,61,77,80,83,86,89,92,95,98,101,104,114,117,122,125,128,131,134,137,142,145,148,151,156,159,162,165,168],[11,12,13],"p",{},"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。",[11,15,16],{},"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。",[11,18,19],{},"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：",[21,22,23],"blockquote",{},[11,24,25],{},"什么才是 AGI？",[11,27,28],{},"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。",[30,31,32],"h2",{"id":32},"理论框架",[11,34,35],{},"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：",[21,37,38],{},[11,39,40],{},"任何能用有效程序计算的函数，都能被图灵机计算。",[11,42,43],{},"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：",[21,45,46],{},[11,47,48],{},"并非所有问题都是可计算的。",[11,50,51],{},"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。",[30,53,54],{"id":54},"理论的局限",[11,56,57],{},"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。",[11,59,60],{},"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：",[62,63,64,68,71,74],"ul",{},[65,66,67],"li",{},"什么是智能？",[65,69,70],{},"智能能被形式化吗？",[65,72,73],{},"一般智能的必要条件是什么？",[65,75,76],{},"智能有理论极限吗？",[11,78,79],{},"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。",[11,81,82],{},"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。",[30,84,85],{"id":85},"计算主义的内在边界",[11,87,88],{},"当今几乎所有主流 AI 方法，核心都建立在计算主义之上。",[11,90,91],{},"也就是说：智能就是计算。",[11,93,94],{},"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）。",[11,96,97],{},"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。",[11,99,100],{},"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。",[11,102,103],{},"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。",[11,105,106,113],{},[107,108,112],"a",{"href":109,"rel":110},"https:\u002F\u002Farxiv.org\u002Fabs\u002Fcs\u002F0004001",[111],"nofollow","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。",[11,115,116],{},"AIXI 把智能定义为：",[21,118,119],{},[11,120,121],{},"在所有可计算环境中，最大化长期期望回报的最优智能体。",[11,123,124],{},"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。",[11,126,127],{},"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。",[11,129,130],{},"这不是工程上的困难，而是理论上的局限。",[11,132,133],{},"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。",[11,135,136],{},"然而，我认为有一个关键公理可能是缺失的：",[21,138,139],{},[11,140,141],{},"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。",[11,143,144],{},"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。",[11,146,147],{},"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。",[11,149,150],{},"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：",[21,152,153],{},[11,154,155],{},"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？",[11,157,158],{},"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，",[11,160,161],{},"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。",[11,163,164],{},"它不会宣告 AI 的失败。",[11,166,167],{},"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。",[11,169,170],{},"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":172,"searchDepth":173,"depth":173,"links":174},"",2,[175,176,177],{"id":32,"depth":173,"text":32},{"id":54,"depth":173,"text":54},{"id":85,"depth":173,"text":85},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F2026-07-30-what-agi-needs-may-not-be-a-larger-model-but-a-set-of-theories",{"title":6,"description":13},"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。",[188,189],"ai","thoughts","2026-07-30T00:00:00+08:00","6b3H4Ls4HkUBe5xve0AixDdwujabBMiU5faXjBSLD4U",{"id":193,"title":194,"body":195,"description":199,"draft":178,"enableComment":179,"extension":180,"image":172,"important":178,"location":181,"meta":16380,"navigation":179,"ogImage":181,"path":16381,"seo":16382,"stem":16383,"summary":16384,"tags":16385,"time":16388,"__hash__":16389},"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":196,"toc":16369},[197,200,203,206,209,374,519,1089,1251,1371,1374,2307,2310,2313,2371,2426,2429,2581,4023,4230,4481,4908,5086,5118,5303,5685,5994,6089,6238,6322,6502,6777,7916,7919,7922,7925,8234,8237,8244,8406,8409,8412,8639,8715,8719,8896,9023,9216,9219,9470,9525,10398,10574,10577,11425,11752,11755,12067,12069,12075,12165,12168,12171,12291,12294,12297,12613,12616,12648,12988,13109,13156,13565,13601,13798,14640,14815,14818,15056,15058,15064,15074,15077,15080,15085,15088,15674,15677,15918,15921,15924,15927,16290,16293,16365],[11,198,199],{},"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。",[11,201,202],{},"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。",[30,204,205],{"id":205},"等势",[11,207,208],{},"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？",[11,210,211,212,373],{},"对于有限集，答案很简单：数元素个数即可。但自然数集 ",[213,214,218,283],"span",{"className":215,"translate":217},[216],"katex","no",[213,219,222],{"className":220},[221],"katex-mathml",[223,224,226],"math",{"xmlns":225},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[227,228,229,278],"semantics",{},[230,231,232,237,241,245,249,253,256,258,261,263,266,268,271,275],"mrow",{},[233,234,236],"mi",{"mathvariant":235},"double-struck","N",[238,239,240],"mo",{},"=",[238,242,244],{"stretchy":243},"false","{",[246,247,248],"mn",{},"0",[238,250,252],{"separator":251},"true",",",[246,254,255],{},"1",[238,257,252],{"separator":251},[246,259,260],{},"2",[238,262,252],{"separator":251},[246,264,265],{},"3",[238,267,252],{"separator":251},[238,269,270],{},"…",[272,273,274],"mtext",{}," ",[238,276,277],{"stretchy":243},"}",[279,280,282],"annotation",{"encoding":281},"application\u002Fx-tex","\\mathbb{N} = \\{0, 1, 2, 3, \\dots\\}",[213,284,287,313],{"className":285,"ariaHidden":251},[286],"katex-html",[213,288,291,296,301,306,310],{"className":289},[290],"base",[213,292],{"className":293,"style":295},[294],"strut","height:0.6889em;",[213,297,236],{"className":298},[299,300],"mord","mathbb",[213,302],{"className":303,"style":305},[304],"mspace","margin-right:0.2778em;",[213,307,240],{"className":308},[309],"mrel",[213,311],{"className":312,"style":305},[304],[213,314,316,320,324,327,331,335,338,341,344,347,350,353,356,359,362,366,369],{"className":315},[290],[213,317],{"className":318,"style":319},[294],"height:1em;vertical-align:-0.25em;",[213,321,244],{"className":322},[323],"mopen",[213,325,248],{"className":326},[299],[213,328,252],{"className":329},[330],"mpunct",[213,332],{"className":333,"style":334},[304],"margin-right:0.1667em;",[213,336,255],{"className":337},[299],[213,339,252],{"className":340},[330],[213,342],{"className":343,"style":334},[304],[213,345,260],{"className":346},[299],[213,348,252],{"className":349},[330],[213,351],{"className":352,"style":334},[304],[213,354,265],{"className":355},[299],[213,357,252],{"className":358},[330],[213,360],{"className":361,"style":334},[304],[213,363,270],{"className":364},[365],"minner",[213,367],{"className":368,"style":334},[304],[213,370,277],{"className":371},[372],"mclose"," 是无穷的，无法「数到底」；因此我们需要一种新的工具。",[11,375,376,377,408,409,439,440,518],{},"定义（双射）。给定两个集合 ",[213,378,380,394],{"className":379,"translate":217},[216],[213,381,383],{"className":382},[221],[223,384,385],{"xmlns":225},[227,386,387,392],{},[230,388,389],{},[233,390,391],{},"A",[279,393,391],{"encoding":281},[213,395,397],{"className":396,"ariaHidden":251},[286],[213,398,400,404],{"className":399},[290],[213,401],{"className":402,"style":403},[294],"height:0.6833em;",[213,405,391],{"className":406},[299,407],"mathnormal"," 与 ",[213,410,412,426],{"className":411,"translate":217},[216],[213,413,415],{"className":414},[221],[223,416,417],{"xmlns":225},[227,418,419,424],{},[230,420,421],{},[233,422,423],{},"B",[279,425,423],{"encoding":281},[213,427,429],{"className":428,"ariaHidden":251},[286],[213,430,432,435],{"className":431},[290],[213,433],{"className":434,"style":403},[294],[213,436,423],{"className":437,"style":438},[299,407],"margin-right:0.0502em;","，若存在函数 ",[213,441,443,468],{"className":442,"translate":217},[216],[213,444,446],{"className":445},[221],[223,447,448],{"xmlns":225},[227,449,450,465],{},[230,451,452,455,458,460,463],{},[233,453,454],{},"f",[238,456,457],{},":",[233,459,391],{},[238,461,462],{},"→",[233,464,423],{},[279,466,467],{"encoding":281},"f: A \\to B",[213,469,471,491,509],{"className":470,"ariaHidden":251},[286],[213,472,474,478,482,485,488],{"className":473},[290],[213,475],{"className":476,"style":477},[294],"height:0.8889em;vertical-align:-0.1944em;",[213,479,454],{"className":480,"style":481},[299,407],"margin-right:0.1076em;",[213,483],{"className":484,"style":305},[304],[213,486,457],{"className":487},[309],[213,489],{"className":490,"style":305},[304],[213,492,494,497,500,503,506],{"className":493},[290],[213,495],{"className":496,"style":403},[294],[213,498,391],{"className":499},[299,407],[213,501],{"className":502,"style":305},[304],[213,504,462],{"className":505},[309],[213,507],{"className":508,"style":305},[304],[213,510,512,515],{"className":511},[290],[213,513],{"className":514,"style":403},[294],[213,516,423],{"className":517,"style":438},[299,407]," 满足：",[62,520,521,910],{},[65,522,523,524,909],{},"单射：",[213,525,527,587],{"className":526,"translate":217},[216],[213,528,530],{"className":529},[221],[223,531,532],{"xmlns":225},[227,533,534,584],{},[230,535,536,543,547,553,556,558,561,567,570,572,574,576,582],{},[537,538,539,541],"msub",{},[233,540,107],{},[246,542,255],{},[238,544,546],{"mathvariant":545},"normal","≠",[537,548,549,551],{},[233,550,107],{},[246,552,260],{},[238,554,555],{},"⇒",[233,557,454],{},[238,559,560],{"stretchy":243},"(",[537,562,563,565],{},[233,564,107],{},[246,566,255],{},[238,568,569],{"stretchy":243},")",[238,571,546],{"mathvariant":545},[233,573,454],{},[238,575,560],{"stretchy":243},[537,577,578,580],{},[233,579,107],{},[246,581,260],{},[238,583,569],{"stretchy":243},[279,585,586],{"encoding":281},"a_1 \\neq a_2 \\Rightarrow f(a_1) \\neq f(a_2)",[213,588,590,701,757,854],{"className":589,"ariaHidden":251},[286],[213,591,593,596,652,655,698],{"className":592},[290],[213,594],{"className":595,"style":477},[294],[213,597,599,602],{"className":598},[299],[213,600,107],{"className":601},[299,407],[213,603,606],{"className":604},[605],"msupsub",[213,607,611,643],{"className":608},[609,610],"vlist-t","vlist-t2",[213,612,615,638],{"className":613},[614],"vlist-r",[213,616,620],{"className":617,"style":619},[618],"vlist","height:0.3011em;",[213,621,623,628],{"style":622},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[213,624],{"className":625,"style":627},[626],"pstrut","height:2.7em;",[213,629,635],{"className":630},[631,632,633,634],"sizing","reset-size6","size3","mtight",[213,636,255],{"className":637},[299,634],[213,639,642],{"className":640},[641],"vlist-s","​",[213,644,646],{"className":645},[614],[213,647,650],{"className":648,"style":649},[618],"height:0.15em;",[213,651],{},[213,653],{"className":654,"style":305},[304],[213,656,658,691,695],{"className":657},[309],[213,659,661],{"className":660},[309],[213,662,665],{"className":663},[299,664],"vbox",[213,666,669],{"className":667},[668],"thinbox",[213,670,673,676,687],{"className":671},[672],"rlap",[213,674],{"className":675,"style":477},[294],[213,677,680],{"className":678},[679],"inner",[213,681,683],{"className":682},[299],[213,684,686],{"className":685},[309],"",[213,688],{"className":689},[690],"fix",[213,692],{"className":693},[304,694],"nobreak",[213,696,240],{"className":697},[309],[213,699],{"className":700,"style":305},[304],[213,702,704,708,748,751,754],{"className":703},[290],[213,705],{"className":706,"style":707},[294],"height:0.5806em;vertical-align:-0.15em;",[213,709,711,714],{"className":710},[299],[213,712,107],{"className":713},[299,407],[213,715,717],{"className":716},[605],[213,718,720,740],{"className":719},[609,610],[213,721,723,737],{"className":722},[614],[213,724,726],{"className":725,"style":619},[618],[213,727,728,731],{"style":622},[213,729],{"className":730,"style":627},[626],[213,732,734],{"className":733},[631,632,633,634],[213,735,260],{"className":736},[299,634],[213,738,642],{"className":739},[641],[213,741,743],{"className":742},[614],[213,744,746],{"className":745,"style":649},[618],[213,747],{},[213,749],{"className":750,"style":305},[304],[213,752,555],{"className":753},[309],[213,755],{"className":756,"style":305},[304],[213,758,760,763,766,769,809,812,815,851],{"className":759},[290],[213,761],{"className":762,"style":319},[294],[213,764,454],{"className":765,"style":481},[299,407],[213,767,560],{"className":768},[323],[213,770,772,775],{"className":771},[299],[213,773,107],{"className":774},[299,407],[213,776,778],{"className":777},[605],[213,779,781,801],{"className":780},[609,610],[213,782,784,798],{"className":783},[614],[213,785,787],{"className":786,"style":619},[618],[213,788,789,792],{"style":622},[213,790],{"className":791,"style":627},[626],[213,793,795],{"className":794},[631,632,633,634],[213,796,255],{"className":797},[299,634],[213,799,642],{"className":800},[641],[213,802,804],{"className":803},[614],[213,805,807],{"className":806,"style":649},[618],[213,808],{},[213,810,569],{"className":811},[372],[213,813],{"className":814,"style":305},[304],[213,816,818,845,848],{"className":817},[309],[213,819,821],{"className":820},[309],[213,822,824],{"className":823},[299,664],[213,825,827],{"className":826},[668],[213,828,830,833,842],{"className":829},[672],[213,831],{"className":832,"style":477},[294],[213,834,836],{"className":835},[679],[213,837,839],{"className":838},[299],[213,840,686],{"className":841},[309],[213,843],{"className":844},[690],[213,846],{"className":847},[304,694],[213,849,240],{"className":850},[309],[213,852],{"className":853,"style":305},[304],[213,855,857,860,863,866,906],{"className":856},[290],[213,858],{"className":859,"style":319},[294],[213,861,454],{"className":862,"style":481},[299,407],[213,864,560],{"className":865},[323],[213,867,869,872],{"className":868},[299],[213,870,107],{"className":871},[299,407],[213,873,875],{"className":874},[605],[213,876,878,898],{"className":877},[609,610],[213,879,881,895],{"className":880},[614],[213,882,884],{"className":883,"style":619},[618],[213,885,886,889],{"style":622},[213,887],{"className":888,"style":627},[626],[213,890,892],{"className":891},[631,632,633,634],[213,893,260],{"className":894},[299,634],[213,896,642],{"className":897},[641],[213,899,901],{"className":900},[614],[213,902,904],{"className":903,"style":649},[618],[213,905],{},[213,907,569],{"className":908},[372],"（不同的输入产生不同的输出）",[65,911,912,913,967,968,1020,1021,1088],{},"满射：对每个 ",[213,914,916,936],{"className":915,"translate":217},[216],[213,917,919],{"className":918},[221],[223,920,921],{"xmlns":225},[227,922,923,933],{},[230,924,925,928,931],{},[233,926,927],{},"b",[238,929,930],{},"∈",[233,932,423],{},[279,934,935],{"encoding":281},"b \\in B",[213,937,939,958],{"className":938,"ariaHidden":251},[286],[213,940,942,946,949,952,955],{"className":941},[290],[213,943],{"className":944,"style":945},[294],"height:0.7335em;vertical-align:-0.0391em;",[213,947,927],{"className":948},[299,407],[213,950],{"className":951,"style":305},[304],[213,953,930],{"className":954},[309],[213,956],{"className":957,"style":305},[304],[213,959,961,964],{"className":960},[290],[213,962],{"className":963,"style":403},[294],[213,965,423],{"className":966,"style":438},[299,407],"，都存在 ",[213,969,971,989],{"className":970,"translate":217},[216],[213,972,974],{"className":973},[221],[223,975,976],{"xmlns":225},[227,977,978,986],{},[230,979,980,982,984],{},[233,981,107],{},[238,983,930],{},[233,985,391],{},[279,987,988],{"encoding":281},"a \\in A",[213,990,992,1011],{"className":991,"ariaHidden":251},[286],[213,993,995,999,1002,1005,1008],{"className":994},[290],[213,996],{"className":997,"style":998},[294],"height:0.5782em;vertical-align:-0.0391em;",[213,1000,107],{"className":1001},[299,407],[213,1003],{"className":1004,"style":305},[304],[213,1006,930],{"className":1007},[309],[213,1009],{"className":1010,"style":305},[304],[213,1012,1014,1017],{"className":1013},[290],[213,1015],{"className":1016,"style":403},[294],[213,1018,391],{"className":1019},[299,407]," 使得 ",[213,1022,1024,1048],{"className":1023,"translate":217},[216],[213,1025,1027],{"className":1026},[221],[223,1028,1029],{"xmlns":225},[227,1030,1031,1045],{},[230,1032,1033,1035,1037,1039,1041,1043],{},[233,1034,454],{},[238,1036,560],{"stretchy":243},[233,1038,107],{},[238,1040,569],{"stretchy":243},[238,1042,240],{},[233,1044,927],{},[279,1046,1047],{"encoding":281},"f(a) = b",[213,1049,1051,1078],{"className":1050,"ariaHidden":251},[286],[213,1052,1054,1057,1060,1063,1066,1069,1072,1075],{"className":1053},[290],[213,1055],{"className":1056,"style":319},[294],[213,1058,454],{"className":1059,"style":481},[299,407],[213,1061,560],{"className":1062},[323],[213,1064,107],{"className":1065},[299,407],[213,1067,569],{"className":1068},[372],[213,1070],{"className":1071,"style":305},[304],[213,1073,240],{"className":1074},[309],[213,1076],{"className":1077,"style":305},[304],[213,1079,1081,1085],{"className":1080},[290],[213,1082],{"className":1083,"style":1084},[294],"height:0.6944em;",[213,1086,927],{"className":1087},[299,407],"（每个输出都被覆盖）",[11,1090,1091,1092,1120,1121,408,1149,1177,1178,1250],{},"则称 ",[213,1093,1095,1108],{"className":1094,"translate":217},[216],[213,1096,1098],{"className":1097},[221],[223,1099,1100],{"xmlns":225},[227,1101,1102,1106],{},[230,1103,1104],{},[233,1105,454],{},[279,1107,454],{"encoding":281},[213,1109,1111],{"className":1110,"ariaHidden":251},[286],[213,1112,1114,1117],{"className":1113},[290],[213,1115],{"className":1116,"style":477},[294],[213,1118,454],{"className":1119,"style":481},[299,407]," 为双射，并称 ",[213,1122,1124,1137],{"className":1123,"translate":217},[216],[213,1125,1127],{"className":1126},[221],[223,1128,1129],{"xmlns":225},[227,1130,1131,1135],{},[230,1132,1133],{},[233,1134,391],{},[279,1136,391],{"encoding":281},[213,1138,1140],{"className":1139,"ariaHidden":251},[286],[213,1141,1143,1146],{"className":1142},[290],[213,1144],{"className":1145,"style":403},[294],[213,1147,391],{"className":1148},[299,407],[213,1150,1152,1165],{"className":1151,"translate":217},[216],[213,1153,1155],{"className":1154},[221],[223,1156,1157],{"xmlns":225},[227,1158,1159,1163],{},[230,1160,1161],{},[233,1162,423],{},[279,1164,423],{"encoding":281},[213,1166,1168],{"className":1167,"ariaHidden":251},[286],[213,1169,1171,1174],{"className":1170},[290],[213,1172],{"className":1173,"style":403},[294],[213,1175,423],{"className":1176,"style":438},[299,407]," 等势（记为 ",[213,1179,1181,1208],{"className":1180,"translate":217},[216],[213,1182,1184],{"className":1183},[221],[223,1185,1186],{"xmlns":225},[227,1187,1188,1205],{},[230,1189,1190,1193,1195,1197,1199,1201,1203],{},[233,1191,1192],{"mathvariant":545},"∣",[233,1194,391],{},[233,1196,1192],{"mathvariant":545},[238,1198,240],{},[233,1200,1192],{"mathvariant":545},[233,1202,423],{},[233,1204,1192],{"mathvariant":545},[279,1206,1207],{"encoding":281},"|A| = |B|",[213,1209,1211,1235],{"className":1210,"ariaHidden":251},[286],[213,1212,1214,1217,1220,1223,1226,1229,1232],{"className":1213},[290],[213,1215],{"className":1216,"style":319},[294],[213,1218,1192],{"className":1219},[299],[213,1221,391],{"className":1222},[299,407],[213,1224,1192],{"className":1225},[299],[213,1227],{"className":1228,"style":305},[304],[213,1230,240],{"className":1231},[309],[213,1233],{"className":1234,"style":305},[304],[213,1236,1238,1241,1244,1247],{"className":1237},[290],[213,1239],{"className":1240,"style":319},[294],[213,1242,1192],{"className":1243},[299],[213,1245,423],{"className":1246,"style":438},[299,407],[213,1248,1192],{"className":1249},[299],"）——即两者「大小相等」。",[11,1252,1253,1254,1282,1283,1312,1313,1341,1342,1370],{},"定义（可数集）。若集合 ",[213,1255,1257,1270],{"className":1256,"translate":217},[216],[213,1258,1260],{"className":1259},[221],[223,1261,1262],{"xmlns":225},[227,1263,1264,1268],{},[230,1265,1266],{},[233,1267,391],{},[279,1269,391],{"encoding":281},[213,1271,1273],{"className":1272,"ariaHidden":251},[286],[213,1274,1276,1279],{"className":1275},[290],[213,1277],{"className":1278,"style":403},[294],[213,1280,391],{"className":1281},[299,407]," 能与自然数集 ",[213,1284,1286,1300],{"className":1285,"translate":217},[216],[213,1287,1289],{"className":1288},[221],[223,1290,1291],{"xmlns":225},[227,1292,1293,1297],{},[230,1294,1295],{},[233,1296,236],{"mathvariant":235},[279,1298,1299],{"encoding":281},"\\mathbb{N}",[213,1301,1303],{"className":1302,"ariaHidden":251},[286],[213,1304,1306,1309],{"className":1305},[290],[213,1307],{"className":1308,"style":295},[294],[213,1310,236],{"className":1311},[299,300]," 建立双射（或 ",[213,1314,1316,1329],{"className":1315,"translate":217},[216],[213,1317,1319],{"className":1318},[221],[223,1320,1321],{"xmlns":225},[227,1322,1323,1327],{},[230,1324,1325],{},[233,1326,391],{},[279,1328,391],{"encoding":281},[213,1330,1332],{"className":1331,"ariaHidden":251},[286],[213,1333,1335,1338],{"className":1334},[290],[213,1336],{"className":1337,"style":403},[294],[213,1339,391],{"className":1340},[299,407]," 为有限集），则称 ",[213,1343,1345,1358],{"className":1344,"translate":217},[216],[213,1346,1348],{"className":1347},[221],[223,1349,1350],{"xmlns":225},[227,1351,1352,1356],{},[230,1353,1354],{},[233,1355,391],{},[279,1357,391],{"encoding":281},[213,1359,1361],{"className":1360,"ariaHidden":251},[286],[213,1362,1364,1367],{"className":1363},[290],[213,1365],{"className":1366,"style":403},[294],[213,1368,391],{"className":1369},[299,407]," 为可数集。",[11,1372,1373],{},"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：",[62,1375,1376,1552,2230],{},[65,1377,1378,1379,1477,1478,1551],{},"偶数集 ",[213,1380,1382,1420],{"className":1381,"translate":217},[216],[213,1383,1385],{"className":1384},[221],[223,1386,1387],{"xmlns":225},[227,1388,1389,1417],{},[230,1390,1391,1393,1395,1397,1399,1401,1404,1406,1409,1411,1413,1415],{},[238,1392,244],{"stretchy":243},[246,1394,248],{},[238,1396,252],{"separator":251},[246,1398,260],{},[238,1400,252],{"separator":251},[246,1402,1403],{},"4",[238,1405,252],{"separator":251},[246,1407,1408],{},"6",[238,1410,252],{"separator":251},[238,1412,270],{},[272,1414,274],{},[238,1416,277],{"stretchy":243},[279,1418,1419],{"encoding":281},"\\{0, 2, 4, 6, \\dots\\}",[213,1421,1423],{"className":1422,"ariaHidden":251},[286],[213,1424,1426,1429,1432,1435,1438,1441,1444,1447,1450,1453,1456,1459,1462,1465,1468,1471,1474],{"className":1425},[290],[213,1427],{"className":1428,"style":319},[294],[213,1430,244],{"className":1431},[323],[213,1433,248],{"className":1434},[299],[213,1436,252],{"className":1437},[330],[213,1439],{"className":1440,"style":334},[304],[213,1442,260],{"className":1443},[299],[213,1445,252],{"className":1446},[330],[213,1448],{"className":1449,"style":334},[304],[213,1451,1403],{"className":1452},[299],[213,1454,252],{"className":1455},[330],[213,1457],{"className":1458,"style":334},[304],[213,1460,1408],{"className":1461},[299],[213,1463,252],{"className":1464},[330],[213,1466],{"className":1467,"style":334},[304],[213,1469,270],{"className":1470},[365],[213,1472],{"className":1473,"style":334},[304],[213,1475,277],{"className":1476},[372]," 可数：双射为 ",[213,1479,1481,1508],{"className":1480,"translate":217},[216],[213,1482,1484],{"className":1483},[221],[223,1485,1486],{"xmlns":225},[227,1487,1488,1505],{},[230,1489,1490,1492,1494,1497,1499,1501,1503],{},[233,1491,454],{},[238,1493,560],{"stretchy":243},[233,1495,1496],{},"n",[238,1498,569],{"stretchy":243},[238,1500,240],{},[246,1502,260],{},[233,1504,1496],{},[279,1506,1507],{"encoding":281},"f(n) = 2n",[213,1509,1511,1538],{"className":1510,"ariaHidden":251},[286],[213,1512,1514,1517,1520,1523,1526,1529,1532,1535],{"className":1513},[290],[213,1515],{"className":1516,"style":319},[294],[213,1518,454],{"className":1519,"style":481},[299,407],[213,1521,560],{"className":1522},[323],[213,1524,1496],{"className":1525},[299,407],[213,1527,569],{"className":1528},[372],[213,1530],{"className":1531,"style":305},[304],[213,1533,240],{"className":1534},[309],[213,1536],{"className":1537,"style":305},[304],[213,1539,1541,1545,1548],{"className":1540},[290],[213,1542],{"className":1543,"style":1544},[294],"height:0.6444em;",[213,1546,260],{"className":1547},[299],[213,1549,1496],{"className":1550},[299,407],"。",[65,1553,1554,1555,1714,1715,2029,2030,2125,2126,1551],{},"整数集 ",[213,1556,1558,1612],{"className":1557,"translate":217},[216],[213,1559,1561],{"className":1560},[221],[223,1562,1563],{"xmlns":225},[227,1564,1565,1609],{},[230,1566,1567,1570,1572,1574,1576,1578,1581,1583,1585,1587,1589,1591,1593,1595,1597,1599,1601,1603,1605,1607],{},[233,1568,1569],{"mathvariant":235},"Z",[238,1571,240],{},[238,1573,244],{"stretchy":243},[238,1575,270],{},[238,1577,252],{"separator":251},[238,1579,1580],{},"−",[246,1582,260],{},[238,1584,252],{"separator":251},[238,1586,1580],{},[246,1588,255],{},[238,1590,252],{"separator":251},[246,1592,248],{},[238,1594,252],{"separator":251},[246,1596,255],{},[238,1598,252],{"separator":251},[246,1600,260],{},[238,1602,252],{"separator":251},[238,1604,270],{},[272,1606,274],{},[238,1608,277],{"stretchy":243},[279,1610,1611],{"encoding":281},"\\mathbb{Z} = \\{\\dots, -2, -1, 0, 1, 2, \\dots\\}",[213,1613,1615,1633],{"className":1614,"ariaHidden":251},[286],[213,1616,1618,1621,1624,1627,1630],{"className":1617},[290],[213,1619],{"className":1620,"style":295},[294],[213,1622,1569],{"className":1623},[299,300],[213,1625],{"className":1626,"style":305},[304],[213,1628,240],{"className":1629},[309],[213,1631],{"className":1632,"style":305},[304],[213,1634,1636,1639,1642,1645,1648,1651,1654,1657,1660,1663,1666,1669,1672,1675,1678,1681,1684,1687,1690,1693,1696,1699,1702,1705,1708,1711],{"className":1635},[290],[213,1637],{"className":1638,"style":319},[294],[213,1640,244],{"className":1641},[323],[213,1643,270],{"className":1644},[365],[213,1646],{"className":1647,"style":334},[304],[213,1649,252],{"className":1650},[330],[213,1652],{"className":1653,"style":334},[304],[213,1655,1580],{"className":1656},[299],[213,1658,260],{"className":1659},[299],[213,1661,252],{"className":1662},[330],[213,1664],{"className":1665,"style":334},[304],[213,1667,1580],{"className":1668},[299],[213,1670,255],{"className":1671},[299],[213,1673,252],{"className":1674},[330],[213,1676],{"className":1677,"style":334},[304],[213,1679,248],{"className":1680},[299],[213,1682,252],{"className":1683},[330],[213,1685],{"className":1686,"style":334},[304],[213,1688,255],{"className":1689},[299],[213,1691,252],{"className":1692},[330],[213,1694],{"className":1695,"style":334},[304],[213,1697,260],{"className":1698},[299],[213,1700,252],{"className":1701},[330],[213,1703],{"className":1704,"style":334},[304],[213,1706,270],{"className":1707},[365],[213,1709],{"className":1710,"style":334},[304],[213,1712,277],{"className":1713},[372]," 可数：可以用 ",[213,1716,1718,1815],{"className":1717,"translate":217},[216],[213,1719,1721],{"className":1720},[221],[223,1722,1723],{"xmlns":225},[227,1724,1725,1812],{},[230,1726,1727,1729,1731,1733,1735,1737],{},[233,1728,454],{},[238,1730,560],{"stretchy":243},[233,1732,1496],{},[238,1734,569],{"stretchy":243},[238,1736,240],{},[230,1738,1739,1741],{},[238,1740,244],{"fence":251},[1742,1743,1747,1776],"mtable",{"rowspacing":1744,"columnalign":1745,"columnspacing":1746},"0.36em","left left","1em",[1748,1749,1750,1765],"mtr",{},[1751,1752,1753],"mtd",{},[1754,1755,1756],"mstyle",{"scriptlevel":248,"displaystyle":243},[230,1757,1758,1760,1763],{},[233,1759,1496],{},[233,1761,1762],{"mathvariant":545},"\u002F",[246,1764,260],{},[1751,1766,1767],{},[1754,1768,1769],{"scriptlevel":248,"displaystyle":243},[230,1770,1771,1773],{},[233,1772,1496],{},[272,1774,1775],{}," 为偶数",[1748,1777,1778,1801],{},[1751,1779,1780],{},[1754,1781,1782],{"scriptlevel":248,"displaystyle":243},[230,1783,1784,1786,1788,1790,1793,1795,1797,1799],{},[238,1785,1580],{},[238,1787,560],{"stretchy":243},[233,1789,1496],{},[238,1791,1792],{},"+",[246,1794,255],{},[238,1796,569],{"stretchy":243},[233,1798,1762],{"mathvariant":545},[246,1800,260],{},[1751,1802,1803],{},[1754,1804,1805],{"scriptlevel":248,"displaystyle":243},[230,1806,1807,1809],{},[233,1808,1496],{},[272,1810,1811],{}," 为奇数",[279,1813,1814],{"encoding":281},"f(n) = \\begin{cases} n\u002F2 & n \\text{ 为偶数} \\\\ -(n+1)\u002F2 & n \\text{ 为奇数} \\end{cases}",[213,1816,1818,1845],{"className":1817,"ariaHidden":251},[286],[213,1819,1821,1824,1827,1830,1833,1836,1839,1842],{"className":1820},[290],[213,1822],{"className":1823,"style":319},[294],[213,1825,454],{"className":1826,"style":481},[299,407],[213,1828,560],{"className":1829},[323],[213,1831,1496],{"className":1832},[299,407],[213,1834,569],{"className":1835},[372],[213,1837],{"className":1838,"style":305},[304],[213,1840,240],{"className":1841},[309],[213,1843],{"className":1844,"style":305},[304],[213,1846,1848,1852],{"className":1847},[290],[213,1849],{"className":1850,"style":1851},[294],"height:3em;vertical-align:-1.25em;",[213,1853,1855,1865,2025],{"className":1854},[365],[213,1856,1860],{"className":1857,"style":1859},[323,1858],"delimcenter","top:0em;",[213,1861,244],{"className":1862},[1863,1864],"delimsizing","size4",[213,1866,1868],{"className":1867},[299],[213,1869,1871,1952,1957],{"className":1870},[1742],[213,1872,1875],{"className":1873},[1874],"col-align-l",[213,1876,1878,1943],{"className":1877},[609,610],[213,1879,1881,1940],{"className":1880},[614],[213,1882,1885,1902],{"className":1883,"style":1884},[618],"height:1.69em;",[213,1886,1888,1892],{"style":1887},"top:-3.69em;",[213,1889],{"className":1890,"style":1891},[626],"height:3.008em;",[213,1893,1895,1898],{"className":1894},[299],[213,1896,1496],{"className":1897},[299,407],[213,1899,1901],{"className":1900},[299],"\u002F2",[213,1903,1905,1908],{"style":1904},"top:-2.25em;",[213,1906],{"className":1907,"style":1891},[626],[213,1909,1911,1914,1917,1920,1924,1928,1931,1934,1937],{"className":1910},[299],[213,1912,1580],{"className":1913},[299],[213,1915,560],{"className":1916},[323],[213,1918,1496],{"className":1919},[299,407],[213,1921],{"className":1922,"style":1923},[304],"margin-right:0.2222em;",[213,1925,1792],{"className":1926},[1927],"mbin",[213,1929],{"className":1930,"style":1923},[304],[213,1932,255],{"className":1933},[299],[213,1935,569],{"className":1936},[372],[213,1938,1901],{"className":1939},[299],[213,1941,642],{"className":1942},[641],[213,1944,1946],{"className":1945},[614],[213,1947,1950],{"className":1948,"style":1949},[618],"height:1.19em;",[213,1951],{},[213,1953],{"className":1954,"style":1956},[1955],"arraycolsep","width:1em;",[213,1958,1960],{"className":1959},[1874],[213,1961,1963,2017],{"className":1962},[609,610],[213,1964,1966,2014],{"className":1965},[614],[213,1967,1969,1993],{"className":1968,"style":1884},[618],[213,1970,1971,1974],{"style":1887},[213,1972],{"className":1973,"style":1891},[626],[213,1975,1977,1980],{"className":1976},[299],[213,1978,1496],{"className":1979},[299,407],[213,1981,1984,1988],{"className":1982},[299,1983],"text",[213,1985,1987],{"className":1986},[299]," ",[213,1989,1992],{"className":1990},[299,1991],"cjk_fallback","为偶数",[213,1994,1995,1998],{"style":1904},[213,1996],{"className":1997,"style":1891},[626],[213,1999,2001,2004],{"className":2000},[299],[213,2002,1496],{"className":2003},[299,407],[213,2005,2007,2010],{"className":2006},[299,1983],[213,2008,1987],{"className":2009},[299],[213,2011,2013],{"className":2012},[299,1991],"为奇数",[213,2015,642],{"className":2016},[641],[213,2018,2020],{"className":2019},[614],[213,2021,2023],{"className":2022,"style":1949},[618],[213,2024],{},[213,2026],{"className":2027},[372,2028],"nulldelimiter"," 把 ",[213,2031,2033,2067],{"className":2032,"translate":217},[216],[213,2034,2036],{"className":2035},[221],[223,2037,2038],{"xmlns":225},[227,2039,2040,2064],{},[230,2041,2042,2044,2046,2048,2050,2052,2054,2056,2058,2060,2062],{},[246,2043,248],{},[238,2045,252],{"separator":251},[246,2047,255],{},[238,2049,252],{"separator":251},[246,2051,260],{},[238,2053,252],{"separator":251},[246,2055,265],{},[238,2057,252],{"separator":251},[246,2059,1403],{},[238,2061,252],{"separator":251},[238,2063,270],{},[279,2065,2066],{"encoding":281},"0,1,2,3,4,\\dots",[213,2068,2070],{"className":2069,"ariaHidden":251},[286],[213,2071,2073,2077,2080,2083,2086,2089,2092,2095,2098,2101,2104,2107,2110,2113,2116,2119,2122],{"className":2072},[290],[213,2074],{"className":2075,"style":2076},[294],"height:0.8389em;vertical-align:-0.1944em;",[213,2078,248],{"className":2079},[299],[213,2081,252],{"className":2082},[330],[213,2084],{"className":2085,"style":334},[304],[213,2087,255],{"className":2088},[299],[213,2090,252],{"className":2091},[330],[213,2093],{"className":2094,"style":334},[304],[213,2096,260],{"className":2097},[299],[213,2099,252],{"className":2100},[330],[213,2102],{"className":2103,"style":334},[304],[213,2105,265],{"className":2106},[299],[213,2108,252],{"className":2109},[330],[213,2111],{"className":2112,"style":334},[304],[213,2114,1403],{"className":2115},[299],[213,2117,252],{"className":2118},[330],[213,2120],{"className":2121,"style":334},[304],[213,2123,270],{"className":2124},[365]," 映射到 ",[213,2127,2129,2167],{"className":2128,"translate":217},[216],[213,2130,2132],{"className":2131},[221],[223,2133,2134],{"xmlns":225},[227,2135,2136,2164],{},[230,2137,2138,2140,2142,2144,2146,2148,2150,2152,2154,2156,2158,2160,2162],{},[246,2139,248],{},[238,2141,252],{"separator":251},[238,2143,1580],{},[246,2145,255],{},[238,2147,252],{"separator":251},[246,2149,255],{},[238,2151,252],{"separator":251},[238,2153,1580],{},[246,2155,260],{},[238,2157,252],{"separator":251},[246,2159,260],{},[238,2161,252],{"separator":251},[238,2163,270],{},[279,2165,2166],{"encoding":281},"0,-1,1,-2,2,\\dots",[213,2168,2170],{"className":2169,"ariaHidden":251},[286],[213,2171,2173,2176,2179,2182,2185,2188,2191,2194,2197,2200,2203,2206,2209,2212,2215,2218,2221,2224,2227],{"className":2172},[290],[213,2174],{"className":2175,"style":2076},[294],[213,2177,248],{"className":2178},[299],[213,2180,252],{"className":2181},[330],[213,2183],{"className":2184,"style":334},[304],[213,2186,1580],{"className":2187},[299],[213,2189,255],{"className":2190},[299],[213,2192,252],{"className":2193},[330],[213,2195],{"className":2196,"style":334},[304],[213,2198,255],{"className":2199},[299],[213,2201,252],{"className":2202},[330],[213,2204],{"className":2205,"style":334},[304],[213,2207,1580],{"className":2208},[299],[213,2210,260],{"className":2211},[299],[213,2213,252],{"className":2214},[330],[213,2216],{"className":2217,"style":334},[304],[213,2219,260],{"className":2220},[299],[213,2222,252],{"className":2223},[330],[213,2225],{"className":2226,"style":334},[304],[213,2228,270],{"className":2229},[365],[65,2231,2232,2233,2264,2265,2306],{},"有理数集 ",[213,2234,2236,2251],{"className":2235,"translate":217},[216],[213,2237,2239],{"className":2238},[221],[223,2240,2241],{"xmlns":225},[227,2242,2243,2248],{},[230,2244,2245],{},[233,2246,2247],{"mathvariant":235},"Q",[279,2249,2250],{"encoding":281},"\\mathbb{Q}",[213,2252,2254],{"className":2253,"ariaHidden":251},[286],[213,2255,2257,2261],{"className":2256},[290],[213,2258],{"className":2259,"style":2260},[294],"height:0.8556em;vertical-align:-0.1667em;",[213,2262,2247],{"className":2263},[299,300]," 同样可数（可以用「对角线枚举法」把所有分数 ",[213,2266,2268,2287],{"className":2267,"translate":217},[216],[213,2269,2271],{"className":2270},[221],[223,2272,2273],{"xmlns":225},[227,2274,2275,2284],{},[230,2276,2277,2279,2281],{},[233,2278,11],{},[233,2280,1762],{"mathvariant":545},[233,2282,2283],{},"q",[279,2285,2286],{"encoding":281},"p\u002Fq",[213,2288,2290],{"className":2289,"ariaHidden":251},[286],[213,2291,2293,2296,2299,2302],{"className":2292},[290],[213,2294],{"className":2295,"style":319},[294],[213,2297,11],{"className":2298},[299,407],[213,2300,1762],{"className":2301},[299],[213,2303,2283],{"className":2304,"style":2305},[299,407],"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）。",[11,2308,2309],{},"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。",[30,2311,2312],{"id":2312},"实数的不可数",[21,2314,2315],{},[11,2316,2317,2318,2370],{},"定理（康托尔，1891）。