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第 18 期 · 回应 Ⅰ·04「数学是发现，还是发明？」
Limits of Logic: The Gödel Legacy
The Flame of Reason章节 · 点击跳转视频
0:05
引子：从悖论到数学符号的诞生
2:47
《数学原理》与消灭自指的雄心
5:47
形式系统：公理、推理规则与无意义操作
11:23
两个证明与「起伏的红线」
15:26
合式公式 vs 定理：可判定与不可预测
23:03
斐波那契数与柯拉茨数的类比
30:51
哥德尔的洞见：证明本身是数学对象
36:05
哥德尔配数：素因数分解编码符号串
41:48
定理数像柯拉茨数：可证性进入算术
49:34
哥德尔句 G：真而不可证
54:43
不可判定性侵入常规数学
本期讲者
Douglas Hofstadter美国认知科学家、印第安纳大学教授，1979 年出版《哥德尔、埃舍尔、巴赫：集异璧之大成》并获普利策奖，以对自指、类比与意识的研究闻名。
01引子：从悖论到数学符号的诞生
0:05
Crysta asked me when he invited me to speak about girdle and I thought that maybe I was going to speak not only about girdle but about other things that are unknowable but then I decided that for my presentation this morning I would talk just about girdles incompleteness theorem and and try to give an overview of it I'm not going to try to talk very much about philosophical consequences we can talk about that in discussion but just to give an overview of of what girdle did and ah so as Marcus said in his presentation girdle was inspired by paradoxical statements like this sentence is false and um he took that a kind of structure or construction and imported it into into mathematics in a way that nobody could have ever expected I have broken my lecture up into so-called chapters I don't know what that means but it's just for the humor of calling things ah chapters so um mathematics you know about four or five hundred years ago was there was no symbols practically at all it was all done in sentences there were not even
Crysta 邀请我来讲哥德尔的时候问过我，我当时想，也许我不光要讲哥德尔，还要讲讲其他那些不可知的东西，但后来我决定，今天上午的演讲我就只讲哥德尔不完备性定理，试着给大家一个总体的介绍。我不打算过多地谈哲学层面的推论，那些我们可以在讨论环节再聊，这里只是概述一下哥德尔做了什么。啊，正如 Marcus 在他的演讲里说的，哥德尔的灵感来自这类悖论式的句子，比如「这句话是假的」。嗯，他把这种结构或者说构造拿了过来，用一种谁都想不到的方式引入到了数学当中。我把这次讲座分成了所谓的「章节」，我也不知道那意味着什么，纯粹是为了把东西叫作「章」这么个玩笑。嗯，数学，你知道，大概四五百年前基本上一个符号都没有，全都是用句子写出来的，甚至连方程这种东西都没有，也没有用字母来表示变量名
1:35
such a thing as an equation or letters of the alphabet used for variable names or constants those things were invented Descartes I think it was invented the idea of using letters of the alphabet to represent numbers letters the beginning of the alphabet to represent things that were constant letters toward the end of the alphabet to represent things that were variable and there was a Scottish mathematician whose name I've forgotten his first name oh I think was Robert but I don't remember his last name who invented the equals sign four or five hundred years ago it's kind of hard to imagine mathematics without any symbols at all but that tells us that we've come a long way in in the 17th 18th 19th centuries mathematics became extremely formal but people didn't they still didn't a lot of it was done in language as it is today but they didn't know exactly what was rigorous and what wasn't huh lots of things that seemed paradoxical were discovered by people like Euler and many other people ah
或者常数。这些都是后来发明的。我想是笛卡尔发明了用字母来表示数的想法——用字母表开头的字母表示常量，用字母表末尾的字母表示变量。还有一位苏格兰数学家，我忘了他的名字，他的名字，哦，我想是 Robert，但姓我记不得了，是他在四五百年前发明了等号。要想象完全没有符号的数学是挺难的，但这也告诉我们，我们已经走了很长一段路。在 17、18、19 世纪，数学变得极其形式化，但人们其实并不……很多内容仍然是用语言表述的，就像今天一样，但他们并不确切知道什么是严格的、什么不是。呃，很多看起来很悖谬的东西被欧拉之类的人以及许多其他人发现，啊，这些悖论引发了大量
02《数学原理》与消灭自指的雄心
2:47
and these paradoxes gave rise to a lot of concern about the reliability of mathematical reasoning even though it was felt very rigorous and very precise the the idea was they wanted to try to pin it down in sort of a formal way so that for once and for all they would know what was mathematical truth now since this is a short lecture I'm not going to try to summarize any of that there were lots of people who contributed I could name people like a bull George Boole Augustus de Morgan goat Loeb Vega and David Hilbert and many many others epic piano there were many people but the people that I want to talk about in particular this is Alfred North Whitehead and hisses Bertrand Russell a rather humorous photograph of Bertrand Russell and their great opus principia mathematica which came out in three volumes in years 1910 to 1913 was an attempt to to formalize all of what mathematics was and to unite it with logic and artifact to ground mathematical reasoning in logic and the theory of sets and this was a noble