区间 ",[213,2319,2321,2343],{"className":2320,"translate":217},[216],[213,2322,2324],{"className":2323},[221],[223,2325,2326],{"xmlns":225},[227,2327,2328,2340],{},[230,2329,2330,2332,2334,2336,2338],{},[238,2331,560],{"stretchy":243},[246,2333,248],{},[238,2335,252],{"separator":251},[246,2337,255],{},[238,2339,569],{"stretchy":243},[279,2341,2342],{"encoding":281},"(0,1)",[213,2344,2346],{"className":2345,"ariaHidden":251},[286],[213,2347,2349,2352,2355,2358,2361,2364,2367],{"className":2348},[290],[213,2350],{"className":2351,"style":319},[294],[213,2353,560],{"className":2354},[323],[213,2356,248],{"className":2357},[299],[213,2359,252],{"className":2360},[330],[213,2362],{"className":2363,"style":334},[304],[213,2365,255],{"className":2366},[299],[213,2368,569],{"className":2369},[372]," 中的实数是不可数的。",[11,2372,2373,2374,2425],{},"这意味着：不存在任何方法能把 ",[213,2375,2377,2398],{"className":2376,"translate":217},[216],[213,2378,2380],{"className":2379},[221],[223,2381,2382],{"xmlns":225},[227,2383,2384,2396],{},[230,2385,2386,2388,2390,2392,2394],{},[238,2387,560],{"stretchy":243},[246,2389,248],{},[238,2391,252],{"separator":251},[246,2393,255],{},[238,2395,569],{"stretchy":243},[279,2397,2342],{"encoding":281},[213,2399,2401],{"className":2400,"ariaHidden":251},[286],[213,2402,2404,2407,2410,2413,2416,2419,2422],{"className":2403},[290],[213,2405],{"className":2406,"style":319},[294],[213,2408,560],{"className":2409},[323],[213,2411,248],{"className":2412},[299],[213,2414,252],{"className":2415},[330],[213,2417],{"className":2418,"style":334},[304],[213,2420,255],{"className":2421},[299],[213,2423,569],{"className":2424},[372]," 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。",[11,2427,2428],{},"我们采用反证法。",[11,2430,2431,2432,2483,2484,2580],{},"假设：",[213,2433,2435,2456],{"className":2434,"translate":217},[216],[213,2436,2438],{"className":2437},[221],[223,2439,2440],{"xmlns":225},[227,2441,2442,2454],{},[230,2443,2444,2446,2448,2450,2452],{},[238,2445,560],{"stretchy":243},[246,2447,248],{},[238,2449,252],{"separator":251},[246,2451,255],{},[238,2453,569],{"stretchy":243},[279,2455,2342],{"encoding":281},[213,2457,2459],{"className":2458,"ariaHidden":251},[286],[213,2460,2462,2465,2468,2471,2474,2477,2480],{"className":2461},[290],[213,2463],{"className":2464,"style":319},[294],[213,2466,560],{"className":2467},[323],[213,2469,248],{"className":2470},[299],[213,2472,252],{"className":2473},[330],[213,2475],{"className":2476,"style":334},[304],[213,2478,255],{"className":2479},[299],[213,2481,569],{"className":2482},[372]," 中的实数可数。即存在双射 ",[213,2485,2487,2517],{"className":2486,"translate":217},[216],[213,2488,2490],{"className":2489},[221],[223,2491,2492],{"xmlns":225},[227,2493,2494,2514],{},[230,2495,2496,2498,2500,2502,2504,2506,2508,2510,2512],{},[233,2497,454],{},[238,2499,457],{},[233,2501,236],{"mathvariant":235},[238,2503,462],{},[238,2505,560],{"stretchy":243},[246,2507,248],{},[238,2509,252],{"separator":251},[246,2511,255],{},[238,2513,569],{"stretchy":243},[279,2515,2516],{"encoding":281},"f: \\mathbb{N} \\to (0,1)",[213,2518,2520,2538,2556],{"className":2519,"ariaHidden":251},[286],[213,2521,2523,2526,2529,2532,2535],{"className":2522},[290],[213,2524],{"className":2525,"style":477},[294],[213,2527,454],{"className":2528,"style":481},[299,407],[213,2530],{"className":2531,"style":305},[304],[213,2533,457],{"className":2534},[309],[213,2536],{"className":2537,"style":305},[304],[213,2539,2541,2544,2547,2550,2553],{"className":2540},[290],[213,2542],{"className":2543,"style":295},[294],[213,2545,236],{"className":2546},[299,300],[213,2548],{"className":2549,"style":305},[304],[213,2551,462],{"className":2552},[309],[213,2554],{"className":2555,"style":305},[304],[213,2557,2559,2562,2565,2568,2571,2574,2577],{"className":2558},[290],[213,2560],{"className":2561,"style":319},[294],[213,2563,560],{"className":2564},[323],[213,2566,248],{"className":2567},[299],[213,2569,252],{"className":2570},[330],[213,2572],{"className":2573,"style":334},[304],[213,2575,255],{"className":2576},[299],[213,2578,569],{"className":2579},[372],"，使得我们可以把这些实数全部排成一个无穷列表：",[213,2582,2585],{"className":2583,"translate":217},[2584],"katex-display",[213,2586,2588,2878],{"className":2587,"translate":217},[216],[213,2589,2591],{"className":2590},[221],[223,2592,2594],{"xmlns":225,"display":2593},"block",[227,2595,2596,2875],{},[1742,2597,2601,2664,2724,2784,2844],{"rowspacing":2598,"columnalign":2599,"columnspacing":2600},"0.25em","right left","0em",[1748,2602,2603,2614],{},[1751,2604,2605],{},[1754,2606,2607],{"scriptlevel":248,"displaystyle":251},[537,2608,2609,2612],{},[233,2610,2611],{},"x",[246,2613,255],{},[1751,2615,2616],{},[1754,2617,2618],{"scriptlevel":248,"displaystyle":251},[230,2619,2620,2622,2624,2627,2635,2637,2644,2646,2653,2655,2662],{},[230,2621],{},[238,2623,240],{},[246,2625,2626],{},"0.",[537,2628,2629,2632],{},[233,2630,2631],{},"d",[246,2633,2634],{},"11",[272,2636,274],{},[537,2638,2639,2641],{},[233,2640,2631],{},[246,2642,2643],{},"12",[272,2645,274],{},[537,2647,2648,2650],{},[233,2649,2631],{},[246,2651,2652],{},"13",[272,2654,274],{},[537,2656,2657,2659],{},[233,2658,2631],{},[246,2660,2661],{},"14",[238,2663,270],{},[1748,2665,2666,2676],{},[1751,2667,2668],{},[1754,2669,2670],{"scriptlevel":248,"displaystyle":251},[537,2671,2672,2674],{},[233,2673,2611],{},[246,2675,260],{},[1751,2677,2678],{},[1754,2679,2680],{"scriptlevel":248,"displaystyle":251},[230,2681,2682,2684,2686,2688,2695,2697,2704,2706,2713,2715,2722],{},[230,2683],{},[238,2685,240],{},[246,2687,2626],{},[537,2689,2690,2692],{},[233,2691,2631],{},[246,2693,2694],{},"21",[272,2696,274],{},[537,2698,2699,2701],{},[233,2700,2631],{},[246,2702,2703],{},"22",[272,2705,274],{},[537,2707,2708,2710],{},[233,2709,2631],{},[246,2711,2712],{},"23",[272,2714,274],{},[537,2716,2717,2719],{},[233,2718,2631],{},[246,2720,2721],{},"24",[238,2723,270],{},[1748,2725,2726,2736],{},[1751,2727,2728],{},[1754,2729,2730],{"scriptlevel":248,"displaystyle":251},[537,2731,2732,2734],{},[233,2733,2611],{},[246,2735,265],{},[1751,2737,2738],{},[1754,2739,2740],{"scriptlevel":248,"displaystyle":251},[230,2741,2742,2744,2746,2748,2755,2757,2764,2766,2773,2775,2782],{},[230,2743],{},[238,2745,240],{},[246,2747,2626],{},[537,2749,2750,2752],{},[233,2751,2631],{},[246,2753,2754],{},"31",[272,2756,274],{},[537,2758,2759,2761],{},[233,2760,2631],{},[246,2762,2763],{},"32",[272,2765,274],{},[537,2767,2768,2770],{},[233,2769,2631],{},[246,2771,2772],{},"33",[272,2774,274],{},[537,2776,2777,2779],{},[233,2778,2631],{},[246,2780,2781],{},"34",[238,2783,270],{},[1748,2785,2786,2796],{},[1751,2787,2788],{},[1754,2789,2790],{"scriptlevel":248,"displaystyle":251},[537,2791,2792,2794],{},[233,2793,2611],{},[246,2795,1403],{},[1751,2797,2798],{},[1754,2799,2800],{"scriptlevel":248,"displaystyle":251},[230,2801,2802,2804,2806,2808,2815,2817,2824,2826,2833,2835,2842],{},[230,2803],{},[238,2805,240],{},[246,2807,2626],{},[537,2809,2810,2812],{},[233,2811,2631],{},[246,2813,2814],{},"41",[272,2816,274],{},[537,2818,2819,2821],{},[233,2820,2631],{},[246,2822,2823],{},"42",[272,2825,274],{},[537,2827,2828,2830],{},[233,2829,2631],{},[246,2831,2832],{},"43",[272,2834,274],{},[537,2836,2837,2839],{},[233,2838,2631],{},[246,2840,2841],{},"44",[238,2843,270],{},[1748,2845,2846,2852],{},[1751,2847,2848],{},[1754,2849,2850],{"scriptlevel":248,"displaystyle":251},[230,2851],{},[1751,2853,2854],{},[1754,2855,2856],{"scriptlevel":248,"displaystyle":251},[230,2857,2858,2860,2863],{},[230,2859],{},[272,2861,2862],{},"  ",[230,2864,2865,2868],{},[233,2866,2867],{"mathvariant":545},"⋮",[2869,2870,2871],"mpadded",{"height":2600,"voffset":2600},[304,2872],{"mathbackground":2873,"width":2600,"height":2874},"black","1.5em",[279,2876,2877],{"encoding":281},"\\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}",[213,2879,2881],{"className":2880,"ariaHidden":251},[286],[213,2882,2884,2888],{"className":2883},[290],[213,2885],{"className":2886,"style":2887},[294],"height:7.86em;vertical-align:-3.68em;",[213,2889,2891],{"className":2890},[299],[213,2892,2894,3126],{"className":2893},[1742],[213,2895,2898],{"className":2896},[2897],"col-align-r",[213,2899,2901,3117],{"className":2900},[609,610],[213,2902,2904,3114],{"className":2903},[614],[213,2905,2908,2958,3007,3056,3105],{"className":2906,"style":2907},[618],"height:4.18em;",[213,2909,2911,2915],{"style":2910},"top:-6.84em;",[213,2912],{"className":2913,"style":2914},[626],"height:3.5em;",[213,2916,2918],{"className":2917},[299],[213,2919,2921,2924],{"className":2920},[299],[213,2922,2611],{"className":2923},[299,407],[213,2925,2927],{"className":2926},[605],[213,2928,2930,2950],{"className":2929},[609,610],[213,2931,2933,2947],{"className":2932},[614],[213,2934,2936],{"className":2935,"style":619},[618],[213,2937,2938,2941],{"style":622},[213,2939],{"className":2940,"style":627},[626],[213,2942,2944],{"className":2943},[631,632,633,634],[213,2945,255],{"className":2946},[299,634],[213,2948,642],{"className":2949},[641],[213,2951,2953],{"className":2952},[614],[213,2954,2956],{"className":2955,"style":649},[618],[213,2957],{},[213,2959,2961,2964],{"style":2960},"top:-5.34em;",[213,2962],{"className":2963,"style":2914},[626],[213,2965,2967],{"className":2966},[299],[213,2968,2970,2973],{"className":2969},[299],[213,2971,2611],{"className":2972},[299,407],[213,2974,2976],{"className":2975},[605],[213,2977,2979,2999],{"className":2978},[609,610],[213,2980,2982,2996],{"className":2981},[614],[213,2983,2985],{"className":2984,"style":619},[618],[213,2986,2987,2990],{"style":622},[213,2988],{"className":2989,"style":627},[626],[213,2991,2993],{"className":2992},[631,632,633,634],[213,2994,260],{"className":2995},[299,634],[213,2997,642],{"className":2998},[641],[213,3000,3002],{"className":3001},[614],[213,3003,3005],{"className":3004,"style":649},[618],[213,3006],{},[213,3008,3010,3013],{"style":3009},"top:-3.84em;",[213,3011],{"className":3012,"style":2914},[626],[213,3014,3016],{"className":3015},[299],[213,3017,3019,3022],{"className":3018},[299],[213,3020,2611],{"className":3021},[299,407],[213,3023,3025],{"className":3024},[605],[213,3026,3028,3048],{"className":3027},[609,610],[213,3029,3031,3045],{"className":3030},[614],[213,3032,3034],{"className":3033,"style":619},[618],[213,3035,3036,3039],{"style":622},[213,3037],{"className":3038,"style":627},[626],[213,3040,3042],{"className":3041},[631,632,633,634],[213,3043,265],{"className":3044},[299,634],[213,3046,642],{"className":3047},[641],[213,3049,3051],{"className":3050},[614],[213,3052,3054],{"className":3053,"style":649},[618],[213,3055],{},[213,3057,3059,3062],{"style":3058},"top:-2.34em;",[213,3060],{"className":3061,"style":2914},[626],[213,3063,3065],{"className":3064},[299],[213,3066,3068,3071],{"className":3067},[299],[213,3069,2611],{"className":3070},[299,407],[213,3072,3074],{"className":3073},[605],[213,3075,3077,3097],{"className":3076},[609,610],[213,3078,3080,3094],{"className":3079},[614],[213,3081,3083],{"className":3082,"style":619},[618],[213,3084,3085,3088],{"style":622},[213,3086],{"className":3087,"style":627},[626],[213,3089,3091],{"className":3090},[631,632,633,634],[213,3092,1403],{"className":3093},[299,634],[213,3095,642],{"className":3096},[641],[213,3098,3100],{"className":3099},[614],[213,3101,3103],{"className":3102,"style":649},[618],[213,3104],{},[213,3106,3108,3111],{"style":3107},"top:-0.18em;",[213,3109],{"className":3110,"style":2914},[626],[213,3112],{"className":3113},[299],[213,3115,642],{"className":3116},[641],[213,3118,3120],{"className":3119},[614],[213,3121,3124],{"className":3122,"style":3123},[618],"height:3.68em;",[213,3125],{},[213,3127,3129],{"className":3128},[1874],[213,3130,3132,4015],{"className":3131},[609,610],[213,3133,3135,4012],{"className":3134},[614],[213,3136,3138,3350,3561,3772,3983],{"className":3137,"style":2907},[618],[213,3139,3141,3145],{"style":3140},"top:-7.0275em;",[213,3142],{"className":3143,"style":3144},[626],"height:3.6875em;",[213,3146,3148,3151,3154,3157,3160,3163,3206,3209,3252,3255,3298,3301,3344,3347],{"className":3147},[299],[213,3149],{"className":3150},[299],[213,3152],{"className":3153,"style":305},[304],[213,3155,240],{"className":3156},[309],[213,3158],{"className":3159,"style":305},[304],[213,3161,2626],{"className":3162},[299],[213,3164,3166,3169],{"className":3165},[299],[213,3167,2631],{"className":3168},[299,407],[213,3170,3172],{"className":3171},[605],[213,3173,3175,3198],{"className":3174},[609,610],[213,3176,3178,3195],{"className":3177},[614],[213,3179,3181],{"className":3180,"style":619},[618],[213,3182,3183,3186],{"style":622},[213,3184],{"className":3185,"style":627},[626],[213,3187,3189],{"className":3188},[631,632,633,634],[213,3190,3192],{"className":3191},[299,634],[213,3193,2634],{"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",[213,4027,4029,4053],{"className":4028,"translate":217},[216],[213,4030,4032],{"className":4031},[221],[223,4033,4034],{"xmlns":225},[227,4035,4036,4050],{},[230,4037,4038],{},[537,4039,4040,4042],{},[233,4041,2631],{},[230,4043,4044,4047],{},[233,4045,4046],{},"i",[233,4048,4049],{},"j",[279,4051,4052],{"encoding":281},"d_{ij}",[213,4054,4056],{"className":4055,"ariaHidden":251},[286],[213,4057,4059,4063],{"className":4058},[290],[213,4060],{"className":4061,"style":4062},[294],"height:0.9805em;vertical-align:-0.2861em;",[213,4064,4066,4069],{"className":4065},[299],[213,4067,2631],{"className":4068},[299,407],[213,4070,4072],{"className":4071},[605],[213,4073,4075,4101],{"className":4074},[609,610],[213,4076,4078,4098],{"className":4077},[614],[213,4079,4082],{"className":4080,"style":4081},[618],"height:0.3117em;",[213,4083,4084,4087],{"style":622},[213,4085],{"className":4086,"style":627},[626],[213,4088,4090],{"className":4089},[631,632,633,634],[213,4091,4093],{"className":4092},[299,634],[213,4094,4097],{"className":4095,"style":4096},[299,407,634],"margin-right:0.0572em;","ij",[213,4099,642],{"className":4100},[641],[213,4102,4104],{"className":4103},[614],[213,4105,4108],{"className":4106,"style":4107},[618],"height:0.2861em;",[213,4109],{}," 表示第 ",[213,4112,4114,4127],{"className":4113,"translate":217},[216],[213,4115,4117],{"className":4116},[221],[223,4118,4119],{"xmlns":225},[227,4120,4121,4125],{},[230,4122,4123],{},[233,4124,4046],{},[279,4126,4046],{"encoding":281},[213,4128,4130],{"className":4129,"ariaHidden":251},[286],[213,4131,4133,4137],{"className":4132},[290],[213,4134],{"className":4135,"style":4136},[294],"height:0.6595em;",[213,4138,4046],{"className":4139},[299,407]," 个数的第 ",[213,4142,4144,4157],{"className":4143,"translate":217},[216],[213,4145,4147],{"className":4146},[221],[223,4148,4149],{"xmlns":225},[227,4150,4151,4155],{},[230,4152,4153],{},[233,4154,4049],{},[279,4156,4049],{"encoding":281},[213,4158,4160],{"className":4159,"ariaHidden":251},[286],[213,4161,4163,4167],{"className":4162},[290],[213,4164],{"className":4165,"style":4166},[294],"height:0.854em;vertical-align:-0.1944em;",[213,4168,4049],{"className":4169,"style":4096},[299,407]," 位小数（取值 ",[213,4172,4174,4187],{"className":4173,"translate":217},[216],[213,4175,4177],{"className":4176},[221],[223,4178,4179],{"xmlns":225},[227,4180,4181,4185],{},[230,4182,4183],{},[246,4184,248],{},[279,4186,248],{"encoding":281},[213,4188,4190],{"className":4189,"ariaHidden":251},[286],[213,4191,4193,4196],{"className":4192},[290],[213,4194],{"className":4195,"style":1544},[294],[213,4197,248],{"className":4198},[299]," 到 ",[213,4201,4203,4217],{"className":4202,"translate":217},[216],[213,4204,4206],{"className":4205},[221],[223,4207,4208],{"xmlns":225},[227,4209,4210,4215],{},[230,4211,4212],{},[246,4213,4214],{},"9",[279,4216,4214],{"encoding":281},[213,4218,4220],{"className":4219,"ariaHidden":251},[286],[213,4221,4223,4226],{"className":4222},[290],[213,4224],{"className":4225,"style":1544},[294],[213,4227,4214],{"className":4228},[299],"）。",[11,4231,4232,4233,4480],{},"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 ",[213,4234,4236,4282],{"className":4235,"translate":217},[216],[213,4237,4239],{"className":4238},[221],[223,4240,4241],{"xmlns":225},[227,4242,4243,4279],{},[230,4244,4245,4248,4250,4252,4259,4265,4271,4277],{},[233,4246,4247],{},"y",[238,4249,240],{},[246,4251,2626],{},[537,4253,4254,4257],{},[233,4255,4256],{},"e",[246,4258,255],{},[537,4260,4261,4263],{},[233,4262,4256],{},[246,4264,260],{},[537,4266,4267,4269],{},[233,4268,4256],{},[246,4270,265],{},[537,4272,4273,4275],{},[233,4274,4256],{},[246,4276,1403],{},[238,4278,270],{},[279,4280,4281],{"encoding":281},"y = 0.e_1 e_2 e_3 e_4 \\dots",[213,4283,4285,4304],{"className":4284,"ariaHidden":251},[286],[213,4286,4288,4292,4295,4298,4301],{"className":4287},[290],[213,4289],{"className":4290,"style":4291},[294],"height:0.625em;vertical-align:-0.1944em;",[213,4293,4247],{"className":4294,"style":2305},[299,407],[213,4296],{"className":4297,"style":305},[304],[213,4299,240],{"className":4300},[309],[213,4302],{"className":4303,"style":305},[304],[213,4305,4307,4311,4314,4354,4394,4434,4474,4477],{"className":4306},[290],[213,4308],{"className":4309,"style":4310},[294],"height:0.7944em;vertical-align:-0.15em;",[213,4312,2626],{"className":4313},[299],[213,4315,4317,4320],{"className":4316},[299],[213,4318,4256],{"className":4319},[299,407],[213,4321,4323],{"className":4322},[605],[213,4324,4326,4346],{"className":4325},[609,610],[213,4327,4329,4343],{"className":4328},[614],[213,4330,4332],{"className":4331,"style":619},[618],[213,4333,4334,4337],{"style":622},[213,4335],{"className":4336,"style":627},[626],[213,4338,4340],{"className":4339},[631,632,633,634],[213,4341,255],{"className":4342},[299,634],[213,4344,642],{"className":4345},[641],[213,4347,4349],{"className":4348},[614],[213,4350,4352],{"className":4351,"style":649},[618],[213,4353],{},[213,4355,4357,4360],{"className":4356},[299],[213,4358,4256],{"className":4359},[299,407],[213,4361,4363],{"className":4362},[605],[213,4364,4366,4386],{"className":4365},[609,610],[213,4367,4369,4383],{"className":4368},[614],[213,4370,4372],{"className":4371,"style":619},[618],[213,4373,4374,4377],{"style":622},[213,4375],{"className":4376,"style":627},[626],[213,4378,4380],{"className":4379},[631,632,633,634],[213,4381,260],{"className":4382},[299,634],[213,4384,642],{"className":4385},[641],[213,4387,4389],{"className":4388},[614],[213,4390,4392],{"className":4391,"style":649},[618],[213,4393],{},[213,4395,4397,4400],{"className":4396},[299],[213,4398,4256],{"className":4399},[299,407],[213,4401,4403],{"className":4402},[605],[213,4404,4406,4426],{"className":4405},[609,610],[213,4407,4409,4423],{"className":4408},[614],[213,4410,4412],{"className":4411,"style":619},[618],[213,4413,4414,4417],{"style":622},[213,4415],{"className":4416,"style":627},[626],[213,4418,4420],{"className":4419},[631,632,633,634],[213,4421,265],{"className":4422},[299,634],[213,4424,642],{"className":4425},[641],[213,4427,4429],{"className":4428},[614],[213,4430,4432],{"className":4431,"style":649},[618],[213,4433],{},[213,4435,4437,4440],{"className":4436},[299],[213,4438,4256],{"className":4439},[299,407],[213,4441,4443],{"className":4442},[605],[213,4444,4446,4466],{"className":4445},[609,610],[213,4447,4449,4463],{"className":4448},[614],[213,4450,4452],{"className":4451,"style":619},[618],[213,4453,4454,4457],{"style":622},[213,4455],{"className":4456,"style":627},[626],[213,4458,4460],{"className":4459},[631,632,633,634],[213,4461,1403],{"className":4462},[299,634],[213,4464,642],{"className":4465},[641],[213,4467,4469],{"className":4468},[614],[213,4470,4472],{"className":4471,"style":649},[618],[213,4473],{},[213,4475],{"className":4476,"style":334},[304],[213,4478,270],{"className":4479},[365],"：",[213,4482,4484],{"className":4483,"translate":217},[2584],[213,4485,4487,4575],{"className":4486,"translate":217},[216],[213,4488,4490],{"className":4489},[221],[223,4491,4492],{"xmlns":225,"display":2593},[227,4493,4494,4572],{},[230,4495,4496,4502,4504],{},[537,4497,4498,4500],{},[233,4499,4256],{},[233,4501,1496],{},[238,4503,240],{},[230,4505,4506,4508],{},[238,4507,244],{"fence":251},[1742,4509,4510,4542],{"rowspacing":1744,"columnalign":1745,"columnspacing":1746},[1748,4511,4512,4519],{},[1751,4513,4514],{},[1754,4515,4516],{"scriptlevel":248,"displaystyle":243},[246,4517,4518],{},"5",[1751,4520,4521],{},[1754,4522,4523],{"scriptlevel":248,"displaystyle":243},[230,4524,4525,4528,4538,4540],{},[272,4526,4527],{},"若 ",[537,4529,4530,4532],{},[233,4531,2631],{},[230,4533,4534,4536],{},[233,4535,1496],{},[233,4537,1496],{},[238,4539,546],{"mathvariant":545},[246,4541,4518],{},[1748,4543,4544,4550],{},[1751,4545,4546],{},[1754,4547,4548],{"scriptlevel":248,"displaystyle":243},[246,4549,1408],{},[1751,4551,4552],{},[1754,4553,4554],{"scriptlevel":248,"displaystyle":243},[230,4555,4556,4558,4568,4570],{},[272,4557,4527],{},[537,4559,4560,4562],{},[233,4561,2631],{},[230,4563,4564,4566],{},[233,4565,1496],{},[233,4567,1496],{},[238,4569,240],{},[246,4571,4518],{},[279,4573,4574],{"encoding":281},"e_n = \\begin{cases} 5 & \\text{若 } d_{nn} \\neq 5 \\\\ 6 & \\text{若 } d_{nn} = 5 \\end{cases}",[213,4576,4578,4634],{"className":4577,"ariaHidden":251},[286],[213,4579,4581,4584,4625,4628,4631],{"className":4580},[290],[213,4582],{"className":4583,"style":707},[294],[213,4585,4587,4590],{"className":4586},[299],[213,4588,4256],{"className":4589},[299,407],[213,4591,4593],{"className":4592},[605],[213,4594,4596,4617],{"className":4595},[609,610],[213,4597,4599,4614],{"className":4598},[614],[213,4600,4603],{"className":4601,"style":4602},[618],"height:0.1514em;",[213,4604,4605,4608],{"style":622},[213,4606],{"className":4607,"style":627},[626],[213,4609,4611],{"className":4610},[631,632,633,634],[213,4612,1496],{"className":4613},[299,407,634],[213,4615,642],{"className":4616},[641],[213,4618,4620],{"className":4619},[614],[213,4621,4623],{"className":4622,"style":649},[618],[213,4624],{},[213,4626],{"className":4627,"style":305},[304],[213,4629,240],{"className":4630},[309],[213,4632],{"className":4633,"style":305},[304],[213,4635,4637,4640],{"className":4636},[290],[213,4638],{"className":4639,"style":1851},[294],[213,4641,4643,4649,4905],{"className":4642},[365],[213,4644,4646],{"className":4645,"style":1859},[323,1858],[213,4647,244],{"className":4648},[1863,1864],[213,4650,4652],{"className":4651},[299],[213,4653,4655,4700,4703],{"className":4654},[1742],[213,4656,4658],{"className":4657},[1874],[213,4659,4661,4692],{"className":4660},[609,610],[213,4662,4664,4689],{"className":4663},[614],[213,4665,4667,4678],{"className":4666,"style":1884},[618],[213,4668,4669,4672],{"style":1887},[213,4670],{"className":4671,"style":1891},[626],[213,4673,4675],{"className":4674},[299],[213,4676,4518],{"className":4677},[299],[213,4679,4680,4683],{"style":1904},[213,4681],{"className":4682,"style":1891},[626],[213,4684,4686],{"className":4685},[299],[213,4687,1408],{"className":4688},[299],[213,4690,642],{"className":4691},[641],[213,4693,4695],{"className":4694},[614],[213,4696,4698],{"className":4697,"style":1949},[618],[213,4699],{},[213,4701],{"className":4702,"style":1956},[1955],[213,4704,4706],{"className":4705},[1874],[213,4707,4709,4897],{"className":4708},[609,610],[213,4710,4712,4894],{"className":4711},[614],[213,4713,4715,4822],{"className":4714,"style":1884},[618],[213,4716,4717,4720],{"style":1887},[213,4718],{"className":4719,"style":1891},[626],[213,4721,4723,4733,4777,4780,4816,4819],{"className":4722},[299],[213,4724,4726,4730],{"className":4725},[299,1983],[213,4727,4729],{"className":4728},[299,1991],"若",[213,4731,1987],{"className":4732},[299],[213,4734,4736,4739],{"className":4735},[299],[213,4737,2631],{"className":4738},[299,407],[213,4740,4742],{"className":4741},[605],[213,4743,4745,4769],{"className":4744},[609,610],[213,4746,4748,4766],{"className":4747},[614],[213,4749,4751],{"className":4750,"style":4602},[618],[213,4752,4753,4756],{"style":622},[213,4754],{"className":4755,"style":627},[626],[213,4757,4759],{"className":4758},[631,632,633,634],[213,4760,4762],{"className":4761},[299,634],[213,4763,4765],{"className":4764},[299,407,634],"nn",[213,4767,642],{"className":4768},[641],[213,4770,4772],{"className":4771},[614],[213,4773,4775],{"className":4774,"style":649},[618],[213,4776],{},[213,4778],{"className":4779,"style":305},[304],[213,4781,4783,4810,4813],{"className":4782},[309],[213,4784,4786],{"className":4785},[309],[213,4787,4789],{"className":4788},[299,664],[213,4790,4792],{"className":4791},[668],[213,4793,4795,4798,4807],{"className":4794},[672],[213,4796],{"className":4797,"style":477},[294],[213,4799,4801],{"className":4800},[679],[213,4802,4804],{"className":4803},[299],[213,4805,686],{"className":4806},[309],[213,4808],{"className":4809},[690],[213,4811],{"className":4812},[304,694],[213,4814,240],{"className":4815},[309],[213,4817],{"className":4818,"style":305},[304],[213,4820,4518],{"className":4821},[299],[213,4823,4824,4827],{"style":1904},[213,4825],{"className":4826,"style":1891},[626],[213,4828,4830,4839,4882,4885,4888,4891],{"className":4829},[299],[213,4831,4833,4836],{"className":4832},[299,1983],[213,4834,4729],{"className":4835},[299,1991],[213,4837,1987],{"className":4838},[299],[213,4840,4842,4845],{"className":4841},[299],[213,4843,2631],{"className":4844},[299,407],[213,4846,4848],{"className":4847},[605],[213,4849,4851,4874],{"className":4850},[609,610],[213,4852,4854,4871],{"className":4853},[614],[213,4855,4857],{"className":4856,"style":4602},[618],[213,4858,4859,4862],{"style":622},[213,4860],{"className":4861,"style":627},[626],[213,4863,4865],{"className":4864},[631,632,633,634],[213,4866,4868],{"className":4867},[299,634],[213,4869,4765],{"className":4870},[299,407,634],[213,4872,642],{"className":4873},[641],[213,4875,4877],{"className":4876},[614],[213,4878,4880],{"className":4879,"style":649},[618],[213,4881],{},[213,4883],{"className":4884,"style":305},[304],[213,4886,240],{"className":4887},[309],[213,4889],{"className":4890,"style":305},[304],[213,4892,4518],{"className":4893},[299],[213,4895,642],{"className":4896},[641],[213,4898,4900],{"className":4899},[614],[213,4901,4903],{"className":4902,"style":1949},[618],[213,4904],{},[213,4906],{"className":4907},[372,2028],[11,4909,4910,4911,5085],{},"（这条规则的唯一目的是保证 ",[213,4912,4914,4944],{"className":4913,"translate":217},[216],[213,4915,4917],{"className":4916},[221],[223,4918,4919],{"xmlns":225},[227,4920,4921,4941],{},[230,4922,4923,4929,4931],{},[537,4924,4925,4927],{},[233,4926,4256],{},[233,4928,1496],{},[238,4930,546],{"mathvariant":545},[537,4932,4933,4935],{},[233,4934,2631],{},[230,4936,4937,4939],{},[233,4938,1496],{},[233,4940,1496],{},[279,4942,4943],{"encoding":281},"e_n \\neq d_{nn}",[213,4945,4947,5035],{"className":4946,"ariaHidden":251},[286],[213,4948,4950,4953,4993,4996,5032],{"className":4949},[290],[213,4951],{"className":4952,"style":477},[294],[213,4954,4956,4959],{"className":4955},[299],[213,4957,4256],{"className":4958},[299,407],[213,4960,4962],{"className":4961},[605],[213,4963,4965,4985],{"className":4964},[609,610],[213,4966,4968,4982],{"className":4967},[614],[213,4969,4971],{"className":4970,"style":4602},[618],[213,4972,4973,4976],{"style":622},[213,4974],{"className":4975,"style":627},[626],[213,4977,4979],{"className":4978},[631,632,633,634],[213,4980,1496],{"className":4981},[299,407,634],[213,4983,642],{"className":4984},[641],[213,4986,4988],{"className":4987},[614],[213,4989,4991],{"className":4990,"style":649},[618],[213,4992],{},[213,4994],{"className":4995,"style":305},[304],[213,4997,4999,5026,5029],{"className":4998},[309],[213,5000,5002],{"className":5001},[309],[213,5003,5005],{"className":5004},[299,664],[213,5006,5008],{"className":5007},[668],[213,5009,5011,5014,5023],{"className":5010},[672],[213,5012],{"className":5013,"style":477},[294],[213,5015,5017],{"className":5016},[679],[213,5018,5020],{"className":5019},[299],[213,5021,686],{"className":5022},[309],[213,5024],{"className":5025},[690],[213,5027],{"className":5028},[304,694],[213,5030,240],{"className":5031},[309],[213,5033],{"className":5034,"style":305},[304],[213,5036,5038,5042],{"className":5037},[290],[213,5039],{"className":5040,"style":5041},[294],"height:0.8444em;vertical-align:-0.15em;",[213,5043,5045,5048],{"className":5044},[299],[213,5046,2631],{"className":5047},[299,407],[213,5049,5051],{"className":5050},[605],[213,5052,5054,5077],{"className":5053},[609,610],[213,5055,5057,5074],{"className":5056},[614],[213,5058,5060],{"className":5059,"style":4602},[618],[213,5061,5062,5065],{"style":622},[213,5063],{"className":5064,"style":627},[626],[213,5066,5068],{"className":5067},[631,632,633,634],[213,5069,5071],{"className":5070},[299,634],[213,5072,4765],{"className":5073},[299,407,634],[213,5075,642],{"className":5076},[641],[213,5078,5080],{"className":5079},[614],[213,5081,5083],{"className":5082,"style":649},[618],[213,5084],{},"；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。）",[11,5087,5088,5089,5117],{},"现在关键问题出现了：",[213,5090,5092,5105],{"className":5091,"translate":217},[216],[213,5093,5095],{"className":5094},[221],[223,5096,5097],{"xmlns":225},[227,5098,5099,5103],{},[230,5100,5101],{},[233,5102,4247],{},[279,5104,4247],{"encoding":281},[213,5106,5108],{"className":5107,"ariaHidden":251},[286],[213,5109,5111,5114],{"className":5110},[290],[213,5112],{"className":5113,"style":4291},[294],[213,5115,4247],{"className":5116,"style":2305},[299,407]," 会出现在原来的列表里吗？",[11,5119,5120,5121,5149,5150,5178,5179,5274,5275,4229],{},"假设 ",[213,5122,5124,5137],{"className":5123,"translate":217},[216],[213,5125,5127],{"className":5126},[221],[223,5128,5129],{"xmlns":225},[227,5130,5131,5135],{},[230,5132,5133],{},[233,5134,4247],{},[279,5136,4247],{"encoding":281},[213,5138,5140],{"className":5139,"ariaHidden":251},[286],[213,5141,5143,5146],{"className":5142},[290],[213,5144],{"className":5145,"style":4291},[294],[213,5147,4247],{"className":5148,"style":2305},[299,407]," 出现在列表中，那么 ",[213,5151,5153,5166],{"className":5152,"translate":217},[216],[213,5154,5156],{"className":5155},[221],[223,5157,5158],{"xmlns":225},[227,5159,5160,5164],{},[230,5161,5162],{},[233,5163,4247],{},[279,5165,4247],{"encoding":281},[213,5167,5169],{"className":5168,"ariaHidden":251},[286],[213,5170,5172,5175],{"className":5171},[290],[213,5173],{"className":5174,"style":4291},[294],[213,5176,4247],{"className":5177,"style":2305},[299,407]," 必定等于被枚举的某个数，不妨设 ",[213,5180,5182,5205],{"className":5181,"translate":217},[216],[213,5183,5185],{"className":5184},[221],[223,5186,5187],{"xmlns":225},[227,5188,5189,5202],{},[230,5190,5191,5193,5195],{},[233,5192,4247],{},[238,5194,240],{},[537,5196,5197,5199],{},[233,5198,2611],{},[233,5200,5201],{},"k",[279,5203,5204],{"encoding":281},"y = x_k",[213,5206,5208,5226],{"className":5207,"ariaHidden":251},[286],[213,5209,5211,5214,5217,5220,5223],{"className":5210},[290],[213,5212],{"className":5213,"style":4291},[294],[213,5215,4247],{"className":5216,"style":2305},[299,407],[213,5218],{"className":5219,"style":305},[304],[213,5221,240],{"className":5222},[309],[213,5224],{"className":5225,"style":305},[304],[213,5227,5229,5232],{"className":5228},[290],[213,5230],{"className":5231,"style":707},[294],[213,5233,5235,5238],{"className":5234},[299],[213,5236,2611],{"className":5237},[299,407],[213,5239,5241],{"className":5240},[605],[213,5242,5244,5266],{"className":5243},[609,610],[213,5245,5247,5263],{"className":5246},[614],[213,5248,5251],{"className":5249,"style":5250},[618],"height:0.3361em;",[213,5252,5253,5256],{"style":622},[213,5254],{"className":5255,"style":627},[626],[213,5257,5259],{"className":5258},[631,632,633,634],[213,5260,5201],{"className":5261,"style":5262},[299,407,634],"margin-right:0.0315em;",[213,5264,642],{"className":5265},[641],[213,5267,5269],{"className":5268},[614],[213,5270,5272],{"className":5271,"style":649},[618],[213,5273],{},"（对某个确定的指标 ",[213,5276,5278,5291],{"className":5277,"translate":217},[216],[213,5279,5281],{"className":5280},[221],[223,5282,5283],{"xmlns":225},[227,5284,5285,5289],{},[230,5286,5287],{},[233,5288,5201],{},[279,5290,5201],{"encoding":281},[213,5292,5294],{"className":5293,"ariaHidden":251},[286],[213,5295,5297,5300],{"className":5296},[290],[213,5298],{"className":5299,"style":1084},[294],[213,5301,5201],{"className":5302,"style":5262},[299,407],[11,5304,5305,5306,5334,5335,5363,5364,5434,5435,5504,5505,5585,5586,5334,5656,5684],{},"但根据我们的构造规则，",[213,5307,5309,5322],{"className":5308,"translate":217},[216],[213,5310,5312],{"className":5311},[221],[223,5313,5314],{"xmlns":225},[227,5315,5316,5320],{},[230,5317,5318],{},[233,5319,4247],{},[279,5321,4247],{"encoding":281},[213,5323,5325],{"className":5324,"ariaHidden":251},[286],[213,5326,5328,5331],{"className":5327},[290],[213,5329],{"className":5330,"style":4291},[294],[213,5332,4247],{"className":5333,"style":2305},[299,407]," 的第 ",[213,5336,5338,5351],{"className":5337,"translate":217},[216],[213,5339,5341],{"className":5340},[221],[223,5342,5343],{"xmlns":225},[227,5344,5345,5349],{},[230,5346,5347],{},[233,5348,5201],{},[279,5350,5201],{"encoding":281},[213,5352,5354],{"className":5353,"ariaHidden":251},[286],[213,5355,5357,5360],{"className":5356},[290],[213,5358],{"className":5359,"style":1084},[294],[213,5361,5201],{"className":5362,"style":5262},[299,407]," 位小数是 ",[213,5365,5367,5385],{"className":5366,"translate":217},[216],[213,5368,5370],{"className":5369},[221],[223,5371,5372],{"xmlns":225},[227,5373,5374,5382],{},[230,5375,5376],{},[537,5377,5378,5380],{},[233,5379,4256],{},[233,5381,5201],{},[279,5383,5384],{"encoding":281},"e_k",[213,5386,5388],{"className":5387,"ariaHidden":251},[286],[213,5389,5391,5394],{"className":5390},[290],[213,5392],{"className":5393,"style":707},[294],[213,5395,5397,5400],{"className":5396},[299],[213,5398,4256],{"className":5399},[299,407],[213,5401,5403],{"className":5402},[605],[213,5404,5406,5426],{"className":5405},[609,610],[213,5407,5409,5423],{"className":5408},[614],[213,5410,5412],{"className":5411,"style":5250},[618],[213,5413,5414,5417],{"style":622},[213,5415],{"className":5416,"style":627},[626],[213,5418,5420],{"className":5419},[631,632,633,634],[213,5421,5201],{"className":5422,"style":5262},[299,407,634],[213,5424,642],{"className":5425},[641],[213,5427,5429],{"className":5428},[614],[213,5430,5432],{"className":5431,"style":649},[618],[213,5433],{},"，而 ",[213,5436,5438,5455],{"className":5437,"translate":217},[216],[213,5439,5441],{"className":5440},[221],[223,5442,5443],{"xmlns":225},[227,5444,5445,5453],{},[230,5446,5447],{},[537,5448,5449,5451],{},[233,5450,4256],{},[233,5452,5201],{},[279,5454,5384],{"encoding":281},[213,5456,5458],{"className":5457,"ariaHidden":251},[286],[213,5459,5461,5464],{"className":5460},[290],[213,5462],{"className":5463,"style":707},[294],[213,5465,5467,5470],{"className":5466},[299],[213,5468,4256],{"className":5469},[299,407],[213,5471,5473],{"className":5472},[605],[213,5474,5476,5496],{"className":5475},[609,610],[213,5477,5479,5493],{"className":5478},[614],[213,5480,5482],{"className":5481,"style":5250},[618],[213,5483,5484,5487],{"style":622},[213,5485],{"className":5486,"style":627},[626],[213,5488,5490],{"className":5489},[631,632,633,634],[213,5491,5201],{"className":5492,"style":5262},[299,407,634],[213,5494,642],{"className":5495},[641],[213,5497,5499],{"className":5498},[614],[213,5500,5502],{"className":5501,"style":649},[618],[213,5503],{}," 被刻意构造得与 ",[213,5506,5508,5530],{"className":5507,"translate":217},[216],[213,5509,5511],{"className":5510},[221],[223,5512,5513],{"xmlns":225},[227,5514,5515,5527],{},[230,5516,5517],{},[537,5518,5519,5521],{},[233,5520,2631],{},[230,5522,5523,5525],{},[233,5524,5201],{},[233,5526,5201],{},[279,5528,5529],{"encoding":281},"d_{kk}",[213,5531,5533],{"className":5532,"ariaHidden":251},[286],[213,5534,5536,5539],{"className":5535},[290],[213,5537],{"className":5538,"style":5041},[294],[213,5540,5542,5545],{"className":5541},[299],[213,5543,2631],{"className":5544},[299,407],[213,5546,5548],{"className":5547},[605],[213,5549,5551,5577],{"className":5550},[609,610],[213,5552,5554,5574],{"className":5553},[614],[213,5555,5557],{"className":5556,"style":5250},[618],[213,5558,5559,5562],{"style":622},[213,5560],{"className":5561,"style":627},[626],[213,5563,5565],{"className":5564},[631,632,633,634],[213,5566,5568,5571],{"className":5567},[299,634],[213,5569,5201],{"className":5570,"style":5262},[299,407,634],[213,5572,5201],{"className":5573,"style":5262},[299,407,634],[213,5575,642],{"className":5576},[641],[213,5578,5580],{"className":5579},[614],[213,5581,5583],{"className":5582,"style":649},[618],[213,5584],{},"（即 ",[213,5587,5589,5607],{"className":5588,"translate":217},[216],[213,5590,5592],{"className":5591},[221],[223,5593,5594],{"xmlns":225},[227,5595,5596,5604],{},[230,5597,5598],{},[537,5599,5600,5602],{},[233,5601,2611],{},[233,5603,5201],{},[279,5605,5606],{"encoding":281},"x_k",[213,5608,5610],{"className":5609,"ariaHidden":251},[286],[213,5611,5613,5616],{"className":5612},[290],[213,5614],{"className":5615,"style":707},[294],[213,5617,5619,5622],{"className":5618},[299],[213,5620,2611],{"className":5621},[299,407],[213,5623,5625],{"className":5624},[605],[213,5626,5628,5648],{"className":5627},[609,610],[213,5629,5631,5645],{"className":5630},[614],[213,5632,5634],{"className":5633,"style":5250},[618],[213,5635,5636,5639],{"style":622},[213,5637],{"className":5638,"style":627},[626],[213,5640,5642],{"className":5641},[631,632,633,634],[213,5643,5201],{"className":5644,"style":5262},[299,407,634],[213,5646,642],{"className":5647},[641],[213,5649,5651],{"className":5650},[614],[213,5652,5654],{"className":5653,"style":649},[618],[213,5655],{},[213,5657,5659,5672],{"className":5658,"translate":217},[216],[213,5660,5662],{"className":5661},[221],[223,5663,5664],{"xmlns":225},[227,5665,5666,5670],{},[230,5667,5668],{},[233,5669,5201],{},[279,5671,5201],{"encoding":281},[213,5673,5675],{"className":5674,"ariaHidden":251},[286],[213,5676,5678,5681],{"className":5677},[290],[213,5679],{"className":5680,"style":1084},[294],[213,5682,5201],{"className":5683,"style":5262},[299,407]," 位小数）不同：",[213,5686,5688],{"className":5687,"translate":217},[2584],[213,5689,5691,5739],{"className":5690,"translate":217},[216],[213,5692,5694],{"className":5693},[221],[223,5695,5696],{"xmlns":225,"display":2593},[227,5697,5698,5736],{},[230,5699,5700,5706,5708,5718,5721,5724,5726,5728,5730],{},[537,5701,5702,5704],{},[233,5703,4256],{},[233,5705,5201],{},[238,5707,546],{"mathvariant":545},[537,5709,5710,5712],{},[233,5711,2631],{},[230,5713,5714,5716],{},[233,5715,5201],{},[233,5717,5201],{},[272,5719,5720],{},"  ",[238,5722,5723],{},"⟹",[272,5725,5720],{},[233,5727,4247],{},[238,5729,546],{"mathvariant":545},[537,5731,5732,5734],{},[233,5733,2611],{},[233,5735,5201],{},[279,5737,5738],{"encoding":281},"e_k \\neq d_{kk} \\implies y \\neq x_k",[213,5740,5742,5830,5897,5948],{"className":5741,"ariaHidden":251},[286],[213,5743,5745,5748,5788,5791,5827],{"className":5744},[290],[213,5746],{"className":5747,"style":477},[294],[213,5749,5751,5754],{"className":5750},[299],[213,5752,4256],{"className":5753},[299,407],[213,5755,5757],{"className":5756},[605],[213,5758,5760,5780],{"className":5759},[609,610],[213,5761,5763,5777],{"className":5762},[614],[213,5764,5766],{"className":5765,"style":5250},[618],[213,5767,5768,5771],{"style":622},[213,5769],{"className":5770,"style":627},[626],[213,5772,5774],{"className":5773},[631,632,633,634],[213,5775,5201],{"className":5776,"style":5262},[299,407,634],[213,5778,642],{"className":5779},[641],[213,5781,5783],{"className":5782},[614],[213,5784,5786],{"className":5785,"style":649},[618],[213,5787],{},[213,5789],{"className":5790,"style":305},[304],[213,5792,5794,5821,5824],{"className":5793},[309],[213,5795,5797],{"className":5796},[309],[213,5798,5800],{"className":5799},[299,664],[213,5801,5803],{"className":5802},[668],[213,5804,5806,5809,5818],{"className":5805},[672],[213,5807],{"className":5808,"style":477},[294],[213,5810,5812],{"className":5811},[679],[213,5813,5815],{"className":5814},[299],[213,5816,686],{"className":5817},[309],[213,5819],{"className":5820},[690],[213,5822],{"className":5823},[304,694],[213,5825,240],{"className":5826},[309],[213,5828],{"className":5829,"style":305},[304],[213,5831,5833,5836,5882,5885,5888,5891,5894],{"className":5832},[290],[213,5834],{"className":5835,"style":5041},[294],[213,5837,5839,5842],{"className":5838},[299],[213,5840,2631],{"className":5841},[299,407],[213,5843,5845],{"className":5844},[605],[213,5846,5848,5874],{"className":5847},[609,610],[213,5849,5851,5871],{"className":5850},[614],[213,5852,5854],{"className":5853,"style":5250},[618],[213,5855,5856,5859],{"style":622},[213,5857],{"className":5858,"style":627},[626],[213,5860,5862],{"className":5861},[631,632,633,634],[213,5863,5865,5868],{"className":5864},[299,634],[213,5866,5201],{"className":5867,"style":5262},[299,407,634],[213,5869,5201],{"className":5870,"style":5262},[299,407,634],[213,5872,642],{"className":5873},[641],[213,5875,5877],{"className":5876},[614],[213,5878,5880],{"className":5879,"style":649},[618],[213,5881],{},[213,5883],{"className":5884,"style":305},[304],[213,5886],{"className":5887,"style":305},[304],[213,5889,5723],{"className":5890},[309],[213,5892],{"className":5893,"style":305},[304],[213,5895],{"className":5896,"style":305},[304],[213,5898,5900,5903,5906,5909,5945],{"className":5899},[290],[213,5901],{"className":5902,"style":477},[294],[213,5904,4247],{"className":5905,"style":2305},[299,407],[213,5907],{"className":5908,"style":305},[304],[213,5910,5912,5939,5942],{"className":5911},[309],[213,5913,5915],{"className":5914},[309],[213,5916,5918],{"className":5917},[299,664],[213,5919,5921],{"className":5920},[668],[213,5922,5924,5927,5936],{"className":5923},[672],[213,5925],{"className":5926,"style":477},[294],[213,5928,5930],{"className":5929},[679],[213,5931,5933],{"className":5932},[299],[213,5934,686],{"className":5935},[309],[213,5937],{"className":5938},[690],[213,5940],{"className":5941},[304,694],[213,5943,240],{"className":5944},[309],[213,5946],{"className":5947,"style":305},[304],[213,5949,5951,5954],{"className":5950},[290],[213,5952],{"className":5953,"style":707},[294],[213,5955,5957,5960],{"className":5956},[299],[213,5958,2611],{"className":5959},[299,407],[213,5961,5963],{"className":5962},[605],[213,5964,5966,5986],{"className":5965},[609,610],[213,5967,5969,5983],{"className":5968},[614],[213,5970,5972],{"className":5971,"style":5250},[618],[213,5973,5974,5977],{"style":622},[213,5975],{"className":5976,"style":627},[626],[213,5978,5980],{"className":5979},[631,632,633,634],[213,5981,5201],{"className":5982,"style":5262},[299,407,634],[213,5984,642],{"className":5985},[641],[213,5987,5989],{"className":5988},[614],[213,5990,5992],{"className":5991,"style":649},[618],[213,5993],{},[11,5995,5996,5997,6088],{},"这与 ",[213,5998,6000,6021],{"className":5999,"translate":217},[216],[213,6001,6003],{"className":6002},[221],[223,6004,6005],{"xmlns":225},[227,6006,6007,6019],{},[230,6008,6009,6011,6013],{},[233,6010,4247],{},[238,6012,240],{},[537,6014,6015,6017],{},[233,6016,2611],{},[233,6018,5201],{},[279,6020,5204],{"encoding":281},[213,6022,6024,6042],{"className":6023,"ariaHidden":251},[286],[213,6025,6027,6030,6033,6036,6039],{"className":6026},[290],[213,6028],{"className":6029,"style":4291},[294],[213,6031,4247],{"className":6032,"style":2305},[299,407],[213,6034],{"className":6035,"style":305},[304],[213,6037,240],{"className":6038},[309],[213,6040],{"className":6041,"style":305},[304],[213,6043,6045,6048],{"className":6044},[290],[213,6046],{"className":6047,"style":707},[294],[213,6049,6051,6054],{"className":6050},[299],[213,6052,2611],{"className":6053},[299,407],[213,6055,6057],{"className":6056},[605],[213,6058,6060,6080],{"className":6059},[609,610],[213,6061,6063,6077],{"className":6062},[614],[213,6064,6066],{"className":6065,"style":5250},[618],[213,6067,6068,6071],{"style":622},[213,6069],{"className":6070,"style":627},[626],[213,6072,6074],{"className":6073},[631,632,633,634],[213,6075,5201],{"className":6076,"style":5262},[299,407,634],[213,6078,642],{"className":6079},[641],[213,6081,6083],{"className":6082},[614],[213,6084,6086],{"className":6085,"style":649},[618],[213,6087],{}," 的假设矛盾！",[11,6090,6091,6092,6120,6121,6149,6150,6179,6180,6208,6209,6237],{},"而且，这一矛盾对每一个 ",[213,6093,6095,6108],{"className":6094,"translate":217},[216],[213,6096,6098],{"className":6097},[221],[223,6099,6100],{"xmlns":225},[227,6101,6102,6106],{},[230,6103,6104],{},[233,6105,5201],{},[279,6107,5201],{"encoding":281},[213,6109,6111],{"className":6110,"ariaHidden":251},[286],[213,6112,6114,6117],{"className":6113},[290],[213,6115],{"className":6116,"style":1084},[294],[213,6118,5201],{"className":6119,"style":5262},[299,407]," 都成立——",[213,6122,6124,6137],{"className":6123,"translate":217},[216],[213,6125,6127],{"className":6126},[221],[223,6128,6129],{"xmlns":225},[227,6130,6131,6135],{},[230,6132,6133],{},[233,6134,4247],{},[279,6136,4247],{"encoding":281},[213,6138,6140],{"className":6139,"ariaHidden":251},[286],[213,6141,6143,6146],{"className":6142},[290],[213,6144],{"className":6145,"style":4291},[294],[213,6147,4247],{"className":6148,"style":2305},[299,407]," 在第 1 位与第 1 个数不同，在第 2 位与第 2 个数不同，在第 ",[213,6151,6153,6166],{"className":6152,"translate":217},[216],[213,6154,6156],{"className":6155},[221],[223,6157,6158],{"xmlns":225},[227,6159,6160,6164],{},[230,6161,6162],{},[233,6163,1496],{},[279,6165,1496],{"encoding":281},[213,6167,6169],{"className":6168,"ariaHidden":251},[286],[213,6170,6172,6176],{"className":6171},[290],[213,6173],{"className":6174,"style":6175},[294],"height:0.4306em;",[213,6177,1496],{"className":6178},[299,407]," 位与第 ",[213,6181,6183,6196],{"className":6182,"translate":217},[216],[213,6184,6186],{"className":6185},[221],[223,6187,6188],{"xmlns":225},[227,6189,6190,6194],{},[230,6191,6192],{},[233,6193,1496],{},[279,6195,1496],{"encoding":281},[213,6197,6199],{"className":6198,"ariaHidden":251},[286],[213,6200,6202,6205],{"className":6201},[290],[213,6203],{"className":6204,"style":6175},[294],[213,6206,1496],{"className":6207},[299,407]," 个数不同……因此 ",[213,6210,6212,6225],{"className":6211,"translate":217},[216],[213,6213,6215],{"className":6214},[221],[223,6216,6217],{"xmlns":225},[227,6218,6219,6223],{},[230,6220,6221],{},[233,6222,4247],{},[279,6224,4247],{"encoding":281},[213,6226,6228],{"className":6227,"ariaHidden":251},[286],[213,6229,6231,6234],{"className":6230},[290],[213,6232],{"className":6233,"style":4291},[294],[213,6235,4247],{"className":6236,"style":2305},[299,407]," 与列表中的每一个数都不同。",[11,6239,6240,6241,6269,6270,6321],{},"然而 ",[213,6242,6244,6257],{"className":6243,"translate":217},[216],[213,6245,6247],{"className":6246},[221],[223,6248,6249],{"xmlns":225},[227,6250,6251,6255],{},[230,6252,6253],{},[233,6254,4247],{},[279,6256,4247],{"encoding":281},[213,6258,6260],{"className":6259,"ariaHidden":251},[286],[213,6261,6263,6266],{"className":6262},[290],[213,6264],{"className":6265,"style":4291},[294],[213,6267,4247],{"className":6268,"style":2305},[299,407]," 无疑是 ",[213,6271,6273,6294],{"className":6272,"translate":217},[216],[213,6274,6276],{"className":6275},[221],[223,6277,6278],{"xmlns":225},[227,6279,6280,6292],{},[230,6281,6282,6284,6286,6288,6290],{},[238,6283,560],{"stretchy":243},[246,6285,248],{},[238,6287,252],{"separator":251},[246,6289,255],{},[238,6291,569],{"stretchy":243},[279,6293,2342],{"encoding":281},[213,6295,6297],{"className":6296,"ariaHidden":251},[286],[213,6298,6300,6303,6306,6309,6312,6315,6318],{"className":6299},[290],[213,6301],{"className":6302,"style":319},[294],[213,6304,560],{"className":6305},[323],[213,6307,248],{"className":6308},[299],[213,6310,252],{"className":6311},[330],[213,6313],{"className":6314,"style":334},[304],[213,6316,255],{"className":6317},[299],[213,6319,569],{"className":6320},[372]," 中的实数（它的每一位都是合法的十进制数字）。",[11,6323,6324,6325,6353,6354,6449,6450,6501],{},"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 ",[213,6326,6328,6341],{"className":6327,"translate":217},[216],[213,6329,6331],{"className":6330},[221],[223,6332,6333],{"xmlns":225},[227,6334,6335,6339],{},[230,6336,6337],{},[233,6338,4247],{},[279,6340,4247],{"encoding":281},[213,6342,6344],{"className":6343,"ariaHidden":251},[286],[213,6345,6347,6350],{"className":6346},[290],[213,6348],{"className":6349,"style":4291},[294],[213,6351,4247],{"className":6352,"style":2305},[299,407],"。这就说明最初的假设——存在双射 ",[213,6355,6357,6386],{"className":6356,"translate":217},[216],[213,6358,6360],{"className":6359},[221],[223,6361,6362],{"xmlns":225},[227,6363,6364,6384],{},[230,6365,6366,6368,6370,6372,6374,6376,6378,6380,6382],{},[233,6367,454],{},[238,6369,457],{},[233,6371,236],{"mathvariant":235},[238,6373,462],{},[238,6375,560],{"stretchy":243},[246,6377,248],{},[238,6379,252],{"separator":251},[246,6381,255],{},[238,6383,569],{"stretchy":243},[279,6385,2516],{"encoding":281},[213,6387,6389,6407,6425],{"className":6388,"ariaHidden":251},[286],[213,6390,6392,6395,6398,6401,6404],{"className":6391},[290],[213,6393],{"className":6394,"style":477},[294],[213,6396,454],{"className":6397,"style":481},[299,407],[213,6399],{"className":6400,"style":305},[304],[213,6402,457],{"className":6403},[309],[213,6405],{"className":6406,"style":305},[304],[213,6408,6410,6413,6416,6419,6422],{"className":6409},[290],[213,6411],{"className":6412,"style":295},[294],[213,6414,236],{"className":6415},[299,300],[213,6417],{"className":6418,"style":305},[304],[213,6420,462],{"className":6421},[309],[213,6423],{"className":6424,"style":305},[304],[213,6426,6428,6431,6434,6437,6440,6443,6446],{"className":6427},[290],[213,6429],{"className":6430,"style":319},[294],[213,6432,560],{"className":6433},[323],[213,6435,248],{"className":6436},[299],[213,6438,252],{"className":6439},[330],[213,6441],{"className":6442,"style":334},[304],[213,6444,255],{"className":6445},[299],[213,6447,569],{"className":6448},[372],"——是假的。",[213,6451,6453,6474],{"className":6452,"translate":217},[216],[213,6454,6456],{"className":6455},[221],[223,6457,6458],{"xmlns":225},[227,6459,6460,6472],{},[230,6461,6462,6464,6466,6468,6470],{},[238,6463,560],{"stretchy":243},[246,6465,248],{},[238,6467,252],{"separator":251},[246,6469,255],{},[238,6471,569],{"stretchy":243},[279,6473,2342],{"encoding":281},[213,6475,6477],{"className":6476,"ariaHidden":251},[286],[213,6478,6480,6483,6486,6489,6492,6495,6498],{"className":6479},[290],[213,6481],{"className":6482,"style":319},[294],[213,6484,560],{"className":6485},[323],[213,6487,248],{"className":6488},[299],[213,6490,252],{"className":6491},[330],[213,6493],{"className":6494,"style":334},[304],[213,6496,255],{"className":6497},[299],[213,6499,569],{"className":6500},[372]," 中的实数不可数。",[11,6503,6504,6505,6705,6706,6776],{},"为什么叫「对角线」？这个名称源于证明中选取数字 ",[213,6506,6508,6546],{"className":6507,"translate":217},[216],[213,6509,6511],{"className":6510},[221],[223,6512,6513],{"xmlns":225},[227,6514,6515,6543],{},[230,6516,6517,6523,6525,6531,6533,6539,6541],{},[537,6518,6519,6521],{},[233,6520,2631],{},[246,6522,2634],{},[238,6524,252],{"separator":251},[537,6526,6527,6529],{},[233,6528,2631],{},[246,6530,2703],{},[238,6532,252],{"separator":251},[537,6534,6535,6537],{},[233,6536,2631],{},[246,6538,2772],{},[238,6540,252],{"separator":251},[238,6542,270],{},[279,6544,6545],{"encoding":281},"d_{11}, d_{22}, d_{33}, \\dots",[213,6547,6549],{"className":6548,"ariaHidden":251},[286],[213,6550,6552,6555,6598,6601,6604,6647,6650,6653,6696,6699,6702],{"className":6551},[290],[213,6553],{"className":6554,"style":477},[294],[213,6556,6558,6561],{"className":6557},[299],[213,6559,2631],{"className":6560},[299,407],[213,6562,6564],{"className":6563},[605],[213,6565,6567,6590],{"className":6566},[609,610],[213,6568,6570,6587],{"className":6569},[614],[213,6571,6573],{"className":6572,"style":619},[618],[213,6574,6575,6578],{"style":622},[213,6576],{"className":6577,"style":627},[626],[213,6579,6581],{"className":6580},[631,632,633,634],[213,6582,6584],{"className":6583},[299,634],[213,6585,2634],{"className":6586},[299,634],[213,6588,642],{"className":6589},[641],[213,6591,6593],{"className":6592},[614],[213,6594,6596],{"className":6595,"style":649},[618],[213,6597],{},[213,6599,252],{"className":6600},[330],[213,6602],{"className":6603,"style":334},[304],[213,6605,6607,6610],{"className":6606},[299],[213,6608,2631],{"className":6609},[299,407],[213,6611,6613],{"className":6612},[605],[213,6614,6616,6639],{"className":6615},[609,610],[213,6617,6619,6636],{"className":6618},[614],[213,6620,6622],{"className":6621,"style":619},[618],[213,6623,6624,6627],{"style":622},[213,6625],{"className":6626,"style":627},[626],[213,6628,6630],{"className":6629},[631,632,633,634],[213,6631,6633],{"className":6632},[299,634],[213,6634,2703],{"className":6635},[299,634],[213,6637,642],{"className":6638},[641],[213,6640,6642],{"className":6641},[614],[213,6643,6645],{"className":6644,"style":649},[618],[213,6646],{},[213,6648,252],{"className":6649},[330],[213,6651],{"className":6652,"style":334},[304],[213,6654,6656,6659],{"className":6655},[299],[213,6657,2631],{"className":6658},[299,407],[213,6660,6662],{"className":6661},[605],[213,6663,6665,6688],{"className":6664},[609,610],[213,6666,6668,6685],{"className":6667},[614],[213,6669,6671],{"className":6670,"style":619},[618],[213,6672,6673,6676],{"style":622},[213,6674],{"className":6675,"style":627},[626],[213,6677,6679],{"className":6678},[631,632,633,634],[213,6680,6682],{"className":6681},[299,634],[213,6683,2772],{"className":6684},[299,634],[213,6686,642],{"className":6687},[641],[213,6689,6691],{"className":6690},[614],[213,6692,6694],{"className":6693,"style":649},[618],[213,6695],{},[213,6697,252],{"className":6698},[330],[213,6700],{"className":6701,"style":334},[304],[213,6703,270],{"className":6704},[365]," 的方式——若把所有 ",[213,6707,6709,6727],{"className":6708,"translate":217},[216],[213,6710,6712],{"className":6711},[221],[223,6713,6714],{"xmlns":225},[227,6715,6716,6724],{},[230,6717,6718],{},[537,6719,6720,6722],{},[233,6721,2611],{},[233,6723,4046],{},[279,6725,6726],{"encoding":281},"x_i",[213,6728,6730],{"className":6729,"ariaHidden":251},[286],[213,6731,6733,6736],{"className":6732},[290],[213,6734],{"className":6735,"style":707},[294],[213,6737,6739,6742],{"className":6738},[299],[213,6740,2611],{"className":6741},[299,407],[213,6743,6745],{"className":6744},[605],[213,6746,6748,6768],{"className":6747},[609,610],[213,6749,6751,6765],{"className":6750},[614],[213,6752,6754],{"className":6753,"style":4081},[618],[213,6755,6756,6759],{"style":622},[213,6757],{"className":6758,"style":627},[626],[213,6760,6762],{"className":6761},[631,632,633,634],[213,6763,4046],{"className":6764},[299,407,634],[213,6766,642],{"className":6767},[641],[213,6769,6771],{"className":6770},[614],[213,6772,6774],{"className":6773,"style":649},[618],[213,6775],{}," 纵向排成无穷矩阵，这些位置恰好构成矩阵的主对角线：",[213,6778,6780],{"className":6779,"translate":217},[2584],[213,6781,6783,6989],{"className":6782,"translate":217},[216],[213,6784,6786],{"className":6785},[221],[223,6787,6788],{"xmlns":225,"display":2593},[227,6789,6790,6986],{},[230,6791,6792,6794,6984],{},[238,6793,560],{"fence":251},[1742,6795,6798,6847,6893,6939],{"rowspacing":6796,"columnalign":6797,"columnspacing":1746},"0.16em","center center center center",[1748,6799,6800,6820,6830,6840],{},[1751,6801,6802],{},[1754,6803,6804],{"scriptlevel":248,"displaystyle":243},[6805,6806,6808],"menclose",{"notation":6807},"box",[1754,6809,6810],{"scriptlevel":248,"displaystyle":243},[1754,6811,6812],{"scriptlevel":248,"displaystyle":243},[1754,6813,6814],{"scriptlevel":248,"displaystyle":251},[537,6815,6816,6818],{},[233,6817,2631],{},[246,6819,2634],{},[1751,6821,6822],{},[1754,6823,6824],{"scriptlevel":248,"displaystyle":243},[537,6825,6826,6828],{},[233,6827,2631],{},[246,6829,2643],{},[1751,6831,6832],{},[1754,6833,6834],{"scriptlevel":248,"displaystyle":243},[537,6835,6836,6838],{},[233,6837,2631],{},[246,6839,2652],{},[1751,6841,6842],{},[1754,6843,6844],{"scriptlevel":248,"displaystyle":243},[238,6845,6846],{"lspace":2600,"rspace":2600},"⋯",[1748,6848,6849,6859,6877,6887],{},[1751,6850,6851],{},[1754,6852,6853],{"scriptlevel":248,"displaystyle":243},[537,6854,6855,6857],{},[233,6856,2631],{},[246,6858,2694],{},[1751,6860,6861],{},[1754,6862,6863],{"scriptlevel":248,"displaystyle":243},[6805,6864,6865],{"notation":6807},[1754,6866,6867],{"scriptlevel":248,"displaystyle":243},[1754,6868,6869],{"scriptlevel":248,"displaystyle":243},[1754,6870,6871],{"scriptlevel":248,"displaystyle":251},[537,6872,6873,6875],{},[233,6874,2631],{},[246,6876,2703],{},[1751,6878,6879],{},[1754,6880,6881],{"scriptlevel":248,"displaystyle":243},[537,6882,6883,6885],{},[233,6884,2631],{},[246,6886,2712],{},[1751,6888,6889],{},[1754,6890,6891],{"scriptlevel":248,"displaystyle":243},[238,6892,6846],{"lspace":2600,"rspace":2600},[1748,6894,6895,6905,6915,6933],{},[1751,6896,6897],{},[1754,6898,6899],{"scriptlevel":248,"displaystyle":243},[537,6900,6901,6903],{},[233,6902,2631],{},[246,6904,2754],{},[1751,6906,6907],{},[1754,6908,6909],{"scriptlevel":248,"displaystyle":243},[537,6910,6911,6913],{},[233,6912,2631],{},[246,6914,2763],{},[1751,6916,6917],{},[1754,6918,6919],{"scriptlevel":248,"displaystyle":243},[6805,6920,6921],{"notation":6807},[1754,6922,6923],{"scriptlevel":248,"displaystyle":243},[1754,6924,6925],{"scriptlevel":248,"displaystyle":243},[1754,6926,6927],{"scriptlevel":248,"displaystyle":251},[537,6928,6929,6931],{},[233,6930,2631],{},[246,6932,2772],{},[1751,6934,6935],{},[1754,6936,6937],{"scriptlevel":248,"displaystyle":243},[238,6938,6846],{"lspace":2600,"rspace":2600},[1748,6940,6941,6953,6965,6977],{},[1751,6942,6943],{},[1754,6944,6945],{"scriptlevel":248,"displaystyle":243},[230,6946,6947,6949],{},[233,6948,2867],{"mathvariant":545},[2869,6950,6951],{"height":2600,"voffset":2600},[304,6952],{"mathbackground":2873,"width":2600,"height":2874},[1751,6954,6955],{},[1754,6956,6957],{"scriptlevel":248,"displaystyle":243},[230,6958,6959,6961],{},[233,6960,2867],{"mathvariant":545},[2869,6962,6963],{"height":2600,"voffset":2600},[304,6964],{"mathbackground":2873,"width":2600,"height":2874},[1751,6966,6967],{},[1754,6968,6969],{"scriptlevel":248,"displaystyle":243},[230,6970,6971,6973],{},[233,6972,2867],{"mathvariant":545},[2869,6974,6975],{"height":2600,"voffset":2600},[304,6976],{"mathbackground":2873,"width":2600,"height":2874},[1751,6978,6979],{},[1754,6980,6981],{"scriptlevel":248,"displaystyle":243},[238,6982,6983],{"lspace":2600,"rspace":2600},"⋱",[238,6985,569],{"fence":251},[279,6987,6988],{"encoding":281},"\\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}",[213,6990,6992],{"className":6991,"ariaHidden":251},[286],[213,6993,6995,6999],{"className":6994},[290],[213,6996],{"className":6997,"style":6998},[294],"height:6.4333em;vertical-align:-2.9667em;",[213,7000,7002,7052,7878],{"className":7001},[365],[213,7003,7005],{"className":7004},[323],[213,7006,7009],{"className":7007},[1863,7008],"mult",[213,7010,7012,7043],{"className":7011},[609,610],[213,7013,7015,7040],{"className":7014},[614],[213,7016,7019],{"className":7017,"style":7018},[618],"height:3.25em;",[213,7020,7022,7026],{"style":7021},"top:-5.25em;",[213,7023],{"className":7024,"style":7025},[626],"height:8em;",[213,7027,7029],{"style":7028},"width:0.875em;height:6em;",[7030,7031,7036],"svg",{"xmlns":7032,"width":7033,"height":7034,"viewBox":7035},"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",[7932,7933,7934,7959,7965,7971,7977,7982,7993,8011,8039,8057,8085,8097,8115,8121,8126,8137,8162,8185,8198,8204,8209,8220],"code",{"__ignoreMap":172},[213,7935,7938,7942,7946,7950,7952,7956],{"class":7936,"line":7937},"line",1,[213,7939,7941],{"class":7940},"snl16","def",[213,7943,7945],{"class":7944},"svObZ"," diagonal_construct",[213,7947,7949],{"class":7948},"s95oV","(number_list, digits",[213,7951,240],{"class":7940},[213,7953,7955],{"class":7954},"sDLfK","10",[213,7957,7958],{"class":7948},"):\n",[213,7960,7961],{"class":7936,"line":173},[213,7962,7964],{"class":7963},"sU2Wk","    \"\"\"\n",[213,7966,7968],{"class":7936,"line":7967},3,[213,7969,7970],{"class":7963},"    number_list：字符串列表，每个形如 \"0.d1d2d3...\"\n",[213,7972,7974],{"class":7936,"line":7973},4,[213,7975,7976],{"class":7963},"    返回：通过对角线方法构造出的新数字字符串\n",[213,7978,7980],{"class":7936,"line":7979},5,[213,7981,7964],{"class":7963},[213,7983,7985,7988,7990],{"class":7936,"line":7984},6,[213,7986,7987],{"class":7948},"    new_digits ",[213,7989,240],{"class":7940},[213,7991,7992],{"class":7948}," []\n",[213,7994,7996,7999,8002,8005,8008],{"class":7936,"line":7995},7,[213,7997,7998],{"class":7940},"    for",[213,8000,8001],{"class":7948}," i, num_str ",[213,8003,8004],{"class":7940},"in",[213,8006,8007],{"class":7954}," enumerate",[213,8009,8010],{"class":7948},"(number_list):\n",[213,8012,8014,8017,8019,8022,8025,8028,8030,8033,8036],{"class":7936,"line":8013},8,[213,8015,8016],{"class":7948},"        decimal_part ",[213,8018,240],{"class":7940},[213,8020,8021],{"class":7948}," num_str.split(",[213,8023,8024],{"class":7963},"'.'",[213,8026,8027],{"class":7948},")[",[213,8029,255],{"class":7954},[213,8031,8032],{"class":7948},"].ljust(digits, ",[213,8034,8035],{"class":7963},"'0'",[213,8037,8038],{"class":7948},")\n",[213,8040,8042,8045,8047,8050,8053],{"class":7936,"line":8041},9,[213,8043,8044],{"class":7948},"        d ",[213,8046,240],{"class":7940},[213,8048,8049],{"class":7954}," int",[213,8051,8052],{"class":7948},"(decimal_part[i])       ",[213,8054,8056],{"class":8055},"sAwPA","# 第 i 个数的第 i 位（对角线数字）\n",[213,8058,8060,8063,8065,8068,8070,8073,8076,8079,8082],{"class":7936,"line":8059},10,[213,8061,8062],{"class":7948},"        new_d ",[213,8064,240],{"class":7940},[213,8066,8067],{"class":7948}," (d ",[213,8069,1792],{"class":7940},[213,8071,8072],{"class":7954}," 5",[213,8074,8075],{"class":7948},") ",[213,8077,8078],{"class":7940},"%",[213,8080,8081],{"class":7954}," 10",[213,8083,8084],{"class":8055},"           # 规则：使新数字与原数字不同\n",[213,8086,8088,8091,8094],{"class":7936,"line":8087},11,[213,8089,8090],{"class":7948},"        new_digits.append(",[213,8092,8093],{"class":7954},"str",[213,8095,8096],{"class":7948},"(new_d))\n",[213,8098,8100,8103,8106,8109,8112],{"class":7936,"line":8099},12,[213,8101,8102],{"class":7940},"    return",[213,8104,8105],{"class":7963}," \"0.\"",[213,8107,8108],{"class":7940}," +",[213,8110,8111],{"class":7963}," \"\"",[213,8113,8114],{"class":7948},".join(new_digits)\n",[213,8116,8118],{"class":7936,"line":8117},13,[213,8119,8120],{"emptyLinePlaceholder":179},"\n",[213,8122,8124],{"class":7936,"line":8123},14,[213,8125,8120],{"emptyLinePlaceholder":179},[213,8127,8129,8132,8134],{"class":7936,"line":8128},15,[213,8130,8131],{"class":7948},"sample_list ",[213,8133,240],{"class":7940},[213,8135,8136],{"class":7948}," [\n",[213,8138,8140,8143,8146,8149,8151,8154,8156,8159],{"class":7936,"line":8139},16,[213,8141,8142],{"class":7963},"    \"0.1234567890\"",[213,8144,8145],{"class":7948},", ",[213,8147,8148],{"class":7963},"\"0.9876543210\"",[213,8150,8145],{"class":7948},[213,8152,8153],{"class":7963},"\"0.5555555555\"",[213,8155,8145],{"class":7948},[213,8157,8158],{"class":7963},"\"0.1111111111\"",[213,8160,8161],{"class":7948},",\n",[213,8163,8165,8168,8170,8173,8175,8178,8180,8183],{"class":7936,"line":8164},17,[213,8166,8167],{"class":7963},"    \"0.3141592653\"",[213,8169,8145],{"class":7948},[213,8171,8172],{"class":7963},"\"0.2718281828\"",[213,8174,8145],{"class":7948},[213,8176,8177],{"class":7963},"\"0.0000000001\"",[213,8179,8145],{"class":7948},[213,8181,8182],{"class":7963},"\"0.9999999998\"",[213,8184,8161],{"class":7948},[213,8186,8188,8191,8193,8196],{"class":7936,"line":8187},18,[213,8189,8190],{"class":7963},"    \"0.4242424242\"",[213,8192,8145],{"class":7948},[213,8194,8195],{"class":7963},"\"0.6180339887\"",[213,8197,8161],{"class":7948},[213,8199,8201],{"class":7936,"line":8200},19,[213,8202,8203],{"class":7948},"]\n",[213,8205,8207],{"class":7936,"line":8206},20,[213,8208,8120],{"emptyLinePlaceholder":179},[213,8210,8212,8215,8217],{"class":7936,"line":8211},21,[213,8213,8214],{"class":7948},"new_number ",[213,8216,240],{"class":7940},[213,8218,8219],{"class":7948}," diagonal_construct(sample_list)\n",[213,8221,8223,8226,8228,8231],{"class":7936,"line":8222},22,[213,8224,8225],{"class":7954},"print",[213,8227,560],{"class":7948},[213,8229,8230],{"class":7963},"\"对角线构造出的新数：\"",[213,8232,8233],{"class":7948},", new_number)\n",[11,8235,8236],{},"输出：",[7926,8238,8242],{"className":8239,"code":8241,"language":1983},[8240],"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",[7932,8243,8241],{"__ignoreMap":172},[11,8245,8246,8247,8250,8251,8279,8280,8405],{},"可以清楚地看到，新构造的数 ",[7932,8248,8249],{},"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 ",[213,8252,8254,8267],{"className":8253,"translate":217},[216],[213,8255,8257],{"className":8256},[221],[223,8258,8259],{"xmlns":225},[227,8260,8261,8265],{},[230,8262,8263],{},[233,8264,5201],{},[279,8266,5201],{"encoding":281},[213,8268,8270],{"className":8269,"ariaHidden":251},[286],[213,8271,8273,8276],{"className":8272},[290],[213,8274],{"className":8275,"style":1084},[294],[213,8277,5201],{"className":8278,"style":5262},[299,407],"，",[213,8281,8283,8305],{"className":8282,"translate":217},[216],[213,8284,8286],{"className":8285},[221],[223,8287,8288],{"xmlns":225},[227,8289,8290,8302],{},[230,8291,8292,8294,8296],{},[233,8293,4247],{},[238,8295,546],{"mathvariant":545},[537,8297,8298,8300],{},[233,8299,2611],{},[233,8301,5201],{},[279,8303,8304],{"encoding":281},"y \\neq x_k",[213,8306,8308,8359],{"className":8307,"ariaHidden":251},[286],[213,8309,8311,8314,8317,8320,8356],{"className":8310},[290],[213,8312],{"className":8313,"style":477},[294],[213,8315,4247],{"className":8316,"style":2305},[299,407],[213,8318],{"className":8319,"style":305},[304],[213,8321,8323,8350,8353],{"className":8322},[309],[213,8324,8326],{"className":8325},[309],[213,8327,8329],{"className":8328},[299,664],[213,8330,8332],{"className":8331},[668],[213,8333,8335,8338,8347],{"className":8334},[672],[213,8336],{"className":8337,"style":477},[294],[213,8339,8341],{"className":8340},[679],[213,8342,8344],{"className":8343},[299],[213,8345,686],{"className":8346},[309],[213,8348],{"className":8349},[690],[213,8351],{"className":8352},[304,694],[213,8354,240],{"className":8355},[309],[213,8357],{"className":8358,"style":305},[304],[213,8360,8362,8365],{"className":8361},[290],[213,8363],{"className":8364,"style":707},[294],[213,8366,8368,8371],{"className":8367},[299],[213,8369,2611],{"className":8370},[299,407],[213,8372,8374],{"className":8373},[605],[213,8375,8377,8397],{"className":8376},[609,610],[213,8378,8380,8394],{"className":8379},[614],[213,8381,8383],{"className":8382,"style":5250},[618],[213,8384,8385,8388],{"style":622},[213,8386],{"className":8387,"style":627},[626],[213,8389,8391],{"className":8390},[631,632,633,634],[213,8392,5201],{"className":8393,"style":5262},[299,407,634],[213,8395,642],{"className":8396},[641],[213,8398,8400],{"className":8399},[614],[213,8401,8403],{"className":8402,"style":649},[618],[213,8404],{},"——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。",[30,8407,8408],{"id":8408},"康托尔定理",[11,8410,8411],{},"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。",[21,8413,8414],{},[11,8415,8416,8417,8445,8446,5585,8494,8522,8523,8551,8552,1551],{},"康托尔定理。对任意集合 ",[213,8418,8420,8433],{"className":8419,"translate":217},[216],[213,8421,8423],{"className":8422},[221],[223,8424,8425],{"xmlns":225},[227,8426,8427,8431],{},[230,8428,8429],{},[233,8430,391],{},[279,8432,391],{"encoding":281},[213,8434,8436],{"className":8435,"ariaHidden":251},[286],[213,8437,8439,8442],{"className":8438},[290],[213,8440],{"className":8441,"style":403},[294],[213,8443,391],{"className":8444},[299,407],"，其幂集 ",[213,8447,8449,8471],{"className":8448,"translate":217},[216],[213,8450,8452],{"className":8451},[221],[223,8453,8454],{"xmlns":225},[227,8455,8456,8468],{},[230,8457,8458,8462,8464,8466],{},[233,8459,8461],{"mathvariant":8460},"script","P",[238,8463,560],{"stretchy":243},[233,8465,391],{},[238,8467,569],{"stretchy":243},[279,8469,8470],{"encoding":281},"\\mathcal{P}(A)",[213,8472,8474],{"className":8473,"ariaHidden":251},[286],[213,8475,8477,8480,8485,8488,8491],{"className":8476},[290],[213,8478],{"className":8479,"style":319},[294],[213,8481,8461],{"className":8482,"style":8484},[299,8483],"mathcal","margin-right:0.0822em;",[213,8486,560],{"className":8487},[323],[213,8489,391],{"className":8490},[299,407],[213,8492,569],{"className":8493},[372],[213,8495,8497,8510],{"className":8496,"translate":217},[216],[213,8498,8500],{"className":8499},[221],[223,8501,8502],{"xmlns":225},[227,8503,8504,8508],{},[230,8505,8506],{},[233,8507,391],{},[279,8509,391],{"encoding":281},[213,8511,8513],{"className":8512,"ariaHidden":251},[286],[213,8514,8516,8519],{"className":8515},[290],[213,8517],{"className":8518,"style":403},[294],[213,8520,391],{"className":8521},[299,407]," 的所有子集构成的集合）的势严格大于 ",[213,8524,8526,8539],{"className":8525,"translate":217},[216],[213,8527,8529],{"className":8528},[221],[223,8530,8531],{"xmlns":225},[227,8532,8533,8537],{},[230,8534,8535],{},[233,8536,391],{},[279,8538,391],{"encoding":281},[213,8540,8542],{"className":8541,"ariaHidden":251},[286],[213,8543,8545,8548],{"className":8544},[290],[213,8546],{"className":8547,"style":403},[294],[213,8549,391],{"className":8550},[299,407]," 本身，即 ",[213,8553,8555,8588],{"className":8554,"translate":217},[216],[213,8556,8558],{"className":8557},[221],[223,8559,8560],{"xmlns":225},[227,8561,8562,8585],{},[230,8563,8564,8566,8568,8570,8573,8575,8577,8579,8581,8583],{},[233,8565,1192],{"mathvariant":545},[233,8567,391],{},[233,8569,1192],{"mathvariant":545},[238,8571,8572],{},"\u003C",[233,8574,1192],{"mathvariant":545},[233,8576,8461],{"mathvariant":8460},[238,8578,560],{"stretchy":243},[233,8580,391],{},[238,8582,569],{"stretchy":243},[233,8584,1192],{"mathvariant":545},[279,8586,8587],{"encoding":281},"|A| \u003C |\\mathcal{P}(A)|",[213,8589,8591,8615],{"className":8590,"ariaHidden":251},[286],[213,8592,8594,8597,8600,8603,8606,8609,8612],{"className":8593},[290],[213,8595],{"className":8596,"style":319},[294],[213,8598,1192],{"className":8599},[299],[213,8601,391],{"className":8602},[299,407],[213,8604,1192],{"className":8605},[299],[213,8607],{"className":8608,"style":305},[304],[213,8610,8572],{"className":8611},[309],[213,8613],{"className":8614,"style":305},[304],[213,8616,8618,8621,8624,8627,8630,8633,8636],{"className":8617},[290],[213,8619],{"className":8620,"style":319},[294],[213,8622,1192],{"className":8623},[299],[213,8625,8461],{"className":8626,"style":8484},[299,8483],[213,8628,560],{"className":8629},[323],[213,8631,391],{"className":8632},[299,407],[213,8634,569],{"className":8635},[372],[213,8637,1192],{"className":8638},[299],[11,8640,8641,8642,8670,8671,8714],{},"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 ",[213,8643,8645,8658],{"className":8644,"translate":217},[216],[213,8646,8648],{"className":8647},[221],[223,8649,8650],{"xmlns":225},[227,8651,8652,8656],{},[230,8653,8654],{},[233,8655,391],{},[279,8657,391],{"encoding":281},[213,8659,8661],{"className":8660,"ariaHidden":251},[286],[213,8662,8664,8667],{"className":8663},[290],[213,8665],{"className":8666,"style":403},[294],[213,8668,391],{"className":8669},[299,407]," 多大，",[213,8672,8674,8693],{"className":8673,"translate":217},[216],[213,8675,8677],{"className":8676},[221],[223,8678,8679],{"xmlns":225},[227,8680,8681,8691],{},[230,8682,8683,8685,8687,8689],{},[233,8684,8461],{"mathvariant":8460},[238,8686,560],{"stretchy":243},[233,8688,391],{},[238,8690,569],{"stretchy":243},[279,8692,8470],{"encoding":281},[213,8694,8696],{"className":8695,"ariaHidden":251},[286],[213,8697,8699,8702,8705,8708,8711],{"className":8698},[290],[213,8700],{"className":8701,"style":319},[294],[213,8703,8461],{"className":8704,"style":8484},[299,8483],[213,8706,560],{"className":8707},[323],[213,8709,391],{"className":8710},[299,407],[213,8712,569],{"className":8713},[372]," 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。",[8716,8717,8718],"h3",{"id":8718},"证明",[11,8720,8721,8722,8810,8811,4229],{},"假设存在一个满射 ",[213,8723,8725,8753],{"className":8724,"translate":217},[216],[213,8726,8728],{"className":8727},[221],[223,8729,8730],{"xmlns":225},[227,8731,8732,8750],{},[230,8733,8734,8736,8738,8740,8742,8744,8746,8748],{},[233,8735,454],{},[238,8737,457],{},[233,8739,391],{},[238,8741,462],{},[233,8743,8461],{"mathvariant":8460},[238,8745,560],{"stretchy":243},[233,8747,391],{},[238,8749,569],{"stretchy":243},[279,8751,8752],{"encoding":281},"f: A \\to \\mathcal{P}(A)",[213,8754,8756,8774,8792],{"className":8755,"ariaHidden":251},[286],[213,8757,8759,8762,8765,8768,8771],{"className":8758},[290],[213,8760],{"className":8761,"style":477},[294],[213,8763,454],{"className":8764,"style":481},[299,407],[213,8766],{"className":8767,"style":305},[304],[213,8769,457],{"className":8770},[309],[213,8772],{"className":8773,"style":305},[304],[213,8775,8777,8780,8783,8786,8789],{"className":8776},[290],[213,8778],{"className":8779,"style":403},[294],[213,8781,391],{"className":8782},[299,407],[213,8784],{"className":8785,"style":305},[304],[213,8787,462],{"className":8788},[309],[213,8790],{"className":8791,"style":305},[304],[213,8793,8795,8798,8801,8804,8807],{"className":8794},[290],[213,8796],{"className":8797,"style":319},[294],[213,8799,8461],{"className":8800,"style":8484},[299,8483],[213,8802,560],{"className":8803},[323],[213,8805,391],{"className":8806},[299,407],[213,8808,569],{"className":8809},[372],"（我们将证明这样的满射不存在，由此得到 ",[213,8812,8814,8845],{"className":8813,"translate":217},[216],[213,8815,8817],{"className":8816},[221],[223,8818,8819],{"xmlns":225},[227,8820,8821,8843],{},[230,8822,8823,8825,8827,8829,8831,8833,8835,8837,8839,8841],{},[233,8824,1192],{"mathvariant":545},[233,8826,391],{},[233,8828,1192],{"mathvariant":545},[238,8830,8572],{},[233,8832,1192],{"mathvariant":545},[233,8834,8461],{"mathvariant":8460},[238,8836,560],{"stretchy":243},[233,8838,391],{},[238,8840,569],{"stretchy":243},[233,8842,1192],{"mathvariant":545},[279,8844,8587],{"encoding":281},[213,8846,8848,8872],{"className":8847,"ariaHidden":251},[286],[213,8849,8851,8854,8857,8860,8863,8866,8869],{"className":8850},[290],[213,8852],{"className":8853,"style":319},[294],[213,8855,1192],{"className":8856},[299],[213,8858,391],{"className":8859},[299,407],[213,8861,1192],{"className":8862},[299],[213,8864],{"className":8865,"style":305},[304],[213,8867,8572],{"className":8868},[309],[213,8870],{"className":8871,"style":305},[304],[213,8873,8875,8878,8881,8884,8887,8890,8893],{"className":8874},[290],[213,8876],{"className":8877,"style":319},[294],[213,8879,1192],{"className":8880},[299],[213,8882,8461],{"className":8883,"style":8484},[299,8483],[213,8885,560],{"className":8886},[323],[213,8888,391],{"className":8889},[299,407],[213,8891,569],{"className":8892},[372],[213,8894,1192],{"className":8895},[299],[11,8897,8898,8899,8279,8949,8993,8994,9022],{},"对每个元素 ",[213,8900,8902,8919],{"className":8901,"translate":217},[216],[213,8903,8905],{"className":8904},[221],[223,8906,8907],{"xmlns":225},[227,8908,8909,8917],{},[230,8910,8911,8913,8915],{},[233,8912,107],{},[238,8914,930],{},[233,8916,391],{},[279,8918,988],{"encoding":281},[213,8920,8922,8940],{"className":8921,"ariaHidden":251},[286],[213,8923,8925,8928,8931,8934,8937],{"className":8924},[290],[213,8926],{"className":8927,"style":998},[294],[213,8929,107],{"className":8930},[299,407],[213,8932],{"className":8933,"style":305},[304],[213,8935,930],{"className":8936},[309],[213,8938],{"className":8939,"style":305},[304],[213,8941,8943,8946],{"className":8942},[290],[213,8944],{"className":8945,"style":403},[294],[213,8947,391],{"className":8948},[299,407],[213,8950,8952,8972],{"className":8951,"translate":217},[216],[213,8953,8955],{"className":8954},[221],[223,8956,8957],{"xmlns":225},[227,8958,8959,8969],{},[230,8960,8961,8963,8965,8967],{},[233,8962,454],{},[238,8964,560],{"stretchy":243},[233,8966,107],{},[238,8968,569],{"stretchy":243},[279,8970,8971],{"encoding":281},"f(a)",[213,8973,8975],{"className":8974,"ariaHidden":251},[286],[213,8976,8978,8981,8984,8987,8990],{"className":8977},[290],[213,8979],{"className":8980,"style":319},[294],[213,8982,454],{"className":8983,"style":481},[299,407],[213,8985,560],{"className":8986},[323],[213,8988,107],{"className":8989},[299,407],[213,8991,569],{"className":8992},[372]," 都是 ",[213,8995,8997,9010],{"className":8996,"translate":217},[216],[213,8998,9000],{"className":8999},[221],[223,9001,9002],{"xmlns":225},[227,9003,9004,9008],{},[230,9005,9006],{},[233,9007,391],{},[279,9009,391],{"encoding":281},[213,9011,9013],{"className":9012,"ariaHidden":251},[286],[213,9014,9016,9019],{"className":9015},[290],[213,9017],{"className":9018,"style":403},[294],[213,9020,391],{"className":9021},[299,407]," 的一个子集。现在构造一个特殊的子集：",[213,9024,9026],{"className":9025,"translate":217},[2584],[213,9027,9029,9075],{"className":9028,"translate":217},[216],[213,9030,9032],{"className":9031},[221],[223,9033,9034],{"xmlns":225,"display":2593},[227,9035,9036,9072],{},[230,9037,9038,9041,9043,9045,9047,9049,9051,9053,9055,9057,9060,9062,9064,9066,9068,9070],{},[233,9039,9040],{},"D",[238,9042,240],{},[238,9044,244],{"stretchy":243},[272,9046,274],{},[233,9048,107],{},[238,9050,930],{},[233,9052,391],{},[238,9054,457],{},[233,9056,107],{},[238,9058,9059],{"mathvariant":545},"∉",[233,9061,454],{},[238,9063,560],{"stretchy":243},[233,9065,107],{},[238,9067,569],{"stretchy":243},[272,9069,274],{},[238,9071,277],{"stretchy":243},[279,9073,9074],{"encoding":281},"D = \\{\\, a \\in A : a \\notin f(a) \\,\\}",[213,9076,9078,9097,9121,9139,9192],{"className":9077,"ariaHidden":251},[286],[213,9079,9081,9084,9088,9091,9094],{"className":9080},[290],[213,9082],{"className":9083,"style":403},[294],[213,9085,9040],{"className":9086,"style":9087},[299,407],"margin-right:0.0278em;",[213,9089],{"className":9090,"style":305},[304],[213,9092,240],{"className":9093},[309],[213,9095],{"className":9096,"style":305},[304],[213,9098,9100,9103,9106,9109,9112,9115,9118],{"className":9099},[290],[213,9101],{"className":9102,"style":319},[294],[213,9104,244],{"className":9105},[323],[213,9107],{"className":9108,"style":334},[304],[213,9110,107],{"className":9111},[299,407],[213,9113],{"className":9114,"style":305},[304],[213,9116,930],{"className":9117},[309],[213,9119],{"className":9120,"style":305},[304],[213,9122,9124,9127,9130,9133,9136],{"className":9123},[290],[213,9125],{"className":9126,"style":403},[294],[213,9128,391],{"className":9129},[299,407],[213,9131],{"className":9132,"style":305},[304],[213,9134,457],{"className":9135},[309],[213,9137],{"className":9138,"style":305},[304],[213,9140,9142,9145,9148,9151,9189],{"className":9141},[290],[213,9143],{"className":9144,"style":319},[294],[213,9146,107],{"className":9147},[299,407],[213,9149],{"className":9150,"style":305},[304],[213,9152,9154,9160],{"className":9153},[309],[213,9155,9157],{"className":9156},[299],[213,9158,930],{"className":9159},[309],[213,9161,9163],{"className":9162},[299,664],[213,9164,9166],{"className":9165},[668],[213,9167,9170,9173,9186],{"className":9168},[9169],"llap",[213,9171],{"className":9172,"style":319},[294],[213,9174,9176],{"className":9175},[679],[213,9177,9179,9182],{"className":9178},[299],[213,9180,1762],{"className":9181},[299],[213,9183],{"className":9184,"style":9185},[304],"margin-right:0.0556em;",[213,9187],{"className":9188},[690],[213,9190],{"className":9191,"style":305},[304],[213,9193,9195,9198,9201,9204,9207,9210,9213],{"className":9194},[290],[213,9196],{"className":9197,"style":319},[294],[213,9199,454],{"className":9200,"style":481},[299,407],[213,9202,560],{"className":9203},[323],[213,9205,107],{"className":9206},[299,407],[213,9208,569],{"className":9209},[372],[213,9211],{"className":9212,"style":334},[304],[213,9214,277],{"className":9215},[372],[11,9217,9218],{},"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。",[11,9220,9221,9222,9250,9251,9279,9280,9308,9309,9352,9353,1020,9404,1551],{},"由于 ",[213,9223,9225,9238],{"className":9224,"translate":217},[216],[213,9226,9228],{"className":9227},[221],[223,9229,9230],{"xmlns":225},[227,9231,9232,9236],{},[230,9233,9234],{},[233,9235,454],{},[279,9237,454],{"encoding":281},[213,9239,9241],{"className":9240,"ariaHidden":251},[286],[213,9242,9244,9247],{"className":9243},[290],[213,9245],{"className":9246,"style":477},[294],[213,9248,454],{"className":9249,"style":481},[299,407]," 是满射，",[213,9252,9254,9267],{"className":9253,"translate":217},[216],[213,9255,9257],{"className":9256},[221],[223,9258,9259],{"xmlns":225},[227,9260,9261,9265],{},[230,9262,9263],{},[233,9264,9040],{},[279,9266,9040],{"encoding":281},[213,9268,9270],{"className":9269,"ariaHidden":251},[286],[213,9271,9273,9276],{"className":9272},[290],[213,9274],{"className":9275,"style":403},[294],[213,9277,9040],{"className":9278,"style":9087},[299,407],"（作为 ",[213,9281,9283,9296],{"className":9282,"translate":217},[216],[213,9284,9286],{"className":9285},[221],[223,9287,9288],{"xmlns":225},[227,9289,9290,9294],{},[230,9291,9292],{},[233,9293,391],{},[279,9295,391],{"encoding":281},[213,9297,9299],{"className":9298,"ariaHidden":251},[286],[213,9300,9302,9305],{"className":9301},[290],[213,9303],{"className":9304,"style":403},[294],[213,9306,391],{"className":9307},[299,407]," 的子集，因而是 ",[213,9310,9312,9331],{"className":9311,"translate":217},[216],[213,9313,9315],{"className":9314},[221],[223,9316,9317],{"xmlns":225},[227,9318,9319,9329],{},[230,9320,9321,9323,9325,9327],{},[233,9322,8461],{"mathvariant":8460},[238,9324,560],{"stretchy":243},[233,9326,391],{},[238,9328,569],{"stretchy":243},[279,9330,8470],{"encoding":281},[213,9332,9334],{"className":9333,"ariaHidden":251},[286],[213,9335,9337,9340,9343,9346,9349],{"className":9336},[290],[213,9338],{"className":9339,"style":319},[294],[213,9341,8461],{"className":9342,"style":8484},[299,8483],[213,9344,560],{"className":9345},[323],[213,9347,391],{"className":9348},[299,407],[213,9350,569],{"className":9351},[372]," 的元素）必有原像；即存在某个 ",[213,9354,9356,9374],{"className":9355,"translate":217},[216],[213,9357,9359],{"className":9358},[221],[223,9360,9361],{"xmlns":225},[227,9362,9363,9371],{},[230,9364,9365,9367,9369],{},[233,9366,2631],{},[238,9368,930],{},[233,9370,391],{},[279,9372,9373],{"encoding":281},"d \\in A",[213,9375,9377,9395],{"className":9376,"ariaHidden":251},[286],[213,9378,9380,9383,9386,9389,9392],{"className":9379},[290],[213,9381],{"className":9382,"style":945},[294],[213,9384,2631],{"className":9385},[299,407],[213,9387],{"className":9388,"style":305},[304],[213,9390,930],{"className":9391},[309],[213,9393],{"className":9394,"style":305},[304],[213,9396,9398,9401],{"className":9397},[290],[213,9399],{"className":9400,"style":403},[294],[213,9402,391],{"className":9403},[299,407],[213,9405,9407,9431],{"className":9406,"translate":217},[216],[213,9408,9410],{"className":9409},[221],[223,9411,9412],{"xmlns":225},[227,9413,9414,9428],{},[230,9415,9416,9418,9420,9422,9424,9426],{},[233,9417,454],{},[238,9419,560],{"stretchy":243},[233,9421,2631],{},[238,9423,569],{"stretchy":243},[238,9425,240],{},[233,9427,9040],{},[279,9429,9430],{"encoding":281},"f(d) = D",[213,9432,9434,9461],{"className":9433,"ariaHidden":251},[286],[213,9435,9437,9440,9443,9446,9449,9452,9455,9458],{"className":9436},[290],[213,9438],{"className":9439,"style":319},[294],[213,9441,454],{"className":9442,"style":481},[299,407],[213,9444,560],{"className":9445},[323],[213,9447,2631],{"className":9448},[299,407],[213,9450,569],{"className":9451},[372],[213,9453],{"className":9454,"style":305},[304],[213,9456,240],{"className":9457},[309],[213,9459],{"className":9460,"style":305},[304],[213,9462,9464,9467],{"className":9463},[290],[213,9465],{"className":9466,"style":403},[294],[213,9468,9040],{"className":9469,"style":9087},[299,407],[11,9471,9472,9473,9524],{},"现在问：",[213,9474,9476,9494],{"className":9475,"translate":217},[216],[213,9477,9479],{"className":9478},[221],[223,9480,9481],{"xmlns":225},[227,9482,9483,9491],{},[230,9484,9485,9487,9489],{},[233,9486,2631],{},[238,9488,930],{},[233,9490,9040],{},[279,9492,9493],{"encoding":281},"d \\in D",[213,9495,9497,9515],{"className":9496,"ariaHidden":251},[286],[213,9498,9500,9503,9506,9509,9512],{"className":9499},[290],[213,9501],{"className":9502,"style":945},[294],[213,9504,2631],{"className":9505},[299,407],[213,9507],{"className":9508,"style":305},[304],[213,9510,930],{"className":9511},[309],[213,9513],{"className":9514,"style":305},[304],[213,9516,9518,9521],{"className":9517},[290],[213,9519],{"className":9520,"style":403},[294],[213,9522,9040],{"className":9523,"style":9087},[299,407]," 吗？",[62,9526,9527,9971],{},[65,9528,9529,9530,9580,9581,9609,9610,9765,9766,9831,9832,9970],{},"若 ",[213,9531,9533,9550],{"className":9532,"translate":217},[216],[213,9534,9536],{"className":9535},[221],[223,9537,9538],{"xmlns":225},[227,9539,9540,9548],{},[230,9541,9542,9544,9546],{},[233,9543,2631],{},[238,9545,930],{},[233,9547,9040],{},[279,9549,9493],{"encoding":281},[213,9551,9553,9571],{"className":9552,"ariaHidden":251},[286],[213,9554,9556,9559,9562,9565,9568],{"className":9555},[290],[213,9557],{"className":9558,"style":945},[294],[213,9560,2631],{"className":9561},[299,407],[213,9563],{"className":9564,"style":305},[304],[213,9566,930],{"className":9567},[309],[213,9569],{"className":9570,"style":305},[304],[213,9572,9574,9577],{"className":9573},[290],[213,9575],{"className":9576,"style":403},[294],[213,9578,9040],{"className":9579,"style":9087},[299,407],"：根据 ",[213,9582,9584,9597],{"className":9583,"translate":217},[216],[213,9585,9587],{"className":9586},[221],[223,9588,9589],{"xmlns":225},[227,9590,9591,9595],{},[230,9592,9593],{},[233,9594,9040],{},[279,9596,9040],{"encoding":281},[213,9598,9600],{"className":9599,"ariaHidden":251},[286],[213,9601,9603,9606],{"className":9602},[290],[213,9604],{"className":9605,"style":403},[294],[213,9607,9040],{"className":9608,"style":9087},[299,407]," 的定义，",[213,9611,9613,9650],{"className":9612,"translate":217},[216],[213,9614,9616],{"className":9615},[221],[223,9617,9618],{"xmlns":225},[227,9619,9620,9647],{},[230,9621,9622,9624,9626,9628,9630,9633,9635,9637,9639,9641,9643,9645],{},[233,9623,2631],{},[238,9625,930],{},[233,9627,9040],{},[272,9629,5720],{},[238,9631,9632],{},"⟺",[272,9634,5720],{},[233,9636,2631],{},[238,9638,9059],{"mathvariant":545},[233,9640,454],{},[238,9642,560],{"stretchy":243},[233,9644,2631],{},[238,9646,569],{"stretchy":243},[279,9648,9649],{"encoding":281},"d \\in D \\iff d \\notin f(d)",[213,9651,9653,9671,9696,9747],{"className":9652,"ariaHidden":251},[286],[213,9654,9656,9659,9662,9665,9668],{"className":9655},[290],[213,9657],{"className":9658,"style":945},[294],[213,9660,2631],{"className":9661},[299,407],[213,9663],{"className":9664,"style":305},[304],[213,9666,930],{"className":9667},[309],[213,9669],{"className":9670,"style":305},[304],[213,9672,9674,9678,9681,9684,9687,9690,9693],{"className":9673},[290],[213,9675],{"className":9676,"style":9677},[294],"height:0.7073em;vertical-align:-0.024em;",[213,9679,9040],{"className":9680,"style":9087},[299,407],[213,9682],{"className":9683,"style":305},[304],[213,9685],{"className":9686,"style":305},[304],[213,9688,9632],{"className":9689},[309],[213,9691],{"className":9692,"style":305},[304],[213,9694],{"className":9695,"style":305},[304],[213,9697,9699,9702,9705,9708,9744],{"className":9698},[290],[213,9700],{"className":9701,"style":319},[294],[213,9703,2631],{"className":9704},[299,407],[213,9706],{"className":9707,"style":305},[304],[213,9709,9711,9717],{"className":9710},[309],[213,9712,9714],{"className":9713},[299],[213,9715,930],{"className":9716},[309],[213,9718,9720],{"className":9719},[299,664],[213,9721,9723],{"className":9722},[668],[213,9724,9726,9729,9741],{"className":9725},[9169],[213,9727],{"className":9728,"style":319},[294],[213,9730,9732],{"className":9731},[679],[213,9733,9735,9738],{"className":9734},[299],[213,9736,1762],{"className":9737},[299],[213,9739],{"className":9740,"style":9185},[304],[213,9742],{"className":9743},[690],[213,9745],{"className":9746,"style":305},[304],[213,9748,9750,9753,9756,9759,9762],{"className":9749},[290],[213,9751],{"className":9752,"style":319},[294],[213,9754,454],{"className":9755,"style":481},[299,407],[213,9757,560],{"className":9758},[323],[213,9760,2631],{"className":9761},[299,407],[213,9763,569],{"className":9764},[372],"。由于 ",[213,9767,9769,9792],{"className":9768,"translate":217},[216],[213,9770,9772],{"className":9771},[221],[223,9773,9774],{"xmlns":225},[227,9775,9776,9790],{},[230,9777,9778,9780,9782,9784,9786,9788],{},[233,9779,454],{},[238,9781,560],{"stretchy":243},[233,9783,2631],{},[238,9785,569],{"stretchy":243},[238,9787,240],{},[233,9789,9040],{},[279,9791,9430],{"encoding":281},[213,9793,9795,9822],{"className":9794,"ariaHidden":251},[286],[213,9796,9798,9801,9804,9807,9810,9813,9816,9819],{"className":9797},[290],[213,9799],{"className":9800,"style":319},[294],[213,9802,454],{"className":9803,"style":481},[299,407],[213,9805,560],{"className":9806},[323],[213,9808,2631],{"className":9809},[299,407],[213,9811,569],{"className":9812},[372],[213,9814],{"className":9815,"style":305},[304],[213,9817,240],{"className":9818},[309],[213,9820],{"className":9821,"style":305},[304],[213,9823,9825,9828],{"className":9824},[290],[213,9826],{"className":9827,"style":403},[294],[213,9829,9040],{"className":9830,"style":9087},[299,407],"，就有 ",[213,9833,9835,9865],{"className":9834,"translate":217},[216],[213,9836,9838],{"className":9837},[221],[223,9839,9840],{"xmlns":225},[227,9841,9842,9862],{},[230,9843,9844,9846,9848,9850,9852,9854,9856,9858,9860],{},[233,9845,2631],{},[238,9847,930],{},[233,9849,9040],{},[272,9851,5720],{},[238,9853,9632],{},[272,9855,5720],{},[233,9857,2631],{},[238,9859,9059],{"mathvariant":545},[233,9861,9040],{},[279,9863,9864],{"encoding":281},"d \\in D \\iff d \\notin D",[213,9866,9868,9886,9910,9961],{"className":9867,"ariaHidden":251},[286],[213,9869,9871,9874,9877,9880,9883],{"className":9870},[290],[213,9872],{"className":9873,"style":945},[294],[213,9875,2631],{"className":9876},[299,407],[213,9878],{"className":9879,"style":305},[304],[213,9881,930],{"className":9882},[309],[213,9884],{"className":9885,"style":305},[304],[213,9887,9889,9892,9895,9898,9901,9904,9907],{"className":9888},[290],[213,9890],{"className":9891,"style":9677},[294],[213,9893,9040],{"className":9894,"style":9087},[299,407],[213,9896],{"className":9897,"style":305},[304],[213,9899],{"className":9900,"style":305},[304],[213,9902,9632],{"className":9903},[309],[213,9905],{"className":9906,"style":305},[304],[213,9908],{"className":9909,"style":305},[304],[213,9911,9913,9916,9919,9922,9958],{"className":9912},[290],[213,9914],{"className":9915,"style":319},[294],[213,9917,2631],{"className":9918},[299,407],[213,9920],{"className":9921,"style":305},[304],[213,9923,9925,9931],{"className":9924},[309],[213,9926,9928],{"className":9927},[299],[213,9929,930],{"className":9930},[309],[213,9932,9934],{"className":9933},[299,664],[213,9935,9937],{"className":9936},[668],[213,9938,9940,9943,9955],{"className":9939},[9169],[213,9941],{"className":9942,"style":319},[294],[213,9944,9946],{"className":9945},[679],[213,9947,9949,9952],{"className":9948},[299],[213,9950,1762],{"className":9951},[299],[213,9953],{"className":9954,"style":9185},[304],[213,9956],{"className":9957},[690],[213,9959],{"className":9960,"style":305},[304],[213,9962,9964,9967],{"className":9963},[290],[213,9965],{"className":9966,"style":403},[294],[213,9968,9040],{"className":9969,"style":9087},[299,407],"。矛盾！",[65,9972,9529,9973,10057,10058,10258,10259,10347,10348,9970],{},[213,9974,9976,9994],{"className":9975,"translate":217},[216],[213,9977,9979],{"className":9978},[221],[223,9980,9981],{"xmlns":225},[227,9982,9983,9991],{},[230,9984,9985,9987,9989],{},[233,9986,2631],{},[238,9988,9059],{"mathvariant":545},[233,9990,9040],{},[279,9992,9993],{"encoding":281},"d \\notin D",[213,9995,9997,10048],{"className":9996,"ariaHidden":251},[286],[213,9998,10000,10003,10006,10009,10045],{"className":9999},[290],[213,10001],{"className":10002,"style":319},[294],[213,10004,2631],{"className":10005},[299,407],[213,10007],{"className":10008,"style":305},[304],[213,10010,10012,10018],{"className":10011},[309],[213,10013,10015],{"className":10014},[299],[213,10016,930],{"className":10017},[309],[213,10019,10021],{"className":10020},[299,664],[213,10022,10024],{"className":10023},[668],[213,10025,10027,10030,10042],{"className":10026},[9169],[213,10028],{"className":10029,"style":319},[294],[213,10031,10033],{"className":10032},[679],[213,10034,10036,10039],{"className":10035},[299],[213,10037,1762],{"className":10038},[299],[213,10040],{"className":10041,"style":9185},[304],[213,10043],{"className":10044},[690],[213,10046],{"className":10047,"style":305},[304],[213,10049,10051,10054],{"className":10050},[290],[213,10052],{"className":10053,"style":403},[294],[213,10055,9040],{"className":10056,"style":9087},[299,407],"：同理，",[213,10059,10061,10104],{"className":10060,"translate":217},[216],[213,10062,10064],{"className":10063},[221],[223,10065,10066],{"xmlns":225},[227,10067,10068,10101],{},[230,10069,10070,10072,10074,10076,10078,10080,10082,10085,10087,10089,10091,10093,10095,10097,10099],{},[233,10071,2631],{},[238,10073,9059],{"mathvariant":545},[233,10075,9040],{},[272,10077,5720],{},[238,10079,9632],{},[272,10081,5720],{},[233,10083,10084],{"mathvariant":545},"¬",[238,10086,560],{"stretchy":243},[233,10088,2631],{},[238,10090,9059],{"mathvariant":545},[233,10092,454],{},[238,10094,560],{"stretchy":243},[233,10096,2631],{},[238,10098,569],{"stretchy":243},[238,10100,569],{"stretchy":243},[279,10102,10103],{"encoding":281},"d \\notin D \\iff \\neg(d \\notin f(d))",[213,10105,10107,10158,10182,10239],{"className":10106,"ariaHidden":251},[286],[213,10108,10110,10113,10116,10119,10155],{"className":10109},[290],[213,10111],{"className":10112,"style":319},[294],[213,10114,2631],{"className":10115},[299,407],[213,10117],{"className":10118,"style":305},[304],[213,10120,10122,10128],{"className":10121},[309],[213,10123,10125],{"className":10124},[299],[213,10126,930],{"className":10127},[309],[213,10129,10131],{"className":10130},[299,664],[213,10132,10134],{"className":10133},[668],[213,10135,10137,10140,10152],{"className":10136},[9169],[213,10138],{"className":10139,"style":319},[294],[213,10141,10143],{"className":10142},[679],[213,10144,10146,10149],{"className":10145},[299],[213,10147,1762],{"className":10148},[299],[213,10150],{"className":10151,"style":9185},[304],[213,10153],{"className":10154},[690],[213,10156],{"className":10157,"style":305},[304],[213,10159,10161,10164,10167,10170,10173,10176,10179],{"className":10160},[290],[213,10162],{"className":10163,"style":9677},[294],[213,10165,9040],{"className":10166,"style":9087},[299,407],[213,10168],{"className":10169,"style":305},[304],[213,10171],{"className":10172,"style":305},[304],[213,10174,9632],{"className":10175},[309],[213,10177],{"className":10178,"style":305},[304],[213,10180],{"className":10181,"style":305},[304],[213,10183,10185,10188,10191,10194,10197,10200,10236],{"className":10184},[290],[213,10186],{"className":10187,"style":319},[294],[213,10189,10084],{"className":10190},[299],[213,10192,560],{"className":10193},[323],[213,10195,2631],{"className":10196},[299,407],[213,10198],{"className":10199,"style":305},[304],[213,10201,10203,10209],{"className":10202},[309],[213,10204,10206],{"className":10205},[299],[213,10207,930],{"className":10208},[309],[213,10210,10212],{"className":10211},[299,664],[213,10213,10215],{"className":10214},[668],[213,10216,10218,10221,10233],{"className":10217},[9169],[213,10219],{"className":10220,"style":319},[294],[213,10222,10224],{"className":10223},[679],[213,10225,10227,10230],{"className":10226},[299],[213,10228,1762],{"className":10229},[299],[213,10231],{"className":10232,"style":9185},[304],[213,10234],{"className":10235},[690],[213,10237],{"className":10238,"style":305},[304],[213,10240,10242,10245,10248,10251,10254],{"className":10241},[290],[213,10243],{"className":10244,"style":319},[294],[213,10246,454],{"className":10247,"style":481},[299,407],[213,10249,560],{"className":10250},[323],[213,10252,2631],{"className":10253},[299,407],[213,10255,10257],{"className":10256},[372],"))","，即 ",[213,10260,10262,10290],{"className":10261,"translate":217},[216],[213,10263,10265],{"className":10264},[221],[223,10266,10267],{"xmlns":225},[227,10268,10269,10287],{},[230,10270,10271,10273,10275,10277,10279,10281,10283,10285],{},[233,10272,2631],{},[238,10274,930],{},[233,10276,454],{},[238,10278,560],{"stretchy":243},[233,10280,2631],{},[238,10282,569],{"stretchy":243},[238,10284,240],{},[233,10286,9040],{},[279,10288,10289],{"encoding":281},"d \\in f(d) = D",[213,10291,10293,10311,10338],{"className":10292,"ariaHidden":251},[286],[213,10294,10296,10299,10302,10305,10308],{"className":10295},[290],[213,10297],{"className":10298,"style":945},[294],[213,10300,2631],{"className":10301},[299,407],[213,10303],{"className":10304,"style":305},[304],[213,10306,930],{"className":10307},[309],[213,10309],{"className":10310,"style":305},[304],[213,10312,10314,10317,10320,10323,10326,10329,10332,10335],{"className":10313},[290],[213,10315],{"className":10316,"style":319},[294],[213,10318,454],{"className":10319,"style":481},[299,407],[213,10321,560],{"className":10322},[323],[213,10324,2631],{"className":10325},[299,407],[213,10327,569],{"className":10328},[372],[213,10330],{"className":10331,"style":305},[304],[213,10333,240],{"className":10334},[309],[213,10336],{"className":10337,"style":305},[304],[213,10339,10341,10344],{"className":10340},[290],[213,10342],{"className":10343,"style":403},[294],[213,10345,9040],{"className":10346,"style":9087},[299,407],"，从而 ",[213,10349,10351,10368],{"className":10350,"translate":217},[216],[213,10352,10354],{"className":10353},[221],[223,10355,10356],{"xmlns":225},[227,10357,10358,10366],{},[230,10359,10360,10362,10364],{},[233,10361,2631],{},[238,10363,930],{},[233,10365,9040],{},[279,10367,9493],{"encoding":281},[213,10369,10371,10389],{"className":10370,"ariaHidden":251},[286],[213,10372,10374,10377,10380,10383,10386],{"className":10373},[290],[213,10375],{"className":10376,"style":945},[294],[213,10378,2631],{"className":10379},[299,407],[213,10381],{"className":10382,"style":305},[304],[213,10384,930],{"className":10385},[309],[213,10387],{"className":10388,"style":305},[304],[213,10390,10392,10395],{"className":10391},[290],[213,10393],{"className":10394,"style":403},[294],[213,10396,9040],{"className":10397,"style":9087},[299,407],[11,10399,10400,10401,10488,10489,1551],{},"两种情况都导致自相矛盾，说明「存在满射 ",[213,10402,10404,10431],{"className":10403,"translate":217},[216],[213,10405,10407],{"className":10406},[221],[223,10408,10409],{"xmlns":225},[227,10410,10411,10429],{},[230,10412,10413,10415,10417,10419,10421,10423,10425,10427],{},[233,10414,454],{},[238,10416,457],{},[233,10418,391],{},[238,10420,462],{},[233,10422,8461],{"mathvariant":8460},[238,10424,560],{"stretchy":243},[233,10426,391],{},[238,10428,569],{"stretchy":243},[279,10430,8752],{"encoding":281},[213,10432,10434,10452,10470],{"className":10433,"ariaHidden":251},[286],[213,10435,10437,10440,10443,10446,10449],{"className":10436},[290],[213,10438],{"className":10439,"style":477},[294],[213,10441,454],{"className":10442,"style":481},[299,407],[213,10444],{"className":10445,"style":305},[304],[213,10447,457],{"className":10448},[309],[213,10450],{"className":10451,"style":305},[304],[213,10453,10455,10458,10461,10464,10467],{"className":10454},[290],[213,10456],{"className":10457,"style":403},[294],[213,10459,391],{"className":10460},[299,407],[213,10462],{"className":10463,"style":305},[304],[213,10465,462],{"className":10466},[309],[213,10468],{"className":10469,"style":305},[304],[213,10471,10473,10476,10479,10482,10485],{"className":10472},[290],[213,10474],{"className":10475,"style":319},[294],[213,10477,8461],{"className":10478,"style":8484},[299,8483],[213,10480,560],{"className":10481},[323],[213,10483,391],{"className":10484},[299,407],[213,10486,569],{"className":10487},[372],"」这一假设站不住脚。因此 ",[213,10490,10492,10523],{"className":10491,"translate":217},[216],[213,10493,10495],{"className":10494},[221],[223,10496,10497],{"xmlns":225},[227,10498,10499,10521],{},[230,10500,10501,10503,10505,10507,10509,10511,10513,10515,10517,10519],{},[233,10502,1192],{"mathvariant":545},[233,10504,391],{},[233,10506,1192],{"mathvariant":545},[238,10508,8572],{},[233,10510,1192],{"mathvariant":545},[233,10512,8461],{"mathvariant":8460},[238,10514,560],{"stretchy":243},[233,10516,391],{},[238,10518,569],{"stretchy":243},[233,10520,1192],{"mathvariant":545},[279,10522,8587],{"encoding":281},[213,10524,10526,10550],{"className":10525,"ariaHidden":251},[286],[213,10527,10529,10532,10535,10538,10541,10544,10547],{"className":10528},[290],[213,10530],{"className":10531,"style":319},[294],[213,10533,1192],{"className":10534},[299],[213,10536,391],{"className":10537},[299,407],[213,10539,1192],{"className":10540},[299],[213,10542],{"className":10543,"style":305},[304],[213,10545,8572],{"className":10546},[309],[213,10548],{"className":10549,"style":305},[304],[213,10551,10553,10556,10559,10562,10565,10568,10571],{"className":10552},[290],[213,10554],{"className":10555,"style":319},[294],[213,10557,1192],{"className":10558},[299],[213,10560,8461],{"className":10561,"style":8484},[299,8483],[213,10563,560],{"className":10564},[323],[213,10566,391],{"className":10567},[299,407],[213,10569,569],{"className":10570},[372],[213,10572,1192],{"className":10573},[299],[11,10575,10576],{},"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：",[10578,10579,10580,10592],"table",{},[10581,10582,10583],"thead",{},[10584,10585,10586,10590],"tr",{},[10587,10588,10589],"th",{},"实数的不可数性",[10587,10591,8408],{},[10593,10594,10595,10786,11073,11361],"tbody",{},[10584,10596,10597,10696],{},[10598,10599,10600,10601],"td",{},"假设双射 ",[213,10602,10604,10633],{"className":10603,"translate":217},[216],[213,10605,10607],{"className":10606},[221],[223,10608,10609],{"xmlns":225},[227,10610,10611,10631],{},[230,10612,10613,10615,10617,10619,10621,10623,10625,10627,10629],{},[233,10614,454],{},[238,10616,457],{},[233,10618,236],{"mathvariant":235},[238,10620,462],{},[238,10622,560],{"stretchy":243},[246,10624,248],{},[238,10626,252],{"separator":251},[246,10628,255],{},[238,10630,569],{"stretchy":243},[279,10632,2516],{"encoding":281},[213,10634,10636,10654,10672],{"className":10635,"ariaHidden":251},[286],[213,10637,10639,10642,10645,10648,10651],{"className":10638},[290],[213,10640],{"className":10641,"style":477},[294],[213,10643,454],{"className":10644,"style":481},[299,407],[213,10646],{"className":10647,"style":305},[304],[213,10649,457],{"className":10650},[309],[213,10652],{"className":10653,"style":305},[304],[213,10655,10657,10660,10663,10666,10669],{"className":10656},[290],[213,10658],{"className":10659,"style":295},[294],[213,10661,236],{"className":10662},[299,300],[213,10664],{"className":10665,"style":305},[304],[213,10667,462],{"className":10668},[309],[213,10670],{"className":10671,"style":305},[304],[213,10673,10675,10678,10681,10684,10687,10690,10693],{"className":10674},[290],[213,10676],{"className":10677,"style":319},[294],[213,10679,560],{"className":10680},[323],[213,10682,248],{"className":10683},[299],[213,10685,252],{"className":10686},[330],[213,10688],{"className":10689,"style":334},[304],[213,10691,255],{"className":10692},[299],[213,10694,569],{"className":10695},[372],[10598,10697,10698,10699],{},"假设满射 ",[213,10700,10702,10729],{"className":10701,"translate":217},[216],[213,10703,10705],{"className":10704},[221],[223,10706,10707],{"xmlns":225},[227,10708,10709,10727],{},[230,10710,10711,10713,10715,10717,10719,10721,10723,10725],{},[233,10712,454],{},[238,10714,457],{},[233,10716,391],{},[238,10718,462],{},[233,10720,8461],{"mathvariant":8460},[238,10722,560],{"stretchy":243},[233,10724,391],{},[238,10726,569],{"stretchy":243},[279,10728,8752],{"encoding":281},[213,10730,10732,10750,10768],{"className":10731,"ariaHidden":251},[286],[213,10733,10735,10738,10741,10744,10747],{"className":10734},[290],[213,10736],{"className":10737,"style":477},[294],[213,10739,454],{"className":10740,"style":481},[299,407],[213,10742],{"className":10743,"style":305},[304],[213,10745,457],{"className":10746},[309],[213,10748],{"className":10749,"style":305},[304],[213,10751,10753,10756,10759,10762,10765],{"className":10752},[290],[213,10754],{"className":10755,"style":403},[294],[213,10757,391],{"className":10758},[299,407],[213,10760],{"className":10761,"style":305},[304],[213,10763,462],{"className":10764},[309],[213,10766],{"className":10767,"style":305},[304],[213,10769,10771,10774,10777,10780,10783],{"className":10770},[290],[213,10772],{"className":10773,"style":319},[294],[213,10775,8461],{"className":10776,"style":8484},[299,8483],[213,10778,560],{"className":10779},[323],[213,10781,391],{"className":10782},[299,407],[213,10784,569],{"className":10785},[372],[10584,10787,10788,10919],{},[10598,10789,10790,10791,10819,10820,5334,10890,10918],{},"构造对角数 ",[213,10792,10794,10807],{"className":10793,"translate":217},[216],[213,10795,10797],{"className":10796},[221],[223,10798,10799],{"xmlns":225},[227,10800,10801,10805],{},[230,10802,10803],{},[233,10804,4247],{},[279,10806,4247],{"encoding":281},[213,10808,10810],{"className":10809,"ariaHidden":251},[286],[213,10811,10813,10816],{"className":10812},[290],[213,10814],{"className":10815,"style":4291},[294],[213,10817,4247],{"className":10818,"style":2305},[299,407],"，每一位都与 ",[213,10821,10823,10841],{"className":10822,"translate":217},[216],[213,10824,10826],{"className":10825},[221],[223,10827,10828],{"xmlns":225},[227,10829,10830,10838],{},[230,10831,10832],{},[537,10833,10834,10836],{},[233,10835,2611],{},[233,10837,1496],{},[279,10839,10840],{"encoding":281},"x_n",[213,10842,10844],{"className":10843,"ariaHidden":251},[286],[213,10845,10847,10850],{"className":10846},[290],[213,10848],{"className":10849,"style":707},[294],[213,10851,10853,10856],{"className":10852},[299],[213,10854,2611],{"className":10855},[299,407],[213,10857,10859],{"className":10858},[605],[213,10860,10862,10882],{"className":10861},[609,610],[213,10863,10865,10879],{"className":10864},[614],[213,10866,10868],{"className":10867,"style":4602},[618],[213,10869,10870,10873],{"style":622},[213,10871],{"className":10872,"style":627},[626],[213,10874,10876],{"className":10875},[631,632,633,634],[213,10877,1496],{"className":10878},[299,407,634],[213,10880,642],{"className":10881},[641],[213,10883,10885],{"className":10884},[614],[213,10886,10888],{"className":10887,"style":649},[618],[213,10889],{},[213,10891,10893,10906],{"className":10892,"translate":217},[216],[213,10894,10896],{"className":10895},[221],[223,10897,10898],{"xmlns":225},[227,10899,10900,10904],{},[230,10901,10902],{},[233,10903,1496],{},[279,10905,1496],{"encoding":281},[213,10907,10909],{"className":10908,"ariaHidden":251},[286],[213,10910,10912,10915],{"className":10911},[290],[213,10913],{"className":10914,"style":6175},[294],[213,10916,1496],{"className":10917},[299,407]," 位不同",[10598,10920,10921,10922],{},"构造对角集 ",[213,10923,10925,10961],{"className":10924,"translate":217},[216],[213,10926,10928],{"className":10927},[221],[223,10929,10930],{"xmlns":225},[227,10931,10932,10958],{},[230,10933,10934,10936,10938,10940,10942,10944,10946,10948,10950,10952,10954,10956],{},[233,10935,9040],{},[238,10937,240],{},[238,10939,244],{"stretchy":243},[233,10941,107],{},[238,10943,457],{},[233,10945,107],{},[238,10947,9059],{"mathvariant":545},[233,10949,454],{},[238,10951,560],{"stretchy":243},[233,10953,107],{},[238,10955,569],{"stretchy":243},[238,10957,277],{"stretchy":243},[279,10959,10960],{"encoding":281},"D = \\{a : a \\notin f(a)\\}",[213,10962,10964,10982,11003,11054],{"className":10963,"ariaHidden":251},[286],[213,10965,10967,10970,10973,10976,10979],{"className":10966},[290],[213,10968],{"className":10969,"style":403},[294],[213,10971,9040],{"className":10972,"style":9087},[299,407],[213,10974],{"className":10975,"style":305},[304],[213,10977,240],{"className":10978},[309],[213,10980],{"className":10981,"style":305},[304],[213,10983,10985,10988,10991,10994,10997,11000],{"className":10984},[290],[213,10986],{"className":10987,"style":319},[294],[213,10989,244],{"className":10990},[323],[213,10992,107],{"className":10993},[299,407],[213,10995],{"className":10996,"style":305},[304],[213,10998,457],{"className":10999},[309],[213,11001],{"className":11002,"style":305},[304],[213,11004,11006,11009,11012,11015,11051],{"className":11005},[290],[213,11007],{"className":11008,"style":319},[294],[213,11010,107],{"className":11011},[299,407],[213,11013],{"className":11014,"style":305},[304],[213,11016,11018,11024],{"className":11017},[309],[213,11019,11021],{"className":11020},[299],[213,11022,930],{"className":11023},[309],[213,11025,11027],{"className":11026},[299,664],[213,11028,11030],{"className":11029},[668],[213,11031,11033,11036,11048],{"className":11032},[9169],[213,11034],{"className":11035,"style":319},[294],[213,11037,11039],{"className":11038},[679],[213,11040,11042,11045],{"className":11041},[299],[213,11043,1762],{"className":11044},[299],[213,11046],{"className":11047,"style":9185},[304],[213,11049],{"className":11050},[690],[213,11052],{"className":11053,"style":305},[304],[213,11055,11057,11060,11063,11066,11069],{"className":11056},[290],[213,11058],{"className":11059,"style":319},[294],[213,11061,454],{"className":11062,"style":481},[299,407],[213,11064,560],{"className":11065},[323],[213,11067,107],{"className":11068},[299,407],[213,11070,11072],{"className":11071},[372],")}",[10584,11074,11075,11232],{},[10598,11076,11077,11078,11106,11107],{},"对所有 ",[213,11079,11081,11094],{"className":11080,"translate":217},[216],[213,11082,11084],{"className":11083},[221],[223,11085,11086],{"xmlns":225},[227,11087,11088,11092],{},[230,11089,11090],{},[233,11091,1496],{},[279,11093,1496],{"encoding":281},[213,11095,11097],{"className":11096,"ariaHidden":251},[286],[213,11098,11100,11103],{"className":11099},[290],[213,11101],{"className":11102,"style":6175},[294],[213,11104,1496],{"className":11105},[299,407]," 有 ",[213,11108,11110,11132],{"className":11109,"translate":217},[216],[213,11111,11113],{"className":11112},[221],[223,11114,11115],{"xmlns":225},[227,11116,11117,11129],{},[230,11118,11119,11121,11123],{},[233,11120,4247],{},[238,11122,546],{"mathvariant":545},[537,11124,11125,11127],{},[233,11126,2611],{},[233,11128,1496],{},[279,11130,11131],{"encoding":281},"y \\neq x_n",[213,11133,11135,11186],{"className":11134,"ariaHidden":251},[286],[213,11136,11138,11141,11144,11147,11183],{"className":11137},[290],[213,11139],{"className":11140,"style":477},[294],[213,11142,4247],{"className":11143,"style":2305},[299,407],[213,11145],{"className":11146,"style":305},[304],[213,11148,11150,11177,11180],{"className":11149},[309],[213,11151,11153],{"className":11152},[309],[213,11154,11156],{"className":11155},[299,664],[213,11157,11159],{"className":11158},[668],[213,11160,11162,11165,11174],{"className":11161},[672],[213,11163],{"className":11164,"style":477},[294],[213,11166,11168],{"className":11167},[679],[213,11169,11171],{"className":11170},[299],[213,11172,686],{"className":11173},[309],[213,11175],{"className":11176},[690],[213,11178],{"className":11179},[304,694],[213,11181,240],{"className":11182},[309],[213,11184],{"className":11185,"style":305},[304],[213,11187,11189,11192],{"className":11188},[290],[213,11190],{"className":11191,"style":707},[294],[213,11193,11195,11198],{"className":11194},[299],[213,11196,2611],{"className":11197},[299,407],[213,11199,11201],{"className":11200},[605],[213,11202,11204,11224],{"className":11203},[609,610],[213,11205,11207,11221],{"className":11206},[614],[213,11208,11210],{"className":11209,"style":4602},[618],[213,11211,11212,11215],{"style":622},[213,11213],{"className":11214,"style":627},[626],[213,11216,11218],{"className":11217},[631,632,633,634],[213,11219,1496],{"className":11220},[299,407,634],[213,11222,642],{"className":11223},[641],[213,11225,11227],{"className":11226},[614],[213,11228,11230],{"className":11229,"style":649},[618],[213,11231],{},[10598,11233,11077,11234,11106,11262],{},[213,11235,11237,11250],{"className":11236,"translate":217},[216],[213,11238,11240],{"className":11239},[221],[223,11241,11242],{"xmlns":225},[227,11243,11244,11248],{},[230,11245,11246],{},[233,11247,107],{},[279,11249,107],{"encoding":281},[213,11251,11253],{"className":11252,"ariaHidden":251},[286],[213,11254,11256,11259],{"className":11255},[290],[213,11257],{"className":11258,"style":6175},[294],[213,11260,107],{"className":11261},[299,407],[213,11263,11265,11289],{"className":11264,"translate":217},[216],[213,11266,11268],{"className":11267},[221],[223,11269,11270],{"xmlns":225},[227,11271,11272,11286],{},[230,11273,11274,11276,11278,11280,11282,11284],{},[233,11275,9040],{},[238,11277,546],{"mathvariant":545},[233,11279,454],{},[238,11281,560],{"stretchy":243},[233,11283,107],{},[238,11285,569],{"stretchy":243},[279,11287,11288],{"encoding":281},"D \\neq f(a)",[213,11290,11292,11343],{"className":11291,"ariaHidden":251},[286],[213,11293,11295,11298,11301,11304,11340],{"className":11294},[290],[213,11296],{"className":11297,"style":477},[294],[213,11299,9040],{"className":11300,"style":9087},[299,407],[213,11302],{"className":11303,"style":305},[304],[213,11305,11307,11334,11337],{"className":11306},[309],[213,11308,11310],{"className":11309},[309],[213,11311,11313],{"className":11312},[299,664],[213,11314,11316],{"className":11315},[668],[213,11317,11319,11322,11331],{"className":11318},[672],[213,11320],{"className":11321,"style":477},[294],[213,11323,11325],{"className":11324},[679],[213,11326,11328],{"className":11327},[299],[213,11329,686],{"className":11330},[309],[213,11332],{"className":11333},[690],[213,11335],{"className":11336},[304,694],[213,11338,240],{"className":11339},[309],[213,11341],{"className":11342,"style":305},[304],[213,11344,11346,11349,11352,11355,11358],{"className":11345},[290],[213,11347],{"className":11348,"style":319},[294],[213,11350,454],{"className":11351,"style":481},[299,407],[213,11353,560],{"className":11354},[323],[213,11356,107],{"className":11357},[299,407],[213,11359,569],{"className":11360},[372],[10584,11362,11363,11394],{},[10598,11364,11365,11393],{},[213,11366,11368,11381],{"className":11367,"translate":217},[216],[213,11369,11371],{"className":11370},[221],[223,11372,11373],{"xmlns":225},[227,11374,11375,11379],{},[230,11376,11377],{},[233,11378,4247],{},[279,11380,4247],{"encoding":281},[213,11382,11384],{"className":11383,"ariaHidden":251},[286],[213,11385,11387,11390],{"className":11386},[290],[213,11388],{"className":11389,"style":4291},[294],[213,11391,4247],{"className":11392,"style":2305},[299,407]," 无法被列表覆盖；矛盾",[10598,11395,11396,11424],{},[213,11397,11399,11412],{"className":11398,"translate":217},[216],[213,11400,11402],{"className":11401},[221],[223,11403,11404],{"xmlns":225},[227,11405,11406,11410],{},[230,11407,11408],{},[233,11409,9040],{},[279,11411,9040],{"encoding":281},[213,11413,11415],{"className":11414,"ariaHidden":251},[286],[213,11416,11418,11421],{"className":11417},[290],[213,11419],{"className":11420,"style":403},[294],[213,11422,9040],{"className":11423,"style":9087},[299,407]," 无法被映射覆盖；矛盾",[11,11426,11427,11428,11479,11480,11569,11570,11654,11655,11699,11700,11751],{},"事实上，",[213,11429,11431,11452],{"className":11430,"translate":217},[216],[213,11432,11434],{"className":11433},[221],[223,11435,11436],{"xmlns":225},[227,11437,11438,11450],{},[230,11439,11440,11442,11444,11446,11448],{},[238,11441,560],{"stretchy":243},[246,11443,248],{},[238,11445,252],{"separator":251},[246,11447,255],{},[238,11449,569],{"stretchy":243},[279,11451,2342],{"encoding":281},[213,11453,11455],{"className":11454,"ariaHidden":251},[286],[213,11456,11458,11461,11464,11467,11470,11473,11476],{"className":11457},[290],[213,11459],{"className":11460,"style":319},[294],[213,11462,560],{"className":11463},[323],[213,11465,248],{"className":11466},[299],[213,11468,252],{"className":11469},[330],[213,11471],{"className":11472,"style":334},[304],[213,11474,255],{"className":11475},[299],[213,11477,569],{"className":11478},[372]," 中的实数可以与 ",[213,11481,11483,11510],{"className":11482,"translate":217},[216],[213,11484,11486],{"className":11485},[221],[223,11487,11488],{"xmlns":225},[227,11489,11490,11507],{},[230,11491,11492,11494,11496,11498,11500],{},[238,11493,244],{"stretchy":243},[246,11495,248],{},[238,11497,252],{"separator":251},[246,11499,255],{},[11501,11502,11503,11505],"msup",{},[238,11504,277],{"stretchy":243},[233,11506,236],{"mathvariant":235},[279,11508,11509],{"encoding":281},"\\{0,1\\}^{\\mathbb{N}}",[213,11511,11513],{"className":11512,"ariaHidden":251},[286],[213,11514,11516,11520,11523,11526,11529,11532,11535],{"className":11515},[290],[213,11517],{"className":11518,"style":11519},[294],"height:1.0952em;vertical-align:-0.25em;",[213,11521,244],{"className":11522},[323],[213,11524,248],{"className":11525},[299],[213,11527,252],{"className":11528},[330],[213,11530],{"className":11531,"style":334},[304],[213,11533,255],{"className":11534},[299],[213,11536,11538,11541],{"className":11537},[372],[213,11539,277],{"className":11540},[372],[213,11542,11544],{"className":11543},[605],[213,11545,11547],{"className":11546},[609],[213,11548,11550],{"className":11549},[614],[213,11551,11554],{"className":11552,"style":11553},[618],"height:0.8452em;",[213,11555,11557,11560],{"style":11556},"top:-3.063em;margin-right:0.05em;",[213,11558],{"className":11559,"style":627},[626],[213,11561,11563],{"className":11562},[631,632,633,634],[213,11564,11566],{"className":11565},[299,634],[213,11567,236],{"className":11568},[299,300,634],"（由 0 和 1 组成的无穷二进制序列集）近似对应，而 ",[213,11571,11573,11598],{"className":11572,"translate":217},[216],[213,11574,11576],{"className":11575},[221],[223,11577,11578],{"xmlns":225},[227,11579,11580,11596],{},[230,11581,11582,11584,11586,11588,11590],{},[238,11583,244],{"stretchy":243},[246,11585,248],{},[238,11587,252],{"separator":251},[246,11589,255],{},[11501,11591,11592,11594],{},[238,11593,277],{"stretchy":243},[233,11595,236],{"mathvariant":235},[279,11597,11509],{"encoding":281},[213,11599,11601],{"className":11600,"ariaHidden":251},[286],[213,11602,11604,11607,11610,11613,11616,11619,11622],{"className":11603},[290],[213,11605],{"className":11606,"style":11519},[294],[213,11608,244],{"className":11609},[323],[213,11611,248],{"className":11612},[299],[213,11614,252],{"className":11615},[330],[213,11617],{"className":11618,"style":334},[304],[213,11620,255],{"className":11621},[299],[213,11623,11625,11628],{"className":11624},[372],[213,11626,277],{"className":11627},[372],[213,11629,11631],{"className":11630},[605],[213,11632,11634],{"className":11633},[609],[213,11635,11637],{"className":11636},[614],[213,11638,11640],{"className":11639,"style":11553},[618],[213,11641,11642,11645],{"style":11556},[213,11643],{"className":11644,"style":627},[626],[213,11646,11648],{"className":11647},[631,632,633,634],[213,11649,11651],{"className":11650},[299,634],[213,11652,236],{"className":11653},[299,300,634]," 本质上就是 ",[213,11656,11658,11678],{"className":11657,"translate":217},[216],[213,11659,11661],{"className":11660},[221],[223,11662,11663],{"xmlns":225},[227,11664,11665,11675],{},[230,11666,11667,11669,11671,11673],{},[233,11668,8461],{"mathvariant":8460},[238,11670,560],{"stretchy":243},[233,11672,236],{"mathvariant":235},[238,11674,569],{"stretchy":243},[279,11676,11677],{"encoding":281},"\\mathcal{P}(\\mathbb{N})",[213,11679,11681],{"className":11680,"ariaHidden":251},[286],[213,11682,11684,11687,11690,11693,11696],{"className":11683},[290],[213,11685],{"className":11686,"style":319},[294],[213,11688,8461],{"className":11689,"style":8484},[299,8483],[213,11691,560],{"className":11692},[323],[213,11694,236],{"className":11695},[299,300],[213,11697,569],{"className":11698},[372],"（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 ",[213,11701,11703,11721],{"className":11702,"translate":217},[216],[213,11704,11706],{"className":11705},[221],[223,11707,11708],{"xmlns":225},[227,11709,11710,11718],{},[230,11711,11712,11714,11716],{},[233,11713,391],{},[238,11715,240],{},[233,11717,236],{"mathvariant":235},[279,11719,11720],{"encoding":281},"A = \\mathbb{N}",[213,11722,11724,11742],{"className":11723,"ariaHidden":251},[286],[213,11725,11727,11730,11733,11736,11739],{"className":11726},[290],[213,11728],{"className":11729,"style":403},[294],[213,11731,391],{"className":11732},[299,407],[213,11734],{"className":11735,"style":305},[304],[213,11737,240],{"className":11738},[309],[213,11740],{"className":11741,"style":305},[304],[213,11743,11745,11748],{"className":11744},[290],[213,11746],{"className":11747,"style":295},[294],[213,11749,236],{"className":11750},[299,300]," 这一特例下的具体体现。",[11,11753,11754],{},"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：",[7926,11756,11758],{"className":7928,"code":11757,"language":7930,"meta":172,"style":172},"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",[7932,11759,11760,11770,11775,11785,11797,11845,11849,11867,11877,11911,11915,11920,11953,11975,11979,12008,12041,12045],{"__ignoreMap":172},[213,11761,11762,11764,11767],{"class":7936,"line":7937},[213,11763,7941],{"class":7940},[213,11765,11766],{"class":7944}," cantor_diagonal_demo",[213,11768,11769],{"class":7948},"(A):\n",[213,11771,11772],{"class":7936,"line":173},[213,11773,11774],{"class":8055},"    # 构造一个「任意」的映射 f: A -> P(A) 用于演示\n",[213,11776,11777,11780,11782],{"class":7936,"line":7967},[213,11778,11779],{"class":7948},"    f ",[213,11781,240],{"class":7940},[213,11783,11784],{"class":7948}," {}\n",[213,11786,11787,11789,11792,11794],{"class":7936,"line":7973},[213,11788,7998],{"class":7940},[213,11790,11791],{"class":7948}," i ",[213,11793,8004],{"class":7940},[213,11795,11796],{"class":7948}," A:\n",[213,11798,11799,11802,11804,11807,11810,11813,11816,11818,11821,11824,11827,11829,11832,11834,11837,11840,11843],{"class":7936,"line":7979},[213,11800,11801],{"class":7948},"        f[i] ",[213,11803,240],{"class":7940},[213,11805,11806],{"class":7954}," frozenset",[213,11808,11809],{"class":7948},"(j ",[213,11811,11812],{"class":7940},"for",[213,11814,11815],{"class":7948}," j ",[213,11817,8004],{"class":7940},[213,11819,11820],{"class":7948}," A ",[213,11822,11823],{"class":7940},"if",[213,11825,11826],{"class":7948}," (i ",[213,11828,1792],{"class":7940},[213,11830,11831],{"class":7948}," j) ",[213,11833,8078],{"class":7940},[213,11835,11836],{"class":7954}," 2",[213,11838,11839],{"class":7940}," ==",[213,11841,11842],{"class":7954}," 0",[213,11844,8038],{"class":7948},[213,11846,11847],{"class":7936,"line":7984},[213,11848,8120],{"emptyLinePlaceholder":179},[213,11850,11851,11854,11856,11859,11861,11864],{"class":7936,"line":7995},[213,11852,11853],{"class":7954},"    print",[213,11855,560],{"class":7948},[213,11857,11858],{"class":7963},"\"集合 A =\"",[213,11860,8145],{"class":7948},[213,11862,11863],{"class":7954},"set",[213,11865,11866],{"class":7948},"(A))\n",[213,11868,11869,11871,11873,11875],{"class":7936,"line":8013},[213,11870,7998],{"class":7940},[213,11872,11791],{"class":7948},[213,11874,8004],{"class":7940},[213,11876,11796],{"class":7948},[213,11878,11879,11882,11884,11886,11889,11891,11893,11895,11898,11901,11904,11906,11909],{"class":7936,"line":8041},[213,11880,11881],{"class":7954},"        print",[213,11883,560],{"class":7948},[213,11885,454],{"class":7940},[213,11887,11888],{"class":7963},"\"  f(",[213,11890,244],{"class":7954},[213,11892,4046],{"class":7948},[213,11894,277],{"class":7954},[213,11896,11897],{"class":7963},") = ",[213,11899,11900],{"class":7954},"{set",[213,11902,11903],{"class":7948},"(f[i])",[213,11905,277],{"class":7954},[213,11907,11908],{"class":7963},"\"",[213,11910,8038],{"class":7948},[213,11912,11913],{"class":7936,"line":8059},[213,11914,8120],{"emptyLinePlaceholder":179},[213,11916,11917],{"class":7936,"line":8087},[213,11918,11919],{"class":8055},"    # 对角线构造：D = { i in A : i not in f(i) }\n",[213,11921,11922,11925,11927,11929,11932,11934,11936,11938,11940,11942,11944,11947,11950],{"class":7936,"line":8099},[213,11923,11924],{"class":7948},"    D ",[213,11926,240],{"class":7940},[213,11928,11806],{"class":7954},[213,11930,11931],{"class":7948},"(i ",[213,11933,11812],{"class":7940},[213,11935,11791],{"class":7948},[213,11937,8004],{"class":7940},[213,11939,11820],{"class":7948},[213,11941,11823],{"class":7940},[213,11943,11791],{"class":7948},[213,11945,11946],{"class":7940},"not",[213,11948,11949],{"class":7940}," in",[213,11951,11952],{"class":7948}," f[i])\n",[213,11954,11955,11957,11959,11961,11964,11966,11969,11971,11973],{"class":7936,"line":8117},[213,11956,11853],{"class":7954},[213,11958,560],{"class":7948},[213,11960,454],{"class":7940},[213,11962,11963],{"class":7963},"\"对角线构造出的集合 D = ",[213,11965,11900],{"class":7954},[213,11967,11968],{"class":7948},"(D)",[213,11970,277],{"class":7954},[213,11972,11908],{"class":7963},[213,11974,8038],{"class":7948},[213,11976,11977],{"class":7936,"line":8123},[213,11978,8120],{"emptyLinePlaceholder":179},[213,11980,11981,11984,11986,11989,11991,11993,11995,11997,11999,12002,12005],{"class":7936,"line":8128},[213,11982,11983],{"class":7948},"    matches ",[213,11985,240],{"class":7940},[213,11987,11988],{"class":7948}," [i ",[213,11990,11812],{"class":7940},[213,11992,11791],{"class":7948},[213,11994,8004],{"class":7940},[213,11996,11820],{"class":7948},[213,11998,11823],{"class":7940},[213,12000,12001],{"class":7948}," f[i] ",[213,12003,12004],{"class":7940},"==",[213,12006,12007],{"class":7948}," D]\n",[213,12009,12010,12012,12014,12016,12019,12021,12024,12026,12029,12032,12035,12037,12039],{"class":7936,"line":8139},[213,12011,11853],{"class":7954},[213,12013,560],{"class":7948},[213,12015,454],{"class":7940},[213,12017,12018],{"class":7963},"\"是否存在 i 使得 f(i) = D？ -> ",[213,12020,244],{"class":7954},[213,12022,12023],{"class":7948},"matches ",[213,12025,11823],{"class":7940},[213,12027,12028],{"class":7948}," matches ",[213,12030,12031],{"class":7940},"else",[213,12033,12034],{"class":7963}," '不存在（矛盾已建立）'",[213,12036,277],{"class":7954},[213,12038,11908],{"class":7963},[213,12040,8038],{"class":7948},[213,12042,12043],{"class":7936,"line":8164},[213,12044,8120],{"emptyLinePlaceholder":179},[213,12046,12047,12050,12052,12054,12056,12058,12060,12062,12064],{"class":7936,"line":8187},[213,12048,12049],{"class":7948},"cantor_diagonal_demo([",[213,12051,248],{"class":7954},[213,12053,8145],{"class":7948},[213,12055,255],{"class":7954},[213,12057,8145],{"class":7948},[213,12059,260],{"class":7954},[213,12061,8145],{"class":7948},[213,12063,265],{"class":7954},[213,12065,12066],{"class":7948},"])\n",[11,12068,8236],{},[7926,12070,12073],{"className":12071,"code":12072,"language":1983},[8240],"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",[7932,12074,12072],{"__ignoreMap":172},[11,12076,12077,12078,12106,12107,12135,12136,12164],{},"这里构造出的 ",[213,12079,12081,12094],{"className":12080,"translate":217},[216],[213,12082,12084],{"className":12083},[221],[223,12085,12086],{"xmlns":225},[227,12087,12088,12092],{},[230,12089,12090],{},[233,12091,9040],{},[279,12093,9040],{"encoding":281},[213,12095,12097],{"className":12096,"ariaHidden":251},[286],[213,12098,12100,12103],{"className":12099},[290],[213,12101],{"className":12102,"style":403},[294],[213,12104,9040],{"className":12105,"style":9087},[299,407]," 恰好是空集，而这个特定的映射 ",[213,12108,12110,12123],{"className":12109,"translate":217},[216],[213,12111,12113],{"className":12112},[221],[223,12114,12115],{"xmlns":225},[227,12116,12117,12121],{},[230,12118,12119],{},[233,12120,454],{},[279,12122,454],{"encoding":281},[213,12124,12126],{"className":12125,"ariaHidden":251},[286],[213,12127,12129,12132],{"className":12128},[290],[213,12130],{"className":12131,"style":477},[294],[213,12133,454],{"className":12134,"style":481},[299,407]," 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 ",[213,12137,12139,12152],{"className":12138,"translate":217},[216],[213,12140,12142],{"className":12141},[221],[223,12143,12144],{"xmlns":225},[227,12145,12146,12150],{},[230,12147,12148],{},[233,12149,454],{},[279,12151,454],{"encoding":281},[213,12153,12155],{"className":12154,"ariaHidden":251},[286],[213,12156,12158,12161],{"className":12157},[290],[213,12159],{"className":12160,"style":477},[294],[213,12162,454],{"className":12163,"style":481},[299,407]," 都成立的普适论证。",[30,12166,12167],{"id":12167},"停机问题",[11,12169,12170],{},"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。",[11,12172,12173,12174,12203,12204,12232,12233,12261,12262,12290],{},"停机问题。给定一个程序 ",[213,12175,12177,12190],{"className":12176,"translate":217},[216],[213,12178,12180],{"className":12179},[221],[223,12181,12182],{"xmlns":225},[227,12183,12184,12188],{},[230,12185,12186],{},[233,12187,8461],{},[279,12189,8461],{"encoding":281},[213,12191,12193],{"className":12192,"ariaHidden":251},[286],[213,12194,12196,12199],{"className":12195},[290],[213,12197],{"className":12198,"style":403},[294],[213,12200,8461],{"className":12201,"style":12202},[299,407],"margin-right:0.1389em;"," 和一个输入 ",[213,12205,12207,12220],{"className":12206,"translate":217},[216],[213,12208,12210],{"className":12209},[221],[223,12211,12212],{"xmlns":225},[227,12213,12214,12218],{},[230,12215,12216],{},[233,12217,2611],{},[279,12219,2611],{"encoding":281},[213,12221,12223],{"className":12222,"ariaHidden":251},[286],[213,12224,12226,12229],{"className":12225},[290],[213,12227],{"className":12228,"style":6175},[294],[213,12230,2611],{"className":12231},[299,407],"，是否存在一个通用算法，能够判断 ",[213,12234,12236,12249],{"className":12235,"translate":217},[216],[213,12237,12239],{"className":12238},[221],[223,12240,12241],{"xmlns":225},[227,12242,12243,12247],{},[230,12244,12245],{},[233,12246,8461],{},[279,12248,8461],{"encoding":281},[213,12250,12252],{"className":12251,"ariaHidden":251},[286],[213,12253,12255,12258],{"className":12254},[290],[213,12256],{"className":12257,"style":403},[294],[213,12259,8461],{"className":12260,"style":12202},[299,407]," 在输入 ",[213,12263,12265,12278],{"className":12264,"translate":217},[216],[213,12266,12268],{"className":12267},[221],[223,12269,12270],{"xmlns":225},[227,12271,12272,12276],{},[230,12273,12274],{},[233,12275,2611],{},[279,12277,2611],{"encoding":281},[213,12279,12281],{"className":12280,"ariaHidden":251},[286],[213,12282,12284,12287],{"className":12283},[290],[213,12285],{"className":12286,"style":6175},[294],[213,12288,2611],{"className":12289},[299,407]," 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）？",[11,12292,12293],{},"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）。",[11,12295,12296],{},"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：",[213,12298,12300],{"className":12299,"translate":217},[2584],[213,12301,12303,12392],{"className":12302,"translate":217},[216],[213,12304,12306],{"className":12305},[221],[223,12307,12308],{"xmlns":225,"display":2593},[227,12309,12310,12389],{},[230,12311,12312,12315,12317,12319,12321,12323,12325,12327],{},[272,12313,12314],{},"halts",[238,12316,560],{"stretchy":243},[233,12318,8461],{},[238,12320,252],{"separator":251},[233,12322,2611],{},[238,12324,569],{"stretchy":243},[238,12326,240],{},[230,12328,12329,12331],{},[238,12330,244],{"fence":251},[1742,12332,12333,12361],{"rowspacing":1744,"columnalign":1745,"columnspacing":1746},[1748,12334,12335,12342],{},[1751,12336,12337],{},[1754,12338,12339],{"scriptlevel":248,"displaystyle":243},[272,12340,12341],{},"真",[1751,12343,12344],{},[1754,12345,12346],{"scriptlevel":248,"displaystyle":243},[230,12347,12348,12350,12352,12354,12356,12358],{},[272,12349,4527],{},[233,12351,8461],{},[238,12353,560],{"stretchy":243},[233,12355,2611],{},[238,12357,569],{"stretchy":243},[272,12359,12360],{}," 停机",[1748,12362,12363,12370],{},[1751,12364,12365],{},[1754,12366,12367],{"scriptlevel":248,"displaystyle":243},[272,12368,12369],{},"假",[1751,12371,12372],{},[1754,12373,12374],{"scriptlevel":248,"displaystyle":243},[230,12375,12376,12378,12380,12382,12384,12386],{},[272,12377,4527],{},[233,12379,8461],{},[238,12381,560],{"stretchy":243},[233,12383,2611],{},[238,12385,569],{"stretchy":243},[272,12387,12388],{}," 不停机",[279,12390,12391],{"encoding":281},"\\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}",[213,12393,12395,12434],{"className":12394,"ariaHidden":251},[286],[213,12396,12398,12401,12407,12410,12413,12416,12419,12422,12425,12428,12431],{"className":12397},[290],[213,12399],{"className":12400,"style":319},[294],[213,12402,12404],{"className":12403},[299,1983],[213,12405,12314],{"className":12406},[299],[213,12408,560],{"className":12409},[323],[213,12411,8461],{"className":12412,"style":12202},[299,407],[213,12414,252],{"className":12415},[330],[213,12417],{"className":12418,"style":334},[304],[213,12420,2611],{"className":12421},[299,407],[213,12423,569],{"className":12424},[372],[213,12426],{"className":12427,"style":305},[304],[213,12429,240],{"className":12430},[309],[213,12432],{"className":12433,"style":305},[304],[213,12435,12437,12440],{"className":12436},[290],[213,12438],{"className":12439,"style":1851},[294],[213,12441,12443,12449,12610],{"className":12442},[365],[213,12444,12446],{"className":12445,"style":1859},[323,1858],[213,12447,244],{"className":12448},[1863,1864],[213,12450,12452],{"className":12451},[299],[213,12453,12455,12506,12509],{"className":12454},[1742],[213,12456,12458],{"className":12457},[1874],[213,12459,12461,12498],{"className":12460},[609,610],[213,12462,12464,12495],{"className":12463},[614],[213,12465,12467,12481],{"className":12466,"style":1884},[618],[213,12468,12469,12472],{"style":1887},[213,12470],{"className":12471,"style":1891},[626],[213,12473,12475],{"className":12474},[299],[213,12476,12478],{"className":12477},[299,1983],[213,12479,12341],{"className":12480},[299,1991],[213,12482,12483,12486],{"style":1904},[213,12484],{"className":12485,"style":1891},[626],[213,12487,12489],{"className":12488},[299],[213,12490,12492],{"className":12491},[299,1983],[213,12493,12369],{"className":12494},[299,1991],[213,12496,642],{"className":12497},[641],[213,12499,12501],{"className":12500},[614],[213,12502,12504],{"className":12503,"style":1949},[618],[213,12505],{},[213,12507],{"className":12508,"style":1956},[1955],[213,12510,12512],{"className":12511},[1874],[213,12513,12515,12602],{"className":12514},[609,610],[213,12516,12518,12599],{"className":12517},[614],[213,12519,12521,12560],{"className":12520,"style":1884},[618],[213,12522,12523,12526],{"style":1887},[213,12524],{"className":12525,"style":1891},[626],[213,12527,12529,12538,12541,12544,12547,12550],{"className":12528},[299],[213,12530,12532,12535],{"className":12531},[299,1983],[213,12533,4729],{"className":12534},[299,1991],[213,12536,1987],{"className":12537},[299],[213,12539,8461],{"className":12540,"style":12202},[299,407],[213,12542,560],{"className":12543},[323],[213,12545,2611],{"className":12546},[299,407],[213,12548,569],{"className":12549},[372],[213,12551,12553,12556],{"className":12552},[299,1983],[213,12554,1987],{"className":12555},[299],[213,12557,12559],{"className":12558},[299,1991],"停机",[213,12561,12562,12565],{"style":1904},[213,12563],{"className":12564,"style":1891},[626],[213,12566,12568,12577,12580,12583,12586,12589],{"className":12567},[299],[213,12569,12571,12574],{"className":12570},[299,1983],[213,12572,4729],{"className":12573},[299,1991],[213,12575,1987],{"className":12576},[299],[213,12578,8461],{"className":12579,"style":12202},[299,407],[213,12581,560],{"className":12582},[323],[213,12584,2611],{"className":12585},[299,407],[213,12587,569],{"className":12588},[372],[213,12590,12592,12595],{"className":12591},[299,1983],[213,12593,1987],{"className":12594},[299],[213,12596,12598],{"className":12597},[299,1991],"不停机",[213,12600,642],{"className":12601},[641],[213,12603,12605],{"className":12604},[614],[213,12606,12608],{"className":12607,"style":1949},[618],[213,12609],{},[213,12611],{"className":12612},[372,2028],[11,12614,12615],{},"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）。",[11,12617,12618,12619,12647],{},"构造一个对角线程序 ",[213,12620,12622,12635],{"className":12621,"translate":217},[216],[213,12623,12625],{"className":12624},[221],[223,12626,12627],{"xmlns":225},[227,12628,12629,12633],{},[230,12630,12631],{},[233,12632,9040],{},[279,12634,9040],{"encoding":281},[213,12636,12638],{"className":12637,"ariaHidden":251},[286],[213,12639,12641,12644],{"className":12640},[290],[213,12642],{"className":12643,"style":403},[294],[213,12645,9040],{"className":12646,"style":9087},[299,407],"，其行为定义如下：",[213,12649,12651],{"className":12650,"translate":217},[2584],[213,12652,12654,12745],{"className":12653,"translate":217},[216],[213,12655,12657],{"className":12656},[221],[223,12658,12659],{"xmlns":225,"display":2593},[227,12660,12661,12742],{},[230,12662,12663,12665,12667,12669,12671,12673],{},[233,12664,9040],{},[238,12666,560],{"stretchy":243},[233,12668,8461],{},[238,12670,569],{"stretchy":243},[238,12672,240],{},[230,12674,12675,12677],{},[238,12676,244],{"fence":251},[1742,12678,12679,12711],{"rowspacing":1744,"columnalign":1745,"columnspacing":1746},[1748,12680,12681,12688],{},[1751,12682,12683],{},[1754,12684,12685],{"scriptlevel":248,"displaystyle":243},[272,12686,12687],{},"死循环",[1751,12689,12690],{},[1754,12691,12692],{"scriptlevel":248,"displaystyle":243},[230,12693,12694,12697,12699,12701,12703,12705,12707,12709],{},[272,12695,12696],{},"若 halts",[238,12698,560],{"stretchy":243},[233,12700,8461],{},[238,12702,252],{"separator":251},[233,12704,8461],{},[238,12706,569],{"stretchy":243},[238,12708,240],{},[272,12710,12341],{},[1748,12712,12713,12720],{},[1751,12714,12715],{},[1754,12716,12717],{"scriptlevel":248,"displaystyle":243},[272,12718,12719],{},"立即停机",[1751,12721,12722],{},[1754,12723,12724],{"scriptlevel":248,"displaystyle":243},[230,12725,12726,12728,12730,12732,12734,12736,12738,12740],{},[272,12727,12696],{},[238,12729,560],{"stretchy":243},[233,12731,8461],{},[238,12733,252],{"separator":251},[233,12735,8461],{},[238,12737,569],{"stretchy":243},[238,12739,240],{},[272,12741,12369],{},[279,12743,12744],{"encoding":281},"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}",[213,12746,12748,12775],{"className":12747,"ariaHidden":251},[286],[213,12749,12751,12754,12757,12760,12763,12766,12769,12772],{"className":12750},[290],[213,12752],{"className":12753,"style":319},[294],[213,12755,9040],{"className":12756,"style":9087},[299,407],[213,12758,560],{"className":12759},[323],[213,12761,8461],{"className":12762,"style":12202},[299,407],[213,12764,569],{"className":12765},[372],[213,12767],{"className":12768,"style":305},[304],[213,12770,240],{"className":12771},[309],[213,12773],{"className":12774,"style":305},[304],[213,12776,12778,12781],{"className":12777},[290],[213,12779],{"className":12780,"style":1851},[294],[213,12782,12784,12790,12985],{"className":12783},[365],[213,12785,12787],{"className":12786,"style":1859},[323,1858],[213,12788,244],{"className":12789},[1863,1864],[213,12791,12793],{"className":12792},[299],[213,12794,12796,12847,12850],{"className":12795},[1742],[213,12797,12799],{"className":12798},[1874],[213,12800,12802,12839],{"className":12801},[609,610],[213,12803,12805,12836],{"className":12804},[614],[213,12806,12808,12822],{"className":12807,"style":1884},[618],[213,12809,12810,12813],{"style":1887},[213,12811],{"className":12812,"style":1891},[626],[213,12814,12816],{"className":12815},[299],[213,12817,12819],{"className":12818},[299,1983],[213,12820,12687],{"className":12821},[299,1991],[213,12823,12824,12827],{"style":1904},[213,12825],{"className":12826,"style":1891},[626],[213,12828,12830],{"className":12829},[299],[213,12831,12833],{"className":12832},[299,1983],[213,12834,12719],{"className":12835},[299,1991],[213,12837,642],{"className":12838},[641],[213,12840,12842],{"className":12841},[614],[213,12843,12845],{"className":12844,"style":1949},[618],[213,12846],{},[213,12848],{"className":12849,"style":1956},[1955],[213,12851,12853],{"className":12852},[1874],[213,12854,12856,12977],{"className":12855},[609,610],[213,12857,12859,12974],{"className":12858},[614],[213,12860,12862,12918],{"className":12861,"style":1884},[618],[213,12863,12864,12867],{"style":1887},[213,12865],{"className":12866,"style":1891},[626],[213,12868,12870,12879,12885,12888,12891,12894,12897,12900,12903,12906,12909,12912],{"className":12869},[299],[213,12871,12873,12876],{"className":12872},[299,1983],[213,12874,4729],{"className":12875},[299,1991],[213,12877,1987],{"className":12878},[299],[213,12880,12882],{"className":12881},[299,1983],[213,12883,12314],{"className":12884},[299],[213,12886,560],{"className":12887},[323],[213,12889,8461],{"className":12890,"style":12202},[299,407],[213,12892,252],{"className":12893},[330],[213,12895],{"className":12896,"style":334},[304],[213,12898,8461],{"className":12899,"style":12202},[299,407],[213,12901,569],{"className":12902},[372],[213,12904],{"className":12905,"style":305},[304],[213,12907,240],{"className":12908},[309],[213,12910],{"className":12911,"style":305},[304],[213,12913,12915],{"className":12914},[299,1983],[213,12916,12341],{"className":12917},[299,1991],[213,12919,12920,12923],{"style":1904},[213,12921],{"className":12922,"style":1891},[626],[213,12924,12926,12935,12941,12944,12947,12950,12953,12956,12959,12962,12965,12968],{"className":12925},[299],[213,12927,12929,12932],{"className":12928},[299,1983],[213,12930,4729],{"className":12931},[299,1991],[213,12933,1987],{"className":12934},[299],[213,12936,12938],{"className":12937},[299,1983],[213,12939,12314],{"className":12940},[299],[213,12942,560],{"className":12943},[323],[213,12945,8461],{"className":12946,"style":12202},[299,407],[213,12948,252],{"className":12949},[330],[213,12951],{"className":12952,"style":334},[304],[213,12954,8461],{"className":12955,"style":12202},[299,407],[213,12957,569],{"className":12958},[372],[213,12960],{"className":12961,"style":305},[304],[213,12963,240],{"className":12964},[309],[213,12966],{"className":12967,"style":305},[304],[213,12969,12971],{"className":12970},[299,1983],[213,12972,12369],{"className":12973},[299,1991],[213,12975,642],{"className":12976},[641],[213,12978,12980],{"className":12979},[614],[213,12981,12983],{"className":12982,"style":1949},[618],[213,12984],{},[213,12986],{"className":12987},[372,2028],[11,12989,12990,12991,13051,13052,4140,13080,13108],{},"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（",[213,12992,12994,13018],{"className":12993,"translate":217},[216],[213,12995,12997],{"className":12996},[221],[223,12998,12999],{"xmlns":225},[227,13000,13001,13015],{},[230,13002,13003,13005,13007,13009,13011,13013],{},[272,13004,12314],{},[238,13006,560],{"stretchy":243},[233,13008,8461],{},[238,13010,252],{"separator":251},[233,13012,8461],{},[238,13014,569],{"stretchy":243},[279,13016,13017],{"encoding":281},"\\text{halts}(P, P)",[213,13019,13021],{"className":13020,"ariaHidden":251},[286],[213,13022,13024,13027,13033,13036,13039,13042,13045,13048],{"className":13023},[290],[213,13025],{"className":13026,"style":319},[294],[213,13028,13030],{"className":13029},[299,1983],[213,13031,12314],{"className":13032},[299],[213,13034,560],{"className":13035},[323],[213,13037,8461],{"className":13038,"style":12202},[299,407],[213,13040,252],{"className":13041},[330],[213,13043],{"className":13044,"style":334},[304],[213,13046,8461],{"className":13047,"style":12202},[299,407],[213,13049,569],{"className":13050},[372],"）。这正是对角线论证中「取第 ",[213,13053,13055,13068],{"className":13054,"translate":217},[216],[213,13056,13058],{"className":13057},[221],[223,13059,13060],{"xmlns":225},[227,13061,13062,13066],{},[230,13063,13064],{},[233,13065,1496],{},[279,13067,1496],{"encoding":281},[213,13069,13071],{"className":13070,"ariaHidden":251},[286],[213,13072,13074,13077],{"className":13073},[290],[213,13075],{"className":13076,"style":6175},[294],[213,13078,1496],{"className":13079},[299,407],[213,13081,13083,13096],{"className":13082,"translate":217},[216],[213,13084,13086],{"className":13085},[221],[223,13087,13088],{"xmlns":225},[227,13089,13090,13094],{},[230,13091,13092],{},[233,13093,1496],{},[279,13095,1496],{"encoding":281},[213,13097,13099],{"className":13098,"ariaHidden":251},[286],[213,13100,13102,13105],{"className":13101},[290],[213,13103],{"className":13104,"style":6175},[294],[213,13106,1496],{"className":13107},[299,407]," 位」的直接类比——让一个程序「审视」自身的自指动作。",[11,13110,13111,13112,1551],{},"导出矛盾。现在考察具体的执行 ",[213,13113,13115,13135],{"className":13114,"translate":217},[216],[213,13116,13118],{"className":13117},[221],[223,13119,13120],{"xmlns":225},[227,13121,13122,13132],{},[230,13123,13124,13126,13128,13130],{},[233,13125,9040],{},[238,13127,560],{"stretchy":243},[233,13129,9040],{},[238,13131,569],{"stretchy":243},[279,13133,13134],{"encoding":281},"D(D)",[213,13136,13138],{"className":13137,"ariaHidden":251},[286],[213,13139,13141,13144,13147,13150,13153],{"className":13140},[290],[213,13142],{"className":13143,"style":319},[294],[213,13145,9040],{"className":13146,"style":9087},[299,407],[213,13148,560],{"className":13149},[323],[213,13151,9040],{"className":13152,"style":9087},[299,407],[213,13154,569],{"className":13155},[372],[62,13157,13158,13362],{},[65,13159,9529,13160,13245,13246,13289,13290,9609,13318,13361],{},[213,13161,13163,13191],{"className":13162,"translate":217},[216],[213,13164,13166],{"className":13165},[221],[223,13167,13168],{"xmlns":225},[227,13169,13170,13188],{},[230,13171,13172,13174,13176,13178,13180,13182,13184,13186],{},[272,13173,12314],{},[238,13175,560],{"stretchy":243},[233,13177,9040],{},[238,13179,252],{"separator":251},[233,13181,9040],{},[238,13183,569],{"stretchy":243},[238,13185,240],{},[272,13187,12341],{},[279,13189,13190],{"encoding":281},"\\text{halts}(D, D) = \\text{真}",[213,13192,13194,13233],{"className":13193,"ariaHidden":251},[286],[213,13195,13197,13200,13206,13209,13212,13215,13218,13221,13224,13227,13230],{"className":13196},[290],[213,13198],{"className":13199,"style":319},[294],[213,13201,13203],{"className":13202},[299,1983],[213,13204,12314],{"className":13205},[299],[213,13207,560],{"className":13208},[323],[213,13210,9040],{"className":13211,"style":9087},[299,407],[213,13213,252],{"className":13214},[330],[213,13216],{"className":13217,"style":334},[304],[213,13219,9040],{"className":13220,"style":9087},[299,407],[213,13222,569],{"className":13223},[372],[213,13225],{"className":13226,"style":305},[304],[213,13228,240],{"className":13229},[309],[213,13231],{"className":13232,"style":305},[304],[213,13234,13236,13239],{"className":13235},[290],[213,13237],{"className":13238,"style":403},[294],[213,13240,13242],{"className":13241},[299,1983],[213,13243,12341],{"className":13244},[299,1991],"（即判定「",[213,13247,13249,13268],{"className":13248,"translate":217},[216],[213,13250,13252],{"className":13251},[221],[223,13253,13254],{"xmlns":225},[227,13255,13256,13266],{},[230,13257,13258,13260,13262,13264],{},[233,13259,9040],{},[238,13261,560],{"stretchy":243},[233,13263,9040],{},[238,13265,569],{"stretchy":243},[279,13267,13134],{"encoding":281},[213,13269,13271],{"className":13270,"ariaHidden":251},[286],[213,13272,13274,13277,13280,13283,13286],{"className":13273},[290],[213,13275],{"className":13276,"style":319},[294],[213,13278,9040],{"className":13279,"style":9087},[299,407],[213,13281,560],{"className":13282},[323],[213,13284,9040],{"className":13285,"style":9087},[299,407],[213,13287,569],{"className":13288},[372]," 停机」），那么根据 ",[213,13291,13293,13306],{"className":13292,"translate":217},[216],[213,13294,13296],{"className":13295},[221],[223,13297,13298],{"xmlns":225},[227,13299,13300,13304],{},[230,13301,13302],{},[233,13303,9040],{},[279,13305,9040],{"encoding":281},[213,13307,13309],{"className":13308,"ariaHidden":251},[286],[213,13310,13312,13315],{"className":13311},[290],[213,13313],{"className":13314,"style":403},[294],[213,13316,9040],{"className":13317,"style":9087},[299,407],[213,13319,13321,13340],{"className":13320,"translate":217},[216],[213,13322,13324],{"className":13323},[221],[223,13325,13326],{"xmlns":225},[227,13327,13328,13338],{},[230,13329,13330,13332,13334,13336],{},[233,13331,9040],{},[238,13333,560],{"stretchy":243},[233,13335,9040],{},[238,13337,569],{"stretchy":243},[279,13339,13134],{"encoding":281},[213,13341,13343],{"className":13342,"ariaHidden":251},[286],[213,13344,13346,13349,13352,13355,13358],{"className":13345},[290],[213,13347],{"className":13348,"style":319},[294],[213,13350,9040],{"className":13351,"style":9087},[299,407],[213,13353,560],{"className":13354},[323],[213,13356,9040],{"className":13357,"style":9087},[299,407],[213,13359,569],{"className":13360},[372]," 应该死循环。这与「它停机」矛盾。",[65,13363,9529,13364,13245,13449,13492,13493,9609,13521,13564],{},[213,13365,13367,13395],{"className":13366,"translate":217},[216],[213,13368,13370],{"className":13369},[221],[223,13371,13372],{"xmlns":225},[227,13373,13374,13392],{},[230,13375,13376,13378,13380,13382,13384,13386,13388,13390],{},[272,13377,12314],{},[238,13379,560],{"stretchy":243},[233,13381,9040],{},[238,13383,252],{"separator":251},[233,13385,9040],{},[238,13387,569],{"stretchy":243},[238,13389,240],{},[272,13391,12369],{},[279,13393,13394],{"encoding":281},"\\text{halts}(D, D) = \\text{假}",[213,13396,13398,13437],{"className":13397,"ariaHidden":251},[286],[213,13399,13401,13404,13410,13413,13416,13419,13422,13425,13428,13431,13434],{"className":13400},[290],[213,13402],{"className":13403,"style":319},[294],[213,13405,13407],{"className":13406},[299,1983],[213,13408,12314],{"className":13409},[299],[213,13411,560],{"className":13412},[323],[213,13414,9040],{"className":13415,"style":9087},[299,407],[213,13417,252],{"className":13418},[330],[213,13420],{"className":13421,"style":334},[304],[213,13423,9040],{"className":13424,"style":9087},[299,407],[213,13426,569],{"className":13427},[372],[213,13429],{"className":13430,"style":305},[304],[213,13432,240],{"className":13433},[309],[213,13435],{"className":13436,"style":305},[304],[213,13438,13440,13443],{"className":13439},[290],[213,13441],{"className":13442,"style":403},[294],[213,13444,13446],{"className":13445},[299,1983],[213,13447,12369],{"className":13448},[299,1991],[213,13450,13452,13471],{"className":13451,"translate":217},[216],[213,13453,13455],{"className":13454},[221],[223,13456,13457],{"xmlns":225},[227,13458,13459,13469],{},[230,13460,13461,13463,13465,13467],{},[233,13462,9040],{},[238,13464,560],{"stretchy":243},[233,13466,9040],{},[238,13468,569],{"stretchy":243},[279,13470,13134],{"encoding":281},[213,13472,13474],{"className":13473,"ariaHidden":251},[286],[213,13475,13477,13480,13483,13486,13489],{"className":13476},[290],[213,13478],{"className":13479,"style":319},[294],[213,13481,9040],{"className":13482,"style":9087},[299,407],[213,13484,560],{"className":13485},[323],[213,13487,9040],{"className":13488,"style":9087},[299,407],[213,13490,569],{"className":13491},[372]," 不停机」），那么根据 ",[213,13494,13496,13509],{"className":13495,"translate":217},[216],[213,13497,13499],{"className":13498},[221],[223,13500,13501],{"xmlns":225},[227,13502,13503,13507],{},[230,13504,13505],{},[233,13506,9040],{},[279,13508,9040],{"encoding":281},[213,13510,13512],{"className":13511,"ariaHidden":251},[286],[213,13513,13515,13518],{"className":13514},[290],[213,13516],{"className":13517,"style":403},[294],[213,13519,9040],{"className":13520,"style":9087},[299,407],[213,13522,13524,13543],{"className":13523,"translate":217},[216],[213,13525,13527],{"className":13526},[221],[223,13528,13529],{"xmlns":225},[227,13530,13531,13541],{},[230,13532,13533,13535,13537,13539],{},[233,13534,9040],{},[238,13536,560],{"stretchy":243},[233,13538,9040],{},[238,13540,569],{"stretchy":243},[279,13542,13134],{"encoding":281},[213,13544,13546],{"className":13545,"ariaHidden":251},[286],[213,13547,13549,13552,13555,13558,13561],{"className":13548},[290],[213,13550],{"className":13551,"style":319},[294],[213,13553,9040],{"className":13554,"style":9087},[299,407],[213,13556,560],{"className":13557},[323],[213,13559,9040],{"className":13560,"style":9087},[299,407],[213,13562,569],{"className":13563},[372]," 应该立即停机。这与「它不停机」矛盾。",[11,13566,13567,13568,13600],{},"两种情况都导致自相矛盾。因此，「存在通用停机判定器 ",[213,13569,13571,13585],{"className":13570,"translate":217},[216],[213,13572,13574],{"className":13573},[221],[223,13575,13576],{"xmlns":225},[227,13577,13578,13582],{},[230,13579,13580],{},[272,13581,12314],{},[279,13583,13584],{"encoding":281},"\\text{halts}",[213,13586,13588],{"className":13587,"ariaHidden":251},[286],[213,13589,13591,13594],{"className":13590},[290],[213,13592],{"className":13593,"style":1084},[294],[213,13595,13597],{"className":13596},[299,1983],[213,13598,12314],{"className":13599},[299],"」的假设是假的。停机问题不可判定。",[11,13602,13603,13604,13797],{},"若按某种规则把所有程序枚举出来（",[213,13605,13607,13645],{"className":13606,"translate":217},[216],[213,13608,13610],{"className":13609},[221],[223,13611,13612],{"xmlns":225},[227,13613,13614,13642],{},[230,13615,13616,13622,13624,13630,13632,13638,13640],{},[537,13617,13618,13620],{},[233,13619,8461],{},[246,13621,255],{},[238,13623,252],{"separator":251},[537,13625,13626,13628],{},[233,13627,8461],{},[246,13629,260],{},[238,13631,252],{"separator":251},[537,13633,13634,13636],{},[233,13635,8461],{},[246,13637,265],{},[238,13639,252],{"separator":251},[238,13641,270],{},[279,13643,13644],{"encoding":281},"P_1, P_2, P_3, \\dots",[213,13646,13648],{"className":13647,"ariaHidden":251},[286],[213,13649,13651,13655,13696,13699,13702,13742,13745,13748,13788,13791,13794],{"className":13650},[290],[213,13652],{"className":13653,"style":13654},[294],"height:0.8778em;vertical-align:-0.1944em;",[213,13656,13658,13661],{"className":13657},[299],[213,13659,8461],{"className":13660,"style":12202},[299,407],[213,13662,13664],{"className":13663},[605],[213,13665,13667,13688],{"className":13666},[609,610],[213,13668,13670,13685],{"className":13669},[614],[213,13671,13673],{"className":13672,"style":619},[618],[213,13674,13676,13679],{"style":13675},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[213,13677],{"className":13678,"style":627},[626],[213,13680,13682],{"className":13681},[631,632,633,634],[213,13683,255],{"className":13684},[299,634],[213,13686,642],{"className":13687},[641],[213,13689,13691],{"className":13690},[614],[213,13692,13694],{"className":13693,"style":649},[618],[213,13695],{},[213,13697,252],{"className":13698},[330],[213,13700],{"className":13701,"style":334},[304],[213,13703,13705,13708],{"className":13704},[299],[213,13706,8461],{"className":13707,"style":12202},[299,407],[213,13709,13711],{"className":13710},[605],[213,13712,13714,13734],{"className":13713},[609,610],[213,13715,13717,13731],{"className":13716},[614],[213,13718,13720],{"className":13719,"style":619},[618],[213,13721,13722,13725],{"style":13675},[213,13723],{"className":13724,"style":627},[626],[213,13726,13728],{"className":13727},[631,632,633,634],[213,13729,260],{"className":13730},[299,634],[213,13732,642],{"className":13733},[641],[213,13735,13737],{"className":13736},[614],[213,13738,13740],{"className":13739,"style":649},[618],[213,13741],{},[213,13743,252],{"className":13744},[330],[213,13746],{"className":13747,"style":334},[304],[213,13749,13751,13754],{"className":13750},[299],[213,13752,8461],{"className":13753,"style":12202},[299,407],[213,13755,13757],{"className":13756},[605],[213,13758,13760,13780],{"className":13759},[609,610],[213,13761,13763,13777],{"className":13762},[614],[213,13764,13766],{"className":13765,"style":619},[618],[213,13767,13768,13771],{"style":13675},[213,13769],{"className":13770,"style":627},[626],[213,13772,13774],{"className":13773},[631,632,633,634],[213,13775,265],{"className":13776},[299,634],[213,13778,642],{"className":13779},[641],[213,13781,13783],{"className":13782},[614],[213,13784,13786],{"className":13785,"style":649},[618],[213,13787],{},[213,13789,252],{"className":13790},[330],[213,13792],{"className":13793,"style":334},[304],[213,13795,270],{"className":13796},[365],"——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：",[213,13799,13801],{"className":13800,"translate":217},[2584],[213,13802,13804,13969],{"className":13803,"translate":217},[216],[213,13805,13807],{"className":13806},[221],[223,13808,13809],{"xmlns":225,"display":2593},[227,13810,13811,13966],{},[230,13812,13813,13815,13964],{},[238,13814,560],{"fence":251},[1742,13816,13818,13878,13938],{"rowspacing":6796,"columnalign":13817,"columnspacing":1746},"center center center",[1748,13819,13820,13846,13872],{},[1751,13821,13822],{},[1754,13823,13824],{"scriptlevel":248,"displaystyle":243},[230,13825,13826,13828,13830,13836,13838,13844],{},[272,13827,12314],{},[238,13829,560],{"stretchy":243},[537,13831,13832,13834],{},[233,13833,8461],{},[246,13835,255],{},[238,13837,252],{"separator":251},[537,13839,13840,13842],{},[233,13841,8461],{},[246,13843,255],{},[238,13845,569],{"stretchy":243},[1751,13847,13848],{},[1754,13849,13850],{"scriptlevel":248,"displaystyle":243},[230,13851,13852,13854,13856,13862,13864,13870],{},[272,13853,12314],{},[238,13855,560],{"stretchy":243},[537,13857,13858,13860],{},[233,13859,8461],{},[246,13861,255],{},[238,13863,252],{"separator":251},[537,13865,13866,13868],{},[233,13867,8461],{},[246,13869,260],{},[238,13871,569],{"stretchy":243},[1751,13873,13874],{},[1754,13875,13876],{"scriptlevel":248,"displaystyle":243},[238,13877,6846],{"lspace":2600,"rspace":2600},[1748,13879,13880,13906,13932],{},[1751,13881,13882],{},[1754,13883,13884],{"scriptlevel":248,"displaystyle":243},[230,13885,13886,13888,13890,13896,13898,13904],{},[272,13887,12314],{},[238,13889,560],{"stretchy":243},[537,13891,13892,13894],{},[233,13893,8461],{},[246,13895,260],{},[238,13897,252],{"separator":251},[537,13899,13900,13902],{},[233,13901,8461],{},[246,13903,255],{},[238,13905,569],{"stretchy":243},[1751,13907,13908],{},[1754,13909,13910],{"scriptlevel":248,"displaystyle":243},[230,13911,13912,13914,13916,13922,13924,13930],{},[272,13913,12314],{},[238,13915,560],{"stretchy":243},[537,13917,13918,13920],{},[233,13919,8461],{},[246,13921,260],{},[238,13923,252],{"separator":251},[537,13925,13926,13928],{},[233,13927,8461],{},[246,13929,260],{},[238,13931,569],{"stretchy":243},[1751,13933,13934],{},[1754,13935,13936],{"scriptlevel":248,"displaystyle":243},[238,13937,6846],{"lspace":2600,"rspace":2600},[1748,13939,13940,13952,13958],{},[1751,13941,13942],{},[1754,13943,13944],{"scriptlevel":248,"displaystyle":243},[230,13945,13946,13948],{},[233,13947,2867],{"mathvariant":545},[2869,13949,13950],{"height":2600,"voffset":2600},[304,13951],{"mathbackground":2873,"width":2600,"height":2874},[1751,13953,13954],{},[1754,13955,13956],{"scriptlevel":248,"displaystyle":243},[230,13957],{},[1751,13959,13960],{},[1754,13961,13962],{"scriptlevel":248,"displaystyle":243},[238,13963,6983],{"lspace":2600,"rspace":2600},[238,13965,569],{"fence":251},[279,13967,13968],{"encoding":281},"\\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}",[213,13970,13972],{"className":13971,"ariaHidden":251},[286],[213,13973,13975,13979],{"className":13974},[290],[213,13976],{"className":13977,"style":13978},[294],"height:4.26em;vertical-align:-1.88em;",[213,13980,13982,14027,14602],{"className":13981},[365],[213,13983,13985],{"className":13984},[323],[213,13986,13988],{"className":13987},[1863,7008],[213,13989,13991,14018],{"className":13990},[609,610],[213,13992,13994,14015],{"className":13993},[614],[213,13995,13998],{"className":13996,"style":13997},[618],"height:2.35em;",[213,13999,14001,14005],{"style":14000},"top:-4.35em;",[213,14002],{"className":14003,"style":14004},[626],"height:6.2em;",[213,14006,14008],{"style":14007},"width:0.875em;height:4.2em;",[7030,14009,14012],{"xmlns":7032,"width":7033,"height":14010,"viewBox":14011},"4.2em","0 0 875 4200",[7037,14013],{"d":14014},"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,684c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-692c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z",[213,14016,642],{"className":14017},[641],[213,14019,14021],{"className":14020},[614],[213,14022,14025],{"className":14023,"style":14024},[618],"height:1.85em;",[213,14026],{},[213,14028,14030],{"className":14029},[299],[213,14031,14033,14289,14292,14295,14540,14543,14546],{"className":14032},[1742],[213,14034,14036],{"className":14035},[7061],[213,14037,14039,14280],{"className":14038},[609,610],[213,14040,14042,14277],{"className":14041},[614],[213,14043,14046,14153,14259],{"className":14044,"style":14045},[618],"height:2.38em;",[213,14047,14049,14052],{"style":14048},"top:-5.2275em;",[213,14050],{"className":14051,"style":3144},[626],[213,14053,14055,14061,14064,14104,14107,14110,14150],{"className":14054},[299],[213,14056,14058],{"className":14057},[299,1983],[213,14059,12314],{"className":14060},[299],[213,14062,560],{"className":14063},[323],[213,14065,14067,14070],{"className":14066},[299],[213,14068,8461],{"className":14069,"style":12202},[299,407],[213,14071,14073],{"className":14072},[605],[213,14074,14076,14096],{"className":14075},[609,610],[213,14077,14079,14093],{"className":14078},[614],[213,14080,14082],{"className":14081,"style":619},[618],[213,14083,14084,14087],{"style":13675},[213,14085],{"className":14086,"style":627},[626],[213,14088,14090],{"className":14089},[631,632,633,634],[213,14091,255],{"className":14092},[299,634],[213,14094,642],{"className":14095},[641],[213,14097,14099],{"className":14098},[614],[213,14100,14102],{"className":14101,"style":649},[618],[213,14103],{},[213,14105,252],{"className":14106},[330],[213,14108],{"className":14109,"style":334},[304],[213,14111,14113,14116],{"className":14112},[299],[213,14114,8461],{"className":14115,"style":12202},[299,407],[213,14117,14119],{"className":14118},[605],[213,14120,14122,14142],{"className":14121},[609,610],[213,14123,14125,14139],{"className":14124},[614],[213,14126,14128],{"className":14127,"style":619},[618],[213,14129,14130,14133],{"style":13675},[213,14131],{"className":14132,"style":627},[626],[213,14134,14136],{"className":14135},[631,632,633,634],[213,14137,255],{"className":14138},[299,634],[213,14140,642],{"className":14141},[641],[213,14143,14145],{"className":14144},[614],[213,14146,14148],{"className":14147,"style":649},[618],[213,14149],{},[213,14151,569],{"className":14152},[372],[213,14154,14155,14158],{"style":3563},[213,14156],{"className":14157,"style":3144},[626],[213,14159,14161,14167,14170,14210,14213,14216,14256],{"className":14160},[299],[213,14162,14164],{"className":14163},[299,1983],[213,14165,12314],{"className":14166},[299],[213,14168,560],{"className":14169},[323],[213,14171,14173,14176],{"className":14172},[299],[213,14174,8461],{"className":14175,"style":12202},[299,407],[213,14177,14179],{"className":14178},[605],[213,14180,14182,14202],{"className":14181},[609,610],[213,14183,14185,14199],{"className":14184},[614],[213,14186,14188],{"className":14187,"style":619},[618],[213,14189,14190,14193],{"style":13675},[213,14191],{"className":14192,"style":627},[626],[213,14194,14196],{"className":14195},[631,632,633,634],[213,14197,260],{"className":14198},[299,634],[213,14200,642],{"className":14201},[641],[213,14203,14205],{"className":14204},[614],[213,14206,14208],{"className":14207,"style":649},[618],[213,14209],{},[213,14211,252],{"className":14212},[330],[213,14214],{"className":14215,"style":334},[304],[213,14217,14219,14222],{"className":14218},[299],[213,14220,8461],{"className":14221,"style":12202},[299,407],[213,14223,14225],{"className":14224},[605],[213,14226,14228,14248],{"className":14227},[609,610],[213,14229,14231,14245],{"className":14230},[614],[213,14232,14234],{"className":14233,"style":619},[618],[213,14235,14236,14239],{"style":13675},[213,14237],{"className":14238,"style":627},[626],[213,14240,14242],{"className":14241},[631,632,633,634],[213,14243,255],{"className":14244},[299,634],[213,14246,642],{"className":14247},[641],[213,14249,14251],{"className":14250},[614],[213,14252,14254],{"className":14253,"style":649},[618],[213,14255],{},[213,14257,569],{"className":14258},[372],[213,14260,14262,14265],{"style":14261},"top:-2.1675em;",[213,14263],{"className":14264,"style":3144},[626],[213,14266,14268],{"className":14267},[299],[213,14269,14271,14274],{"className":14270},[299],[213,14272,2867],{"className":14273},[299],[213,14275],{"className":14276,"style":4011},[299,4010],[213,14278,642],{"className":14279},[641],[213,14281,14283],{"className":14282},[614],[213,14284,14287],{"className":14285,"style":14286},[618],"height:1.88em;",[213,14288],{},[213,14290],{"className":14291,"sty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3,14592,642],{"className":14593},[641],[213,14595,14597],{"className":14596},[614],[213,14598,14600],{"className":14599,"style":14286},[618],[213,14601],{},[213,14603,14605],{"className":14604},[372],[213,14606,14608],{"className":14607},[1863,7008],[213,14609,14611,14632],{"className":14610},[609,610],[213,14612,14614,14629],{"className":14613},[614],[213,14615,14617],{"className":14616,"style":13997},[618],[213,14618,14619,14622],{"style":14000},[213,14620],{"className":14621,"style":14004},[626],[213,14623,14624],{"style":14007},[7030,14625,14626],{"xmlns":7032,"width":7033,"height":14010,"viewBox":14011},[7037,14627],{"d":14628},"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",[213,14630,642],{"className":14631},[641],[213,14633,14635],{"className":14634},[614],[213,14636,14638],{"className":14637,"style":14024},[618],[213,14639],{},[11,14641,14642,14643,14671,14672,14814],{},"对角线程序 ",[213,14644,14646,14659],{"className":14645,"translate":217},[216],[213,14647,14649],{"className":14648},[221],[223,14650,14651],{"xmlns":225},[227,14652,14653,14657],{},[230,14654,14655],{},[233,14656,9040],{},[279,14658,9040],{"encoding":281},[213,14660,14662],{"className":14661,"ariaHidden":251},[286],[213,14663,14665,14668],{"className":14664},[290],[213,14666],{"className":14667,"style":403},[294],[213,14669,9040],{"className":14670,"style":9087},[299,407]," 的行为恰恰取决于这张表格的主对角线 ",[213,14673,14675,14707],{"className":14674,"translate":217},[216],[213,14676,14678],{"className":14677},[221],[223,14679,14680],{"xmlns":225},[227,14681,14682,14704],{},[230,14683,14684,14686,14688,14694,14696,14702],{},[272,14685,12314],{},[238,14687,560],{"stretchy":243},[537,14689,14690,14692],{},[233,14691,8461],{},[233,14693,4046],{},[238,14695,252],{"separator":251},[537,14697,14698,14700],{},[233,14699,8461],{},[233,14701,4046],{},[238,14703,569],{"stretchy":243},[279,14705,14706],{"encoding":281},"\\text{halts}(P_i, P_i)",[213,14708,14710],{"className":14709,"ariaHidden":251},[286],[213,14711,14713,14716,14722,14725,14765,14768,14771,14811],{"className":14712},[290],[213,14714],{"className":14715,"style":319},[294],[213,14717,14719],{"className":14718},[299,1983],[213,14720,12314],{"className":14721},[299],[213,14723,560],{"className":14724},[323],[213,14726,14728,14731],{"className":14727},[299],[213,14729,8461],{"className":14730,"style":12202},[299,407],[213,14732,14734],{"className":14733},[605],[213,14735,14737,14757],{"className":14736},[609,610],[213,14738,14740,14754],{"className":14739},[614],[213,14741,14743],{"className":14742,"style":4081},[618],[213,14744,14745,14748],{"style":13675},[213,14746],{"className":14747,"style":627},[626],[213,14749,14751],{"className":14750},[631,632,633,634],[213,14752,4046],{"className":14753},[299,407,634],[213,14755,642],{"className":14756},[641],[213,14758,14760],{"className":14759},[614],[213,14761,14763],{"className":14762,"style":649},[618],[213,14764],{},[213,14766,252],{"className":14767},[330],[213,14769],{"className":14770,"style":334},[304],[213,14772,14774,14777],{"className":14773},[299],[213,14775,8461],{"className":14776,"style":12202},[299,407],[213,14778,14780],{"className":14779},[605],[213,14781,14783,14803],{"className":14782},[609,610],[213,14784,14786,14800],{"className":14785},[614],[213,14787,14789],{"className":14788,"style":4081},[618],[213,14790,14791,14794],{"style":13675},[213,14792],{"className":14793,"style":627},[626],[213,14795,14797],{"className":14796},[631,632,633,634],[213,14798,4046],{"className":14799},[299,407,634],[213,14801,642],{"className":14802},[641],[213,14804,14806],{"className":14805},[614],[213,14807,14809],{"className":14808,"style":649},[618],[213,14810],{},[213,14812,569],{"className":14813},[372],"，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。",[11,14816,14817],{},"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：",[7926,14819,14821],{"className":7928,"code":14820,"language":7930,"meta":172,"style":172},"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",[7932,14822,14823,14831,14841,14845,14861,14866,14874,14884,14901,14911,14918,14922,14938,14953,14967,14985,14992,15005,15012,15019,15023,15044],{"__ignoreMap":172},[213,14824,14825,14828],{"class":7936,"line":7937},[213,14826,14827],{"class":7940},"import",[213,14829,14830],{"class":7948}," sys\n",[213,14832,14833,14836,14839],{"class":7936,"line":173},[213,14834,14835],{"class":7948},"sys.setrecursionlimit(",[213,14837,14838],{"class":7954},"50",[213,14840,8038],{"class":7948},[213,14842,14843],{"class":7936,"line":7967},[213,14844,8120],{"emptyLinePlaceholder":179},[213,14846,14847,14849,14852,14855,14857,14859],{"class":7936,"line":7973},[213,14848,7941],{"class":7940},[213,14850,14851],{"class":7944}," hypothetical_halts",[213,14853,14854],{"class":7948},"(f, x, fuel",[213,14856,240],{"class":7940},[213,14858,7955],{"class":7954},[213,14860,7958],{"class":7948},[213,14862,14863],{"class":7936,"line":7979},[213,14864,14865],{"class":7963},"    \"\"\"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",[213,14867,14868,14871],{"class":7936,"line":7984},[213,14869,14870],{"class":7940},"    try",[213,14872,14873],{"class":7948},":\n",[213,14875,14876,14879,14881],{"class":7936,"line":7995},[213,14877,14878],{"class":7948},"        result ",[213,14880,240],{"class":7940},[213,14882,14883],{"class":7948}," f(x, fuel)\n",[213,14885,14886,14889,14892,14895,14898],{"class":7936,"line":8013},[213,14887,14888],{"class":7940},"        return",[213,14890,14891],{"class":7948}," result ",[213,14893,14894],{"class":7940},"is",[213,14896,14897],{"class":7940}," not",[213,14899,14900],{"class":7954}," None\n",[213,14902,14903,14906,14909],{"class":7936,"line":8041},[213,14904,14905],{"class":7940},"    except",[213,14907,14908],{"class":7954}," RecursionError",[213,14910,14873],{"class":7948},[213,14912,14913,14915],{"class":7936,"line":8059},[213,14914,14888],{"class":7940},[213,14916,14917],{"class":7954}," False\n",[213,14919,14920],{"class":7936,"line":8087},[213,14921,8120],{"emptyLinePlaceholder":179},[213,14923,14924,14926,14929,14932,14934,14936],{"class":7936,"line":8099},[213,14925,7941],{"class":7940},[213,14927,14928],{"class":7944}," D",[213,14930,14931],{"class":7948},"(x, fuel",[213,14933,240],{"class":7940},[213,14935,7955],{"class":7954},[213,14937,7958],{"class":7948},[213,14939,14940,14943,14946,14949,14951],{"class":7936,"line":8117},[213,14941,14942],{"class":7940},"    if",[213,14944,14945],{"class":7948}," fuel ",[213,14947,14948],{"class":7940},"\u003C=",[213,14950,11842],{"class":7954},[213,14952,14873],{"class":7948},[213,14954,14955,14958,14960,14962,14965],{"class":7936,"line":8123},[213,14956,14957],{"class":7940},"        raise",[213,14959,14908],{"class":7954},[213,14961,560],{"class":7948},[213,14963,14964],{"class":7963},"\"燃料耗尽，模拟死循环\"",[213,14966,8038],{"class":7948},[213,14968,14969,14972,14974,14977,14980,14983],{"class":7936,"line":8128},[213,14970,14971],{"class":7948},"    will_halt ",[213,14973,240],{"class":7940},[213,14975,14976],{"class":7948}," hypothetical_halts(x, x, fuel ",[213,14978,14979],{"class":7940},"-",[213,14981,14982],{"class":7954}," 1",[213,14984,8038],{"class":7948},[213,14986,14987,14989],{"class":7936,"line":8139},[213,14988,14942],{"class":7940},[213,14990,14991],{"class":7948}," will_halt:\n",[213,14993,14994,14996,14998,15000,15003],{"class":7936,"line":8164},[213,14995,14957],{"class":7940},[213,14997,14908],{"class":7954},[213,14999,560],{"class":7948},[213,15001,15002],{"class":7963},"\"模拟死循环：halts 判定其停机，故 D 刻意不停机\"",[213,15004,8038],{"class":7948},[213,15006,15007,15010],{"class":7936,"line":8187},[213,15008,15009],{"class":7940},"    else",[213,15011,14873],{"class":7948},[213,15013,15014,15016],{"class":7936,"line":8200},[213,15015,14888],{"class":7940},[213,15017,15018],{"class":7963}," \"D 停机\"\n",[213,15020,15021],{"class":7936,"line":8206},[213,15022,8120],{"emptyLinePlaceholder":179},[213,15024,15025,15028,15030,15033,15037,15039,15042],{"class":7936,"line":8211},[213,15026,15027],{"class":7948},"outcome ",[213,15029,240],{"class":7940},[213,15031,15032],{"class":7948}," D(D, ",[213,15034,15036],{"class":15035},"s9osk","fuel",[213,15038,240],{"class":7940},[213,15040,15041],{"class":7954},"8",[213,15043,8038],{"class":7948},[213,15045,15046,15048,15050,15053],{"class":7936,"line":8222},[213,15047,8225],{"class":7954},[213,15049,560],{"class":7948},[213,15051,15052],{"class":7963},"\"D(D) 的执行结果：\"",[213,15054,15055],{"class":7948},", outcome)\n",[11,15057,8236],{},[7926,15059,15062],{"className":15060,"code":15061,"language":1983},[8240],"=== 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",[7932,15063,15061],{"__ignoreMap":172},[11,15065,15066,15067,15070,15071,15073],{},"这段代码里的 ",[7932,15068,15069],{},"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 ",[7932,15072,15036],{},"（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）。",[30,15075,15076],{"id":15076},"哥德尔不完备定理",[11,15078,15079],{},"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：",[21,15081,15082],{},[11,15083,15084],{},"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。",[11,15086,15087],{},"哥德尔的证明大体分为三步：",[15089,15090,15091,15094,15451],"ol",{},[65,15092,15093],{},"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。",[65,15095,15096,15097,15126,15127,15210,15213,15214,15259,15260,15290,15291,15377,15378,15421,15422,15450],{},"构造自指命题。利用编码技巧，构造一个命题 ",[213,15098,15100,15114],{"className":15099,"translate":217},[216],[213,15101,15103],{"className":15102},[221],[223,15104,15105],{"xmlns":225},[227,15106,15107,15112],{},[230,15108,15109],{},[233,15110,15111],{},"G",[279,15113,15111],{"encoding":281},[213,15115,15117],{"className":15116,"ariaHidden":251},[286],[213,15118,15120,15123],{"className":15119},[290],[213,15121],{"className":15122,"style":403},[294],[213,15124,15111],{"className":15125},[299,407],"，其核心内容是：",[213,15128,15130],{"className":15129,"translate":217},[2584],[213,15131,15133,15159],{"className":15132,"translate":217},[216],[213,15134,15136],{"className":15135},[221],[223,15137,15138],{"xmlns":225,"display":2593},[227,15139,15140,15156],{},[230,15141,15142,15144,15146,15148,15151,15153],{},[233,15143,15111],{},[238,15145,457],{},[238,15147,240],{},[272,15149,15150],{},"「命题 ",[233,15152,15111],{},[272,15154,15155],{}," 本身在该系统内不可证明」",[279,15157,15158],{"encoding":281},"G := \\text{「命题 } G \\text{ 本身在该系统内不可证明」}",[213,15160,15162,15181],{"className":15161,"ariaHidden":251},[286],[213,15163,15165,15168,15171,15174,15178],{"className":15164},[290],[213,15166],{"className":15167,"style":403},[294],[213,15169,15111],{"className":15170},[299,407],[213,15172],{"className":15173,"style":305},[304],[213,15175,15177],{"className":15176},[309],":=",[213,15179],{"className":15180,"style":305},[304],[213,15182,15184,15187,15197,15200],{"className":15183},[290],[213,15185],{"className":15186,"style":403},[294],[213,15188,15190,15194],{"className":15189},[299,1983],[213,15191,15193],{"className":15192},[299,1991],"「命题",[213,15195,1987],{"className":15196},[299],[213,15198,15111],{"className":15199},[299,407],[213,15201,15203,15206],{"className":15202},[299,1983],[213,15204,1987],{"className":15205},[299],[213,15207,15209],{"className":15208},[299,1991],"本身在该系统内不可证明」",[15211,15212],"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 ",[213,15215,15217,15238],{"className":15216,"translate":217},[216],[213,15218,15220],{"className":15219},[221],[223,15221,15222],{"xmlns":225},[227,15223,15224,15235],{},[230,15225,15226,15229,15231,15233],{},[233,15227,15228],{},"ϕ",[238,15230,560],{"stretchy":243},[233,15232,2611],{},[238,15234,569],{"stretchy":243},[279,15236,15237],{"encoding":281},"\\phi(x)",[213,15239,15241],{"className":15240,"ariaHidden":251},[286],[213,15242,15244,15247,15250,15253,15256],{"className":15243},[290],[213,15245],{"className":15246,"style":319},[294],[213,15248,15228],{"className":15249},[299,407],[213,15251,560],{"className":15252},[323],[213,15254,2611],{"className":15255},[299,407],[213,15257,569],{"className":15258},[372],"，都能构造一个句子 ",[213,15261,15263,15278],{"className":15262,"translate":217},[216],[213,15264,15266],{"className":15265},[221],[223,15267,15268],{"xmlns":225},[227,15269,15270,15275],{},[230,15271,15272],{},[233,15273,15274],{},"ψ",[279,15276,15277],{"encoding":281},"\\psi",[213,15279,15281],{"className":15280,"ariaHidden":251},[286],[213,15282,15284,15287],{"className":15283},[290],[213,15285],{"className":15286,"style":477},[294],[213,15288,15274],{"className":15289,"style":2305},[299,407],"，使系统证明 ",[213,15292,15294,15329],{"className":15293,"translate":217},[216],[213,15295,15297],{"className":15296},[221],[223,15298,15299],{"xmlns":225},[227,15300,15301,15326],{},[230,15302,15303,15305,15308,15310,15312,15317,15319,15324],{},[233,15304,15274],{},[238,15306,15307],{},"↔",[233,15309,15228],{},[238,15311,560],{"stretchy":243},[238,15313,15314],{},[233,15315,15316],{"mathvariant":545},"⌜",[233,15318,15274],{},[238,15320,15321],{},[233,15322,15323],{"mathvariant":545},"⌝",[238,15325,569],{"stretchy":243},[279,15327,15328],{"encoding":281},"\\psi \\leftrightarrow \\phi(\\ulcorner \\psi \\urcorner)",[213,15330,15332,15350],{"className":15331,"ariaHidden":251},[286],[213,15333,15335,15338,15341,15344,15347],{"className":15334},[290],[213,15336],{"className":15337,"style":477},[294],[213,15339,15274],{"className":15340,"style":2305},[299,407],[213,15342],{"className":15343,"style":305},[304],[213,15345,15307],{"className":15346},[309],[213,15348],{"className":15349,"style":305},[304],[213,15351,15353,15356,15359,15362,15367,15370,15374],{"className":15352},[290],[213,15354],{"className":15355,"style":319},[294],[213,15357,15228],{"className":15358},[299,407],[213,15360,560],{"className":15361},[323],[213,15363,15366],{"className":15364},[323,15365],"amsrm","┌",[213,15368,15274],{"className":15369,"style":2305},[299,407],[213,15371,15373],{"className":15372},[372,15365],"┐",[213,15375,569],{"className":15376},[372],"（其中 ",[213,15379,15381,15403],{"className":15380,"translate":217},[216],[213,15382,15384],{"className":15383},[221],[223,15385,15386],{"xmlns":225},[227,15387,15388,15400],{},[230,15389,15390,15394,15396],{},[238,15391,15392],{},[233,15393,15316],{"mathvariant":545},[233,15395,15274],{},[238,15397,15398],{},[233,15399,15323],{"mathvariant":545},[279,15401,15402],{"encoding":281},"\\ulcorner \\psi \\urcorner",[213,15404,15406],{"className":15405,"ariaHidden":251},[286],[213,15407,15409,15412,15415,15418],{"className":15408},[290],[213,15410],{"className":15411,"style":477},[294],[213,15413,15366],{"className":15414},[323,15365],[213,15416,15274],{"className":15417,"style":2305},[299,407],[213,15419,15373],{"className":15420},[372,15365]," 表示 ",[213,15423,15425,15438],{"className":15424,"translate":217},[216],[213,15426,15428],{"className":15427},[221],[223,15429,15430],{"xmlns":225},[227,15431,15432,15436],{},[230,15433,15434],{},[233,15435,15274],{},[279,15437,15277],{"encoding":281},[213,15439,15441],{"className":15440,"ariaHidden":251},[286],[213,15442,15444,15447],{"className":15443},[290],[213,15445],{"className":15446,"style":477},[294],[213,15448,15274],{"className":15449,"style":2305},[299,407]," 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。",[65,15452,15453,15454],{},"导出矛盾的两难。",[62,15455,15456,15546,15642],{},[65,15457,15458,15459,15487,15488,15516,15517,15545],{},"若系统能证明 ",[213,15460,15462,15475],{"className":15461,"translate":217},[216],[213,15463,15465],{"className":15464},[221],[223,15466,15467],{"xmlns":225},[227,15468,15469,15473],{},[230,15470,15471],{},[233,15472,15111],{},[279,15474,15111],{"encoding":281},[213,15476,15478],{"className":15477,"ariaHidden":251},[286],[213,15479,15481,15484],{"className":15480},[290],[213,15482],{"className":15483,"style":403},[294],[213,15485,15111],{"className":15486},[299,407],"，那么根据 ",[213,15489,15491,15504],{"className":15490,"translate":217},[216],[213,15492,15494],{"className":15493},[221],[223,15495,15496],{"xmlns":225},[227,15497,15498,15502],{},[230,15499,15500],{},[233,15501,15111],{},[279,15503,15111],{"encoding":281},[213,15505,15507],{"className":15506,"ariaHidden":251},[286],[213,15508,15510,15513],{"className":15509},[290],[213,15511],{"className":15512,"style":403},[294],[213,15514,15111],{"className":15515},[299,407]," 的含义（「",[213,15518,15520,15533],{"className":15519,"translate":217},[216],[213,15521,15523],{"className":15522},[221],[223,15524,15525],{"xmlns":225},[227,15526,15527,15531],{},[230,15528,15529],{},[233,15530,15111],{},[279,15532,15111],{"encoding":281},[213,15534,15536],{"className":15535,"ariaHidden":251},[286],[213,15537,15539,15542],{"className":15538},[290],[213,15540],{"className":15541,"style":403},[294],[213,15543,15111],{"className":15544},[299,407]," 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。",[65,15547,15548,15549,15577,15578,15612,15613,15641],{},"若系统能否证 ",[213,15550,15552,15565],{"className":15551,"translate":217},[216],[213,15553,15555],{"className":15554},[221],[223,15556,15557],{"xmlns":225},[227,15558,15559,15563],{},[230,15560,15561],{},[233,15562,15111],{},[279,15564,15111],{"encoding":281},[213,15566,15568],{"className":15567,"ariaHidden":251},[286],[213,15569,15571,15574],{"className":15570},[290],[213,15572],{"className":15573,"style":403},[294],[213,15575,15111],{"className":15576},[299,407],"（即证明 ",[213,15579,15581,15597],{"className":15580,"translate":217},[216],[213,15582,15584],{"className":15583},[221],[223,15585,15586],{"xmlns":225},[227,15587,15588,15594],{},[230,15589,15590,15592],{},[233,15591,10084],{"mathvariant":545},[233,15593,15111],{},[279,15595,15596],{"encoding":281},"\\neg G",[213,15598,15600],{"className":15599,"ariaHidden":251},[286],[213,15601,15603,15606,15609],{"className":15602},[290],[213,15604],{"className":15605,"style":403},[294],[213,15607,10084],{"className":15608},[299],[213,15610,15111],{"className":15611},[299,407],"，相当于证明「",[213,15614,15616,15629],{"className":15615,"translate":217},[216],[213,15617,15619],{"className":15618},[221],[223,15620,15621],{"xmlns":225},[227,15622,15623,15627],{},[230,15624,15625],{},[233,15626,15111],{},[279,15628,15111],{"encoding":281},[213,15630,15632],{"className":15631,"ariaHidden":251},[286],[213,15633,15635,15638],{"className":15634},[290],[213,15636],{"className":15637,"style":403},[294],[213,15639,15111],{"className":15640},[299,407]," 可证明」），那么在合理的补充条件下，这同样会导致矛盾。",[65,15643,15644,15645,15673],{},"因此 ",[213,15646,15648,15661],{"className":15647,"translate":217},[216],[213,15649,15651],{"className":15650},[221],[223,15652,15653],{"xmlns":225},[227,15654,15655,15659],{},[230,15656,15657],{},[233,15658,15111],{},[279,15660,15111],{"encoding":281},[213,15662,15664],{"className":15663,"ariaHidden":251},[286],[213,15665,15667,15670],{"className":15666},[290],[213,15668],{"className":15669,"style":403},[294],[213,15671,15111],{"className":15672},[299,407]," 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」。",[11,15675,15676],{},"与前面两个证明的共通之处：",[10578,15678,15679,15809],{},[10581,15680,15681],{},[10584,15682,15683,15685,15716,15747,15778],{},[10587,15684],{},[10587,15686,15687,15688],{},"对角数 ",[213,15689,15691,15704],{"className":15690,"translate":217},[216],[213,15692,15694],{"className":15693},[221],[223,15695,15696],{"xmlns":225},[227,15697,15698,15702],{},[230,15699,15700],{},[233,15701,4247],{},[279,15703,4247],{"encoding":281},[213,15705,15707],{"className":15706,"ariaHidden":251},[286],[213,15708,15710,15713],{"className":15709},[290],[213,15711],{"className":15712,"style":4291},[294],[213,15714,4247],{"className":15715,"style":2305},[299,407],[10587,15717,15718,15719],{},"对角集 ",[213,15720,15722,15735],{"className":15721,"translate":217},[216],[213,15723,15725],{"className":15724},[221],[223,15726,15727],{"xmlns":225},[227,15728,15729,15733],{},[230,15730,15731],{},[233,15732,9040],{},[279,15734,9040],{"encoding":281},[213,15736,15738],{"className":15737,"ariaHidden":251},[286],[213,15739,15741,15744],{"className":15740},[290],[213,15742],{"className":15743,"style":403},[294],[213,15745,9040],{"className":15746,"style":9087},[299,407],[10587,15748,15749,15750],{},"对角程序 ",[213,15751,15753,15766],{"className":15752,"translate":217},[216],[213,15754,15756],{"className":15755},[221],[223,15757,15758],{"xmlns":225},[227,15759,15760,15764],{},[230,15761,15762],{},[233,15763,9040],{},[279,15765,9040],{"encoding":281},[213,15767,15769],{"className":15768,"ariaHidden":251},[286],[213,15770,15772,15775],{"className":15771},[290],[213,15773],{"className":15774,"style":403},[294],[213,15776,9040],{"className":15777,"style":9087},[299,407],[10587,15779,15780,15781],{},"哥德尔句 ",[213,15782,15784,15797],{"className":15783,"translate":217},[216],[213,15785,15787],{"className":15786},[221],[223,15788,15789],{"xmlns":225},[227,15790,15791,15795],{},[230,15792,15793],{},[233,15794,15111],{},[279,15796,15111],{"encoding":281},[213,15798,15800],{"className":15799,"ariaHidden":251},[286],[213,15801,15803,15806],{"className":15802},[290],[213,15804],{"className":15805,"style":403},[294],[213,15807,15111],{"className":15808},[299,407],[10593,15810,15811,15828,15845],{},[10584,15812,15813,15816,15819,15822,15825],{},[10598,15814,15815],{},"自指的对象",[10598,15817,15818],{},"每一位「避开」自身",[10598,15820,15821],{},"收集「不属于自己」的元素",[10598,15823,15824],{},"用自己的行为否定对自身的判定",[10598,15826,15827],{},"断言「我不可证明」",[10584,15829,15830,15833,15836,15839,15842],{},[10598,15831,15832],{},"依赖的机制",[10598,15834,15835],{},"十进制展开的对角线",[10598,15837,15838],{},"集合属于关系的自指",[10598,15840,15841],{},"程序以自身为输入",[10598,15843,15844],{},"通过哥德尔编号实现的自指",[10584,15846,15847,15850,15881,15912,15915],{},[10598,15848,15849],{},"矛盾的来源",[10598,15851,15852,15853],{},"假定的双射无法覆盖 ",[213,15854,15856,15869],{"className":15855,"translate":217},[216],[213,15857,15859],{"className":15858},[221],[223,15860,15861],{"xmlns":225},[227,15862,15863,15867],{},[230,15864,15865],{},[233,15866,4247],{},[279,15868,4247],{"encoding":281},[213,15870,15872],{"className":15871,"ariaHidden":251},[286],[213,15873,15875,15878],{"className":15874},[290],[213,15876],{"className":15877,"style":4291},[294],[213,15879,4247],{"className":15880,"style":2305},[299,407],[10598,15882,15883,15884],{},"假定的满射无法覆盖 ",[213,15885,15887,15900],{"className":15886,"translate":217},[216],[213,15888,15890],{"className":15889},[221],[223,15891,15892],{"xmlns":225},[227,15893,15894,15898],{},[230,15895,15896],{},[233,15897,9040],{},[279,15899,9040],{"encoding":281},[213,15901,15903],{"className":15902,"ariaHidden":251},[286],[213,15904,15906,15909],{"className":15905},[290],[213,15907],{"className":15908,"style":403},[294],[213,15910,9040],{"className":15911,"style":9087},[299,407],[10598,15913,15914],{},"假定的判定器导致悖论",[10598,15916,15917],{},"假定的可证明性导致悖论",[11,15919,15920],{},"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。",[30,15922,15923],{"id":15923},"延伸",[11,15925,15926],{},"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：",[62,15928,15929,15932,15993,15996],{},[65,15930,15931],{},"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。",[65,15933,15934,15935,15992],{},"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 ",[213,15936,15938,15958],{"className":15937,"translate":217},[216],[213,15939,15941],{"className":15940},[221],[223,15942,15943],{"xmlns":225},[227,15944,15945,15955],{},[230,15946,15947,15949,15952],{},[233,15948,8461],{},[238,15950,15951],{},"⊊",[272,15953,15954],{},"EXP",[279,15956,15957],{"encoding":281},"P \\subsetneq \\text{EXP}",[213,15959,15961,15980],{"className":15960,"ariaHidden":251},[286],[213,15962,15964,15968,15971,15974,15977],{"className":15963},[290],[213,15965],{"className":15966,"style":15967},[294],"height:0.8193em;vertical-align:-0.136em;",[213,15969,8461],{"className":15970,"style":12202},[299,407],[213,15972],{"className":15973,"style":305},[304],[213,15975,15951],{"className":15976},[309,15365],[213,15978],{"className":15979,"style":305},[304],[213,15981,15983,15986],{"className":15982},[290],[213,15984],{"className":15985,"style":403},[294],[213,15987,15989],{"className":15988},[299,1983],[213,15990,15954],{"className":15991},[299]," 等谱系关系的基础）。",[65,15994,15995],{},"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。",[65,15997,15998,15999,16139,16140,16289],{},"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，",[213,16000,16002,16033],{"className":16001,"translate":217},[216],[213,16003,16005],{"className":16004},[221],[223,16006,16007],{"xmlns":225},[227,16008,16009,16030],{},[230,16010,16011,16014,16016,16018,16020,16022,16024,16026,16028],{},[233,16012,16013],{},"R",[238,16015,240],{},[238,16017,244],{"stretchy":243},[233,16019,2611],{},[238,16021,457],{},[233,16023,2611],{},[238,16025,9059],{"mathvariant":545},[233,16027,2611],{},[238,16029,277],{"stretchy":243},[279,16031,16032],{"encoding":281},"R = \\{x : x \\notin x\\}",[213,16034,16036,16055,16076,16127],{"className":16035,"ariaHidden":251},[286],[213,16037,16039,16042,16046,16049,16052],{"className":16038},[290],[213,16040],{"className":16041,"style":403},[294],[213,16043,16013],{"className":16044,"style":16045},[299,407],"margin-right:0.0077em;",[213,16047],{"className":16048,"style":305},[304],[213,16050,240],{"className":16051},[309],[213,16053],{"className":16054,"style":305},[304],[213,16056,16058,16061,16064,16067,16070,16073],{"className":16057},[290],[213,16059],{"className":16060,"style":319},[294],[213,16062,244],{"className":16063},[323],[213,16065,2611],{"className":16066},[299,407],[213,16068],{"className":16069,"style":305},[304],[213,16071,457],{"className":16072},[309],[213,16074],{"className":16075,"style":305},[304],[213,16077,16079,16082,16085,16088,16124],{"className":16078},[290],[213,16080],{"className":16081,"style":319},[294],[213,16083,2611],{"className":16084},[299,407],[213,16086],{"className":16087,"style":305},[304],[213,16089,16091,16097],{"className":16090},[309],[213,16092,16094],{"className":16093},[299],[213,16095,930],{"className":16096},[309],[213,16098,16100],{"className":16099},[299,664],[213,16101,16103],{"className":16102},[668],[213,16104,16106,16109,16121],{"className":16105},[9169],[213,16107],{"className":16108,"style":319},[294],[213,16110,16112],{"className":16111},[679],[213,16113,16115,16118],{"className":16114},[299],[213,16116,1762],{"className":16117},[299],[213,16119],{"className":16120,"style":9185},[304],[213,16122],{"className":16123},[690],[213,16125],{"className":16126,"style":305},[304],[213,16128,16130,16133,16136],{"className":16129},[290],[213,16131],{"className":16132,"style":319},[294],[213,16134,2611],{"className":16135},[299,407],[213,16137,277],{"className":16138},[372],"，与康托尔定理证明中的集合 ",[213,16141,16143,16178],{"className":16142,"translate":217},[216],[213,16144,16146],{"className":16145},[221],[223,16147,16148],{"xmlns":225},[227,16149,16150,16176],{},[230,16151,16152,16154,16156,16158,16160,16162,16164,16166,16168,16170,16172,16174],{},[233,16153,9040],{},[238,16155,240],{},[238,16157,244],{"stretchy":243},[233,16159,107],{},[238,16161,457],{},[233,16163,107],{},[238,16165,9059],{"mathvariant":545},[233,16167,454],{},[238,16169,560],{"stretchy":243},[233,16171,107],{},[238,16173,569],{"stretchy":243},[238,16175,277],{"stretchy":243},[279,16177,10960],{"encoding":281},[213,16179,16181,16199,16220,16271],{"className":16180,"ariaHidden":251},[286],[213,16182,16184,16187,16190,16193,16196],{"className":16183},[290],[213,16185],{"className":16186,"style":403},[294],[213,16188,9040],{"className":16189,"style":9087},[299,407],[213,16191],{"className":16192,"style":305},[304],[213,16194,240],{"className":16195},[309],[213,16197],{"className":16198,"style":305},[304],[213,16200,16202,16205,16208,16211,16214,16217],{"className":16201},[290],[213,16203],{"className":16204,"style":319},[294],[213,16206,244],{"className":16207},[323],[213,16209,107],{"className":16210},[299,407],[213,16212],{"className":16213,"style":305},[304],[213,16215,457],{"className":16216},[309],[213,16218],{"className":16219,"style":305},[304],[213,16221,16223,16226,16229,16232,16268],{"className":16222},[290],[213,16224],{"className":16225,"style":319},[294],[213,16227,107],{"className":16228},[299,407],[213,16230],{"className":16231,"style":305},[304],[213,16233,16235,16241],{"className":16234},[309],[213,16236,16238],{"className":16237},[299],[213,16239,930],{"className":16240},[309],[213,16242,16244],{"className":16243},[299,664],[213,16245,16247],{"className":16246},[668],[213,16248,16250,16253,16265],{"className":16249},[9169],[213,16251],{"className":16252,"style":319},[294],[213,16254,16256],{"className":16255},[679],[213,16257,16259,16262],{"className":16258},[299],[213,16260,1762],{"className":16261},[299],[213,16263],{"className":16264,"style":9185},[304],[213,16266],{"className":16267},[690],[213,16269],{"className":16270,"style":305},[304],[213,16272,16274,16277,16280,16283,16286],{"className":16273},[290],[213,16275],{"className":16276,"style":319},[294],[213,16278,454],{"className":16279,"style":481},[299,407],[213,16281,560],{"className":16282},[323],[213,16284,107],{"className":16285},[299,407],[213,16287,11072],{"className":16288},[372]," 具有完全相同的结构——两者都是「自指 + 自我排除」的产物。",[11,16291,16292],{},"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：",[15089,16294,16295,16298,16301,16362],{},[65,16296,16297],{},"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。",[65,16299,16300],{},"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）。",[65,16302,16303,16304,16332,16333,16361],{},"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 ",[213,16305,16307,16320],{"className":16306,"translate":217},[216],[213,16308,16310],{"className":16309},[221],[223,16311,16312],{"xmlns":225},[227,16313,16314,16318],{},[230,16315,16316],{},[233,16317,1496],{},[279,16319,1496],{"encoding":281},[213,16321,16323],{"className":16322,"ariaHidden":251},[286],[213,16324,16326,16329],{"className":16325},[290],[213,16327],{"className":16328,"style":6175},[294],[213,16330,1496],{"className":16331},[299,407]," 个条目在第 ",[213,16334,16336,16349],{"className":16335,"translate":217},[216],[213,16337,16339],{"className":16338},[221],[223,16340,16341],{"xmlns":225},[227,16342,16343,16347],{},[230,16344,16345],{},[233,16346,1496],{},[279,16348,1496],{"encoding":281},[213,16350,16352],{"className":16351,"ariaHidden":251},[286],[213,16353,16355,16358],{"className":16354},[290],[213,16356],{"className":16357,"style":6175},[294],[213,16359,1496],{"className":16360},[299,407]," 个位置不同」这一模式）。",[65,16363,16364],{},"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。",[16366,16367,16368],"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 .sAwPA{--shiki-default:#6A737D}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .s9osk, html code.shiki .s9osk{--shiki-default:#FFAB70}",{"title":172,"searchDepth":173,"depth":173,"links":16370},[16371,16372,16373,16374,16377,16378,16379],{"id":205,"depth":173,"text":205},{"id":2312,"depth":173,"text":2312},{"id":7921,"depth":173,"text":7921},{"id":8408,"depth":173,"text":8408,"children":16375},[16376],{"id":8718,"depth":7967,"text":8718},{"id":12167,"depth":173,"text":12167},{"id":15076,"depth":173,"text":15076},{"id":15923,"depth":173,"text":15923},{},"\u002Fblog\u002F2026\u002F2026-07-31-the-diagonal-argument-en",{"title":194,"description":199},"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。",[16386,16387],"mathematics","computation","2026-07-31T00:00:00+08:00","ZcA-tfQ1zVCr2HL5QHuWWDVkSCGS-RbN98cAuxLrnw0",{"id":16391,"title":16392,"body":16393,"description":172,"draft":178,"enableComment":179,"extension":180,"image":172,"important":178,"location":181,"meta":17369,"navigation":179,"ogImage":181,"path":17370,"seo":17371,"stem":17372,"summary":17373,"tags":17374,"time":17376,"__hash__":17377},"blog\u002Fblog\u002F2026\u002F2026-06-30-about-the-emergency.md","涌现是幻象吗？",{"type":8,"value":16394,"toc":17352},[16395,16398,16407,16410,16413,16643,16808,16963,16966,16969,16972,16976,17167,17171,17267,17271,17274,17278,17281,17284,17287,17290,17293,17296,17300,17303,17306,17318,17322,17325,17328,17331,17334,17337,17340,17343,17346,17349],[30,16396,16397],{"id":16397},"涌现",[11,16399,16400,16401,16406],{},"「涌现是伪概念」——即复杂系统中貌似质变的现象，只是把非线性的评估指标套在本质上线性或平滑变化的过程上所产生的假象——这一主张自 Schaeffer、Miranda 与 Koyejo 2023 年的论文《",[107,16402,16405],{"href":16403,"rel":16404},"https:\u002F\u002Farxiv.org\u002Fabs\u002F2304.15004",[111],"大语言模型的涌现能力是幻象吗？","》发表以来获得了广泛支持。这个论证的吸引力在于它的简洁：选一个不连续的指标，你就会在一个底层数据生成过程中不存在不连续的地方制造出一个不连续。本文考察这一论证的力量与局限，把它置于物理与生物系统相变研究的更广阔文献之中，并提出一个更具体的问题：当代机器学习理论——尺度定律、损失地形几何、grokking 与机制可解释性——究竟告诉我们，经过训练的网络中出现的涌现，是测量假象、真实的动力学相变，还是根本抗拒这种二分法？",[11,16408,16409],{},"我在下文详细论证的结论是：「幻象」批判作为对某一类基准报告式涌现能力的一个局部、方法论层面的论断是正确的，但它并不能推广为对涌现作为科学概念的彻底否定。过去三年的机器学习研究，已经从训练损失相变、grokking 动力学和电路层级的可解释性等方面独立积累了证据，证明神经网络行为中的某些不连续性是优化动力学自身的属性，与读出指标的选择无关。区分这两种情形需要特定的方法论纪律，本文试图把它明确化。",[30,16411,16412],{"id":16412},"幻象论证的形式化",[11,16414,16415,16416,16444,16445,16473,16474,16503,16504,16642],{},"Schaeffer 等人的核心技术主张建立在一个简单的概率分解之上。考虑一个需要 ",[213,16417,16419,16432],{"className":16418,"translate":217},[216],[213,16420,16422],{"className":16421},[221],[223,16423,16424],{"xmlns":225},[227,16425,16426,16430],{},[230,16427,16428],{},[233,16429,1496],{},[279,16431,1496],{"encoding":281},[213,16433,16435],{"className":16434,"ariaHidden":251},[286],[213,16436,16438,16441],{"className":16437},[290],[213,16439],{"className":16440,"style":6175},[294],[213,16442,1496],{"className":16443},[299,407]," 个独立子步骤全部正确的多步任务——例如多位数字的算术问题，或带有若干中间步骤的思维链推导。如果模型每步的正确率 ",[213,16446,16448,16461],{"className":16447,"translate":217},[216],[213,16449,16451],{"className":16450},[221],[223,16452,16453],{"xmlns":225},[227,16454,16455,16459],{},[230,16456,16457],{},[233,16458,11],{},[279,16460,11],{"encoding":281},[213,16462,16464],{"className":16463,"ariaHidden":251},[286],[213,16465,16467,16470],{"className":16466},[290],[213,16468],{"className":16469,"style":4291},[294],[213,16471,11],{"className":16472},[299,407]," 随参数量 ",[213,16475,16477,16490],{"className":16476,"translate":217},[216],[213,16478,16480],{"className":16479},[221],[223,16481,16482],{"xmlns":225},[227,16483,16484,16488],{},[230,16485,16486],{},[233,16487,236],{},[279,16489,236],{"encoding":281},[213,16491,16493],{"className":16492,"ariaHidden":251},[286],[213,16494,16496,16499],{"className":16495},[290],[213,16497],{"className":16498,"style":403},[294],[213,16500,236],{"className":16501,"style":16502},[299,407],"margin-right:0.109em;"," 平滑、连续地提高（比方说，按照尺度定律理论的标准形式遵循幂律，",[213,16505,16507,16547],{"className":16506,"translate":217},[216],[213,16508,16510],{"className":16509},[221],[223,16511,16512],{"xmlns":225},[227,16513,16514,16544],{},[230,16515,16516,16518,16520,16522,16524,16527,16529,16531,16533],{},[233,16517,11],{},[238,16519,560],{"stretchy":243},[233,16521,236],{},[238,16523,569],{"stretchy":243},[238,16525,16526],{},"≈",[246,16528,255],{},[238,16530,1580],{},[233,16532,107],{},[11501,16534,16535,16537],{},[233,16536,236],{},[230,16538,16539,16541],{},[238,16540,1580],{},[233,16542,16543],{},"α",[279,16545,16546],{"encoding":281},"p(N) \\approx 1 - aN^{-\\alpha}",[213,16548,16550,16577,16596],{"className":16549,"ariaHidden":251},[286],[213,16551,16553,16556,16559,16562,16565,16568,16571,16574],{"className":16552},[290],[213,16554],{"className":16555,"style":319},[294],[213,16557,11],{"className":16558},[299,407],[213,16560,560],{"className":16561},[323],[213,16563,236],{"className":16564,"style":16502},[299,407],[213,16566,569],{"className":16567},[372],[213,16569],{"className":16570,"style":305},[304],[213,16572,16526],{"className":16573},[309],[213,16575],{"className":16576,"style":305},[304],[213,16578,16580,16584,16587,16590,16593],{"className":16579},[290],[213,16581],{"className":16582,"style":16583},[294],"height:0.7278em;vertical-align:-0.0833em;",[213,16585,255],{"className":16586},[299],[213,16588],{"className":16589,"style":1923},[304],[213,16591,1580],{"className":16592},[1927],[213,16594],{"className":16595,"style":1923},[304],[213,16597,16599,16603,16606],{"className":16598},[290],[213,16600],{"className":16601,"style":16602},[294],"height:0.7713em;",[213,16604,107],{"className":16605},[299,407],[213,16607,16609,16612],{"className":16608},[299],[213,16610,236],{"className":16611,"style":16502},[299,407],[213,16613,16615],{"className":16614},[605],[213,16616,16618],{"className":16617},[609],[213,16619,16621],{"className":16620},[614],[213,16622,16624],{"className":16623,"style":16602},[618],[213,16625,16626,16629],{"style":11556},[213,16627],{"className":16628,"style":627},[626],[213,16630,16632],{"className":16631},[631,632,633,634],[213,16633,16635,16638],{"className":16634},[299,634],[213,16636,1580],{"className":16637},[299,634],[213,16639,16543],{"className":16640,"style":16641},[299,407,634],"margin-right:0.0037em;","），那么在精确匹配指标下的任务级准确率就是：",[213,16644,16646],{"className":16645,"translate":217},[2584],[213,16647,16649,16688],{"className":16648,"translate":217},[216],[213,16650,16652],{"className":16651},[221],[223,16653,16654],{"xmlns":225,"display":2593},[227,16655,16656,16685],{},[230,16657,16658,16665,16667,16669,16671,16673,16675,16677,16679],{},[537,16659,16660,16662],{},[233,16661,8461],{},[272,16663,16664],{},"任务",[238,16666,560],{"stretchy":243},[233,16668,236],{},[238,16670,569],{"stretchy":243},[238,16672,240],{},[233,16674,11],{},[238,16676,560],{"stretchy":243},[233,16678,236],{},[11501,16680,16681,16683],{},[238,16682,569],{"stretchy":243},[233,16684,1496],{},[279,16686,16687],{"encoding":281},"P_{\\text{任务}}(N) = p(N)^n",[213,16689,16691,16762],{"className":16690,"ariaHidden":251},[286],[213,16692,16694,16697,16744,16747,16750,16753,16756,16759],{"className":16693},[290],[213,16695],{"className":16696,"style":319},[294],[213,16698,16700,16703],{"className":16699},[299],[213,16701,8461],{"className":16702,"style":12202},[299,407],[213,16704,16706],{"className":16705},[605],[213,16707,16709,16736],{"className":16708},[609,610],[213,16710,16712,16733],{"className":16711},[614],[213,16713,16716],{"className":16714,"style":16715},[618],"height:0.3283em;",[213,16717,16718,16721],{"style":13675},[213,16719],{"className":16720,"style":627},[626],[213,16722,16724],{"className":16723},[631,632,633,634],[213,16725,16727],{"className":16726},[299,634],[213,16728,16730],{"className":16729},[299,1983,634],[213,16731,16664],{"className":16732},[299,1991,634],[213,16734,642],{"className":16735},[641],[213,16737,16739],{"className":16738},[614],[213,16740,16742],{"className":16741,"style":649},[618],[213,16743],{},[213,16745,560],{"className":16746},[323],[213,16748,236],{"className":16749,"style":16502},[299,407],[213,16751,569],{"className":16752},[372],[213,16754],{"className":16755,"style":305},[304],[213,16757,240],{"className":16758},[309],[213,16760],{"className":16761,"style":305},[304],[213,16763,16765,16768,16771,16774,16777],{"className":16764},[290],[213,16766],{"className":16767,"style":319},[294],[213,16769,11],{"className":16770},[299,407],[213,16772,560],{"className":16773},[323],[213,16775,236],{"className":16776,"style":16502},[299,407],[213,16778,16780,16783],{"className":16779},[372],[213,16781,569],{"className":16782},[372],[213,16784,16786],{"className":16785},[605],[213,16787,16789],{"className":16788},[609],[213,16790,16792],{"className":16791},[614],[213,16793,16796],{"className":16794,"style":16795},[618],"height:0.7144em;",[213,16797,16799,16802],{"style":16798},"top:-3.113em;margin-right:0.05em;",[213,16800],{"className":16801,"style":627},[626],[213,16803,16805],{"className":16804},[631,632,633,634],[213,16806,1496],{"className":16807},[299,407,634],[11,16809,16810,16811,16839,16840,16917,16918,16962],{},"由于这是一个趋近于 1 的量的幂，",[213,16812,16814,16827],{"className":16813,"translate":217},[216],[213,16815,16817],{"className":16816},[221],[223,16818,16819],{"xmlns":225},[227,16820,16821,16825],{},[230,16822,16823],{},[233,16824,11],{},[279,16826,11],{"encoding":281},[213,16828,16830],{"className":16829,"ariaHidden":251},[286],[213,16831,16833,16836],{"className":16832},[290],[213,16834],{"className":16835,"style":4291},[294],[213,16837,11],{"className":16838},[299,407]," 在接近上限处的微小绝对增长，会引起 ",[213,16841,16843,16861],{"className":16842,"translate":217},[216],[213,16844,16846],{"className":16845},[221],[223,16847,16848],{"xmlns":225},[227,16849,16850,16858],{},[230,16851,16852],{},[537,16853,16854,16856],{},[233,16855,8461],{},[272,16857,16664],{},[279,16859,16860],{"encoding":281},"P_{\\text{任务}}",[213,16862,16864],{"className":16863,"ariaHidden":251},[286],[213,16865,16867,16871],{"className":16866},[290],[213,16868],{"className":16869,"style":16870},[294],"height:0.8333em;vertical-align:-0.15em;",[213,16872,16874,16877],{"className":16873},[299],[213,16875,8461],{"className":16876,"style":12202},[299,407],[213,16878,16880],{"className":16879},[605],[213,16881,16883,16909],{"className":16882},[609,610],[213,16884,16886,16906],{"className":16885},[614],[213,16887,16889],{"className":16888,"style":16715},[618],[213,16890,16891,16894],{"style":13675},[213,16892],{"className":16893,"style":627},[626],[213,16895,16897],{"className":16896},[631,632,633,634],[213,16898,16900],{"className":16899},[299,634],[213,16901,16903],{"className":16902},[299,1983,634],[213,16904,16664],{"className":16905},[299,1991,634],[213,16907,642],{"className":16908},[641],[213,16910,16912],{"className":16911},[614],[213,16913,16915],{"className":16914,"style":649},[618],[213,16916],{}," 不成比例的巨大相对增长。结果是出现一条先平后陡的曲线——正是 BIG-Bench 上几十个任务报告的那种标志性的「涌现」S 型曲线——尽管 ",[213,16919,16921,16941],{"className":16920,"translate":217},[216],[213,16922,16924],{"className":16923},[221],[223,16925,16926],{"xmlns":225},[227,16927,16928,16938],{},[230,16929,16930,16932,16934,16936],{},[233,16931,11],{},[238,16933,560],{"stretchy":243},[233,16935,236],{},[238,16937,569],{"stretchy":243},[279,16939,16940],{"encoding":281},"p(N)",[213,16942,16944],{"className":16943,"ariaHidden":251},[286],[213,16945,16947,16950,16953,16956,16959],{"className":16946},[290],[213,16948],{"className":16949,"style":319},[294],[213,16951,11],{"className":16952},[299,407],[213,16954,560],{"className":16955},[323],[213,16957,236],{"className":16958,"style":16502},[299,407],[213,16960,569],{"className":16961},[372]," 本身完全平滑。关键在于，作者还表明：换成某种连续的指标（词元级对数似然、布莱尔分数或部分得分），就能在同一批模型、同一批任务上恢复平滑性。这之所以是一个有力的结果，恰恰因为它是一个受控比较：同样的系统、同样的数据，不同的读出函数，得到不同的定性结论。",[11,16964,16965],{},"这推广了统计学与心理物理学中一个早已熟知的观点：连续变量的阶跃函数可以通过取阈值轻而易举地构造出来，而取阈值带来的不连续，告诉你的只是阈值本身，而不是那个变量。同样的逻辑也能解释洞察式问题解决研究中出现的「恍然大悟时刻」——主观的、二值的自我报告掩盖了连续的、低于阈值的信息积累，后者要等眼动追踪等连续仪器才能揭示。它也能解释那些表面上的生态「崩塌」，一旦提高采样频率，便消解为渐进的衰退。这些情形的共同结构缺陷是：把观测的离散性误当成了过程的离散性。",[30,16967,16968],{"id":16968},"论证的越界",[11,16970,16971],{},"从「某些报告的涌现能力是指标假象」推出「涌现是伪概念」，犯了我所谓反例消除谬误的错误：对某一子类主张的一次具体、受控的证伪，被当成了对整个范畴的证伪。有三点考虑限制了幻象论证的适用范围。",[8716,16973,16975],{"id":16974},"组合式任务结构是世界的真实属性而非仅指标的属性","组合式任务结构是世界的真实属性，而非仅指标的属性",[11,16977,16978,16979,17007,17008,17166],{},"如果某个下游应用确实要求全部 ",[213,16980,16982,16995],{"className":16981,"translate":217},[216],[213,16983,16985],{"className":16984},[221],[223,16986,16987],{"xmlns":225},[227,16988,16989,16993],{},[230,16990,16991],{},[233,16992,1496],{},[279,16994,1496],{"encoding":281},[213,16996,16998],{"className":16997,"ariaHidden":251},[286],[213,16999,17001,17004],{"className":17000},[290],[213,17002],{"className":17003,"style":6175},[294],[213,17005,1496],{"className":17006},[299,407]," 步都正确——编译后的程序必须没有语法错误；证明必须没有无效步骤——那么 ",[213,17009,17011,17048],{"className":17010,"translate":217},[216],[213,17012,17014],{"className":17013},[221],[223,17015,17016],{"xmlns":225},[227,17017,17018,17046],{},[230,17019,17020,17026,17028,17030,17032,17034,17036,17038,17040],{},[537,17021,17022,17024],{},[233,17023,8461],{},[272,17025,16664],{},[238,17027,560],{"stretchy":243},[233,17029,236],{},[238,17031,569],{"stretchy":243},[238,17033,240],{},[233,17035,11],{},[238,17037,560],{"stretchy":243},[233,17039,236],{},[11501,17041,17042,17044],{},[238,17043,569],{"stretchy":243},[233,17045,1496],{},[279,17047,16687],{"encoding":281},[213,17049,17051,17121],{"className":17050,"ariaHidden":251},[286],[213,17052,17054,17057,17103,17106,17109,17112,17115,17118],{"className":17053},[290],[213,17055],{"className":17056,"style":319},[294],[213,17058,17060,17063],{"className":17059},[299],[213,17061,8461],{"className":17062,"style":12202},[299,407],[213,17064,17066],{"className":17065},[605],[213,17067,17069,17095],{"className":17068},[609,610],[213,17070,17072,17092],{"className":17071},[614],[213,17073,17075],{"className":17074,"style":16715},[618],[213,17076,17077,17080],{"style":13675},[213,17078],{"className":17079,"style":627},[626],[213,17081,17083],{"className":17082},[631,632,633,634],[213,17084,17086],{"className":17085},[299,634],[213,17087,17089],{"className":17088},[299,1983,634],[213,17090,16664],{"className":17091},[299,1991,634],[213,17093,642],{"className":17094},[641],[213,17096,17098],{"className":17097},[614],[213,17099,17101],{"className":17100,"style":649},[618],[213,17102],{},[213,17104,560],{"className":17105},[323],[213,17107,236],{"className":17108,"style":16502},[299,407],[213,17110,569],{"className":17111},[372],[213,17113],{"className":17114,"style":305},[304],[213,17116,240],{"className":17117},[309],[213,17119],{"className":17120,"style":305},[304],[213,17122,17124,17127,17130,17133,17136],{"className":17123},[290],[213,17125],{"className":17126,"style":319},[294],[213,17128,11],{"className":17129},[299,407],[213,17131,560],{"className":17132},[323],[213,17134,236],{"className":17135,"style":16502},[299,407],[213,17137,17139,17142],{"className":17138},[372],[213,17140,569],{"className":17141},[372],[213,17143,17145],{"className":17144},[605],[213,17146,17148],{"className":17147},[609],[213,17149,17151],{"className":17150},[614],[213,17152,17155],{"className":17153,"style":17154},[618],"height:0.6644em;",[213,17156,17157,17160],{"style":11556},[213,17158],{"className":17159,"style":627},[626],[213,17161,17163],{"className":17162},[631,632,633,634],[213,17164,1496],{"className":17165},[299,407,634]," 就不是测量假象，而是对用户相关的成功概率的正确描述。其机制完全可以解释、毫无神秘可言（这是初等概率，不是隐藏的相变），但现象本身——在一个狭窄的能力区间内可用输出占比的急剧上升——是真实且有后果的。幻象批判正确地揭开了机制的谜团；它并没有消除现象。这个重要的区分在论证的通俗转述中被抹平了：「无法解释」与「虚假」不是同义词。",[8716,17168,17170],{"id":17169},"复杂系统中确实存在真实相变具有指标无关的签名","复杂系统中确实存在真实相变，具有指标无关的签名",[11,17172,17173,17174,8145,17205,8145,17235,17266],{},"统计物理提供了一个严谨的反例类。铁磁体系中的居里点是真正意义上的二级相变：重整化群理论预言了临界指数（",[213,17175,17177,17192],{"className":17176,"translate":217},[216],[213,17178,17180],{"className":17179},[221],[223,17181,17182],{"xmlns":225},[227,17183,17184,17189],{},[230,17185,17186],{},[233,17187,17188],{},"β",[279,17190,17191],{"encoding":281},"\\beta",[213,17193,17195],{"className":17194,"ariaHidden":251},[286],[213,17196,17198,17201],{"className":17197},[290],[213,17199],{"className":17200,"style":477},[294],[213,17202,17188],{"className":17203,"style":17204},[299,407],"margin-right:0.0528em;",[213,17206,17208,17223],{"className":17207,"translate":217},[216],[213,17209,17211],{"className":17210},[221],[223,17212,17213],{"xmlns":225},[227,17214,17215,17220],{},[230,17216,17217],{},[233,17218,17219],{},"γ",[279,17221,17222],{"encoding":281},"\\gamma",[213,17224,17226],{"className":17225,"ariaHidden":251},[286],[213,17227,17229,17232],{"className":17228},[290],[213,17230],{"className":17231,"style":4291},[294],[213,17233,17219],{"className":17234,"style":9185},[299,407],[213,17236,17238,17253],{"className":17237,"translate":217},[216],[213,17239,17241],{"className":17240},[221],[223,17242,17243],{"xmlns":225},[227,17244,17245,17250],{},[230,17246,17247],{},[233,17248,17249],{},"ν",[279,17251,17252],{"encoding":281},"\\nu",[213,17254,17256],{"className":17255,"ariaHidden":251},[286],[213,17257,17259,17262],{"className":17258},[290],[213,17260],{"className":17261,"style":6175},[294],[213,17263,17249],{"className":17264,"style":17265},[299,407],"margin-right:0.0637em;","），并在相互独立的可观测量——磁化强度、磁化率、关联长度、比热——上得到验证，所有这些量都在同一个临界温度处发散或消失，无论你选择测量哪一个。随机图上的渗流转变具有类似的普适性：当边密度跨越阈值时，连通性性质急剧转变，这可以从概率论的第一性原理出发证明，与任何特定的观测读出无关。在这些情形里，那个令 LLM「涌现能力」声名扫地的检验——转变在更换指标后是否依然存在？——被漂亮地通过了。一旦离开幻象论文所在的特定领域（LLM 基准评测），「涌现总是指标假象」这一全称论断在经验上就是错的。",[8716,17268,17270],{"id":17269},"涌现涵盖多种不同的现象其中只有一部分处于争议之中","「涌现」涵盖多种不同的现象，其中只有一部分处于争议之中",[11,17272,17273],{},"科学哲学文献（安德森 1972 年的《多即不同》是标准参考点）至少区分了：（a）认识论涌现——尽管原则上可由微观规则推导，却难以从微观规则预测的性质；（b）本体论\u002F弱涌现——在宏观尺度上新颖、但原则上可经由模拟还原的性质；以及（c）强涌现——被声称原则上也不可还原的性质，这是一个更有争议的概念。幻象论证在其恰当范围内，是一个关于特定认识论错觉——由指标选择诱发的表面上的不可预测性——的论证，而对弱涌现是否存在于神经网络中几乎不置一词。混淆这些含义，正是这场争论产生更多热量而非光亮的缘由之一。",[8716,17275,17277],{"id":17276},"镜像情形线性指标掩盖真实的非线性结构","镜像情形：线性指标掩盖真实的非线性结构",[11,17279,17280],{},"前面的讨论都把指标选择当作产生假象式不连续的来源——LLM 的情形、洞察式问题解决的情形、粗略生态普查的情形。三者共享同一个误差方向：非线性或取阈值的读出函数，从平滑的底层现实中制造出表面上的陡峭。但地震震级里氏标度的发展历史则展示了相反的误差方向，它之所以有教益，恰恰因为它不符合幻象模板。",[11,17282,17283],{},"地震学家最初用地面运动的原始振幅量化地震强度，这是一种线性的仪器读数，与地震仪记录到的物理位移成正比。结果发现，这个指标与研究者真正关心的构念——破坏潜力——匹配得很差，因为地震能量释放随振幅不是线性而是乘性地变化：振幅增大十倍，辐射能量约增大三十二倍，而破坏能力与能量、而非位移成正比。在原始的线性振幅指标下，现象的真实结构——能量释放的古登堡 - 里克特幂律分布，其中灾难性事件与轻微震颤相差许多个数量级——被压缩进一个狭窄的、视觉上波澜不惊的数值区间，系统地掩盖了 8 级事件与 5 级事件之间类别的天壤之别。查尔斯·里希特在 1935 年的解决方案，是把震级定义为振幅以 10 为底的对数（并作距离修正）。这是有意把非线性引入指标，使得新标度上的等量增量对应底层能量释放的等量比值——把一个乘性的、难以沟通的现实，变成一个加性的、可解读的。",[11,17285,17286],{},"地震情形是 LLM 幻象在结构上的镜像：在那里，非线性指标（组合式任务结构下的精确匹配准确率）从一个实际上平滑的底层现实中制造出虚假的不连续。在这里，线性指标（原始振幅）掩盖了一个真实的、性质刻画清楚的乘性结构，而纠正的做法恰恰是采用对数——即非线性——指标，正因为底层物理过程（由断层破裂面积与应力降控制的弹性能量释放）本身在所测量的量上是指数式的。关键在于，这并不是随意的选择：采用对数变换，是因为独立的物理理论（地震矩、破裂尺度与辐射能量之间的关系）预先预言了乘性关系，新标度也正是依据这一理论得到验证——这正是下文判别框架中将被形式化为检验 2 的那种「独立理论预言」标准。",[11,17288,17289],{},"把 LLM 与地震这两个案例放在一起读，可以三角定位出这场争论背后那个更深层的方法论原则：任何指标对现象施加的定性形状都不可能是中立的，而可能的失真方向是双向的。非线性\u002F取阈值的指标可以从平滑现实中制造出虚假的陡峭（LLM 基准）；线性指标同样可以掩盖真实的、有理论根基的非线性结构（里氏标度之前的振幅标度）。因此，「偏好连续指标」本身并不是从幻象论证中应该提取的正确方法论教训——正确的教训是：指标的函数形式必须与被测量的原始量与科学关注的构念之间真实关系的函数形式相匹配，而这种匹配必须独立地（通过理论，或通过与破坏性之类外部准则的验证）得到辩护，而不是在任一方向上默认为之。仅仅因为某个指标是连续的便选择它，却不追问那个特定的连续变换是否追踪了所关注的构念，将和幻象论文所批评的精确匹配错误一样没有原则。",[30,17291,17292],{"id":17292},"机器学习理论的补充",[11,17294,17295],{},"最有趣的近期证据并不来自基准准确率曲线，而是来自神经网络的内部训练动力学——这是幻象论文未曾涉及的领域，因为它只研究跨模型规模的推理期尺度行为，而不研究单个模型在训练时间上的轨迹。",[8716,17297,17299],{"id":17298},"grokking真实的动力学转变","Grokking：真实的动力学转变",[11,17301,17302],{},"Power、Burns、Edwards、Babuschkin 与 Misra（2022）记录了「grokking」现象：在小算法任务（例如模算术训练）上，网络几乎立即记住训练集，随后长时间停留在接近零的测试准确率平台上，再急剧转变到接近完美的泛化——往往比训练损失首次收敛的时间晚好几个数量级。这不是 Schaeffer 意义上的基准指标假象：它在许多随机种子下的测试集准确率上都能观察到，在测试损失等连续指标读出下也成立，而且后来被赋予了一个机制解释——同一网络内部一个记忆电路与一个泛化电路相互竞争，在权重衰减压力下泛化电路最终胜出（Nanda 等，2023，《通过机制可解释性衡量 grokking 的进展指标》）。关键在于，Nanda 等人表明，连续的机制性进展指标——从学习到的嵌入的傅里叶结构导出、追踪用于实现模加法的三角表示的逐渐形成的量——在测试准确率看似平坦的平台上平滑、可预期地上升。这正是 LLM 基准故事在结构上的镜像：在这里，粗粒度指标（准确率）中貌似离散的转变，底层是良好选择的细粒度指标中的连续进展，而这种细粒度进展确实最终促成了网络内部计算一次真实的、在机制上得到刻画的重新组织——一次电路层级的相变，而不只是统计读出的效应。粗与细两幅图景可以同时为真；有趣的问题是细粒度进展指标做了什么工作，而 grokking 研究给出了异常清晰的回答。",[8716,17304,17305],{"id":17305},"归纳头与训练损失相变",[11,17307,17308,17309,17311,17312,17314,17315,17317],{},"Olsson 等（2022，《上下文学习与归纳头》，Anthropic）报告了一个更有资格被称为真实相变的现象：transformer 语言模型的训练损失曲线上出现一个陡峭的隆起，发生在训练早期一个特定的、一致的时点，并在各模型规模与架构上，与「归纳头」的形成同步——归纳头是实现简单复制算法（[A]",[213,17310,423],{}," ... ",[213,17313,391],{}," → 预测 ",[213,17316,423],{},"）的注意力电路，后来被复用于更一般的上下文学习。这个转变在训练损失本身上可见——这是一个连续的、直接被优化的量，不涉及任何取阈值或精确匹配的离散化——这直接击破了幻象解释，因为幻象机制需要一个被人为离散化的读出才能制造出陡峭的外观。此外，损失隆起的时点（通过消融研究）与归纳电路的出现之间存在因果关系，这为转变为何发生提供了一个机制的说明，正如重整化群理论解释了为何居里点在磁化强度上产生陡峭行为一样。这是深度学习目前最接近物理式涌现的东西：一个指标不变的、有机制解释的、可复现的动力系统不连续。",[8716,17319,17321],{"id":17320},"尺度定律损失地形与阈值附近方差的压缩","尺度定律、损失地形与阈值附近方差的压缩",[11,17323,17324],{},"还有一条互补的、更带怀疑色彩的 ML 理论脉络值得摆到台面上，因为它从另一个角度部分支持了幻象阵营。标准的神经网络尺度定律（Kaplan 等，2020；Hoffmann 等，2022）表明，损失本身——可用理论动机最充分的连续指标——随参数、数据与计算量按平滑幂律缩放，跨越多个数量级都没有不连续。如果损失平滑而下游任务准确率不平滑，那么在缺乏内部离散重组证据的情况下，最节俭的解释确实是：任务级不连续是从损失到任务表现的映射（组合式结构、取阈值、指标选择都从这儿进入）所产生的下游假象，而非模型底层能力的不连续。这正是 Schaeffer 等人的论点；在缺乏训练期机制性叙事来回应它时（正如大多数报告的「涌现」基准能力那样，与归纳头或 grokking 形成对照），幻象解释应当被视为默认的、更节俭的假设，符合奥卡姆剃刀式先验。",[30,17326,17327],{"id":17327},"判别框架",[11,17329,17330],{},"把这些线索汇聚起来，我提出三个可操作的检验，用于在任何复杂系统（包括机器学习）中区分真实的动力学涌现与指标诱发的错觉。",[11,17332,17333],{},"指标不变性，方向要正确。转变在更换指标后是否依然存在？该指标的函数形式必须被独立地证明确实追踪了所关注的构念，而不仅仅是连续。正如地震震级案例所示，「连续」并不等于「正确」——线性指标同样容易掩盖真实的乘性结构，正如取阈值指标容易制造虚假结构一样。相关的问题不是「读出是否连续？」而是「读出的函数形式是否与真实生成关系的函数形式相匹配？」通过这一恰当表述的检验（如归纳头的损失隆起、经由多个独立可观测量测得的铁磁临界指数、或对照地震能量理论验证的对数震级），是底层存在真实结构的强证据。未通过（如大多数 BIG-Bench「涌现能力」在对数似然读出下的表现），则是原始指标选择造成假象的强证据。",[11,17335,17336],{},"独立的理论预言对比事后的曲线拟合。转变的位置或存在，是由独立的理论框架（临界现象的重整化群理论；归纳头的电路形成理论）预言的，还是仅仅通过肉眼看出一条恰好弯曲的曲线而事后识别出来的？先验的预言比事后的模式识别具有强得多的认识论效力，后者极易受确认偏误与基准套件中几十个任务上的多重比较问题的影响。",[11,17338,17339],{},"不连续的所在：生成机制对比读出函数。不连续位于生成系统内部状态的过程之中（一个电路形成、一个序参量跨越临界值、一个竞争电路动力学趋于解决），还是位于把内部状态映射到外部报告的函数之中（精确匹配评分器、二值的「解决\u002F未解决」判断、粗糙的采样间隔）？第一类不连续是真实涌现的候选者；第二类不连续在没有反证之前应当按假象处理。",[11,17341,17342],{},"应用这个框架：归纳头的形成与 grokking 的电路竞争通过了全部三项检验，有资格被称为学习动力学中真实的——尽管规模不大——相变。而大语言模型的大多数基准报告的「涌现能力」正如 Schaeffer 等人直接展示的那样，在检验 1 上彻底失败，通常也在检验 2 上失败，因为它们是靠扫描许多任务找不连续形状的曲线识别出来的，而不是由独立的关于能力习得的理论预言出来的。",[30,17344,17345],{"id":17345},"结语",[11,17347,17348],{},"地震震级案例是一个有用的纠偏，可以和上面讨论的机器学习证据并置，因为它防止判别框架坍缩成一个简单启发式——「偏好平滑指标，警惕取阈值指标」。这个启发式是从 Schaeffer 等人那里应该得到的错误教训，因为它会劝告地震学家拒绝（非线性的、对数的）里氏标度，改用它所取代的（线性的、连续的）振幅读数——这恰恰反了，因为对数标度才是追踪物理现实的那个。一般原则与其说是关于线性本身，不如说是关于指标的函数形式是否已从关于底层生成过程的理论中独立推导出来、或依据该理论得到验证，而不是出于惯例或便利被采用。",[11,17350,17351],{},"幻象批判最好的理解方式，不是证明了涌现普遍是虚假的，而是证明了某一特定且流行的取证实践——把精确匹配准确率随模型规模的陡峭跳跃报告为质上崭新能力的证据——在统计上是有混淆的，应当退役或大幅限缩。这是对一处已然漂移的文献的一次真实而有价值的修正，那处文献在有些地方已滑向把基准曲线形状当作窥见模型认知的直接窗口。但机器学习理论本身，经由机制可解释性与训练动力学研究——它们与催生幻象论文的基准尺度文献基本正交——已经独立地提供了符合更严格、指标不变、有机制根基的真实动力学相变标准的例子：grokking 的电路竞争、归纳头的形成及其伴随的训练损失转变。这里的教训不是涌现作为一个笼统命题是真实的还是虚假的，而是这个词一直不加区分地被用来覆盖两类范畴，而这一领域如今第一次拥有了逐案区分二者的经验与理论工具。这种纪律——指标不变性、独立预言、以及对不连续所在之处的关注——才是对不加批判的涌现话语及其笼统否定两者的恰当继承者。",{"title":172,"searchDepth":173,"depth":173,"links":17353},[17354,17355,17356,17362,17367,17368],{"id":16397,"depth":173,"text":16397},{"id":16412,"depth":173,"text":16412},{"id":16968,"depth":173,"text":16968,"children":17357},[17358,17359,17360,17361],{"id":16974,"depth":7967,"text":16975},{"id":17169,"depth":7967,"text":17170},{"id":17269,"depth":7967,"text":17270},{"id":17276,"depth":7967,"text":17277},{"id":17292,"depth":173,"text":17292,"children":17363},[17364,17365,17366],{"id":17298,"depth":7967,"text":17299},{"id":17305,"depth":7967,"text":17305},{"id":17320,"depth":7967,"text":17321},{"id":17327,"depth":173,"text":17327},{"id":17345,"depth":173,"text":17345},{},"\u002Fblog\u002F2026\u002F2026-06-30-about-the-emergency",{"title":16392,"description":172},"blog\u002F2026\u002F2026-06-30-about-the-emergency","Schaeffer 等人表明，LLM 的「涌现能力」往往只是不连续的指标掩盖了平滑的能力增长。但 grokking、归纳头以及居里点式的转变都证明，真实的相变确实存在。",[188,17375,189],"machine-learning","2026-06-30T00:00:00+08:00","J2GOoTqHNfI--QpaxK0Ew-pGsAlVLh4epBhYTlovx48",{"left":17379,"top":17379,"width":17380,"height":17380,"rotate":17379,"vFlip":178,"hFlip":178,"body":17381},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 .238.655c.131.228.325.422.711.808l5.176 5.176c.787.787 1.18 1.18 1.636 1.328c.402.131.835.131 1.237 0c.456-.148.853-.544 1.645-1.336l3.96-3.96c.792-.792 1.187-1.188 1.336-1.644a2 2 0 0 0-.001-1.236c-.148-.457-.543-.853-1.335-1.645l-5.163-5.163c-.39-.39-.584-.584-.813-.716a2 2 0 0 0-.656-.238c-.26-.045-.535-.02-1.084.03l-3.63.33c-.944.086-1.417.129-1.787.335a2 2 0 0 0-.775.775c-.205.368-.248.838-.333 1.773z\"\u002F>\u003Cpath d=\"M9.713 9.713a1 1 0 1 0-1.415-1.414a1 1 0 0 0 1.415 1.414\"\u002F>\u003C\u002Fg>",{"left":17379,"top":17379,"width":17380,"height":17380,"rotate":17379,"vFlip":178,"hFlip":178,"body":17383},"\u003Cpath fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\" d=\"M12 7v5l3 3m6-3a9 9 0 1 1-18 0a9 9 0 0 1 18 0\"\u002F>",1789084286920]