对数学推理可靠性的担忧，尽管数学给人的感觉是非常严格、非常精确的。当时的想法是，他们想设法用某种形式化的方式把它钉死，这样一劳永逸地就能知道什么是数学真理。既然这是一场短讲座，我就不打算去总结这些内容了，有很多人做出过贡献，我可以举出布尔——乔治·布尔、奥古斯都·德·摩根、弗雷格、大卫·希尔伯特，还有许许多多其他人，皮亚诺，人有很多。但我特别想讲的这两位是——这位是阿尔弗雷德·诺思·怀特海，这位是伯特兰·罗素，一张相当滑稽的罗素的照片。他们的巨著《数学原理》在 1910 到 1913 年间分三卷出版，试图把数学的全部内容形式化，并把它和逻辑统一起来，实际上是要把数学推理奠基在逻辑和集合论之上。这是一次高尚的
4:18
attempt and one of the things that lay at the core of their work was the idea of trying to get rid of paradox as I mentioned earlier that was troubling to many people and and to Russell it felt as if the root of all paradoxes was self-reference or sentences that could talk about themselves like this sentence is false or sets that could contain themselves and so he developed what he called the theory of types which was I'm not going to go into it but it was a way of trying to eliminate self reference from coming into a system and it was a very elaborate and and careful Bastion created to prevent self reference from ever coming up from ever arising in such a system and that was something that was very important and for many years ah it was this this is just a page that I took out of it to show the I don't know what you would call it the prick leanness of their notation but for many years uh this their work was taken as being the realization of this dream that of grounding all of mathematics in logic
尝试。而他们工作的核心之一，就是设法消除悖论的想法，正如我前面提到的，那让很多人很困扰。对罗素来说，他觉得一切悖论的根源在于自指，也就是那些能谈论自身的句子，比如「这句话是假的」，或者能包含自身的集合。于是他发展出了他所谓的类型论——我就不展开讲了——那是一种试图阻止自指进入一个系统的办法。那是一道极其精巧、极其小心构筑的堡垒，用来防止自指在这样的系统中出现、产生。一个系统，这是非常重要的一件事，而且很多年来，啊，这就是——这只是我从中截取的一页用来展示那种，我不知道该怎么形容，他们那套记号法的繁琐挑剔，但很多年来呃，他们的这项工作一直被视为实现了这样一个梦想，即把全部数学奠基于逻辑之上
03形式系统：公理、推理规则与无意义操作
5:47
and of formalizing it and of having it be a precise system for once and for all all mathematics was going to come out of this work principia mathematica but in Vienna girdle was caught girdle was in some sense dubious of the of what was of this goal and he felt that there was something wrong about this attempt I'm going to talk about what he did but I wanna I have to first talk a little bit about the nature of mathematical reasoning and how it worked in not only principia mathematica but in any formal system the idea of a formal system is that one has a set of axioms which are written down in a formal notation and uh which are then manipulated in certain formal ways by things called rules of inference I'm certainly not going to detail any of that I hear are some axioms that can be taken for number theory and they're not all of the axioms of number theory but they illustrate things and I'll try to decode this notation for you it's it really doesn't matter very much but the upside-down capital a means for all and
并把它形式化，让它一劳永逸地成为一个精确的系统，所有的数学都将从这部《数学原理》中推导出来。但在维也纳，哥德尔却抓住了——哥德尔在某种意义上对这个目标是持怀疑态度的他觉得这种尝试有什么地方不对劲。我要讲的是他做了什么，但我想——我得先稍微讲一点数学推理的本质，以及它是如何运作的，不仅在《数学原理》中，而是在任何形式系统中。形式系统的想法是，你有一组公理，它们以某种形式记号写下来，呃，然后通过叫做推理规则的东西，以特定的形式方式加以操作。我当然不打算详述这些。这里是一些可以用于数论的公理，它们并不是数论的全部公理，但足以说明问题。我会试着为你们解读这套记号，其实这真的没那么重要，不过那个倒过来的大写 A 表示“对所有”，所以这句话是说：对所有的 a，这个波浪号
7:21
so this says for all a this twiddle means not it's a negation the capital S means successor of so this says for all a it is not the case that the successor of a equals zero what this effectively says is I can translate that I can say there is no number whose successor zero successor being the thing you do when you add one every time so basically this is saying there are there is no such thing as negative one there's no negative numbers this one is sort of defining the property of the number zero it says for all a the sum of a and zero is a zero is an additive identity this one says something about the nature of addition it says for a all a and B if you add a and the successor of B it's the same thing as taking the successor of the sum of a and B and that's a rather trivial statement it would seem but it's enough to allow addition to get off the ground then this is the statement that multiplying by zero of course gives you zero and this is a statement that is sort of getting defines the essence of multiplication
表示“非”，是一个否定；大写 S 表示“……的后继”，所以这句话是说：对所有的 a，并不存在这样的情况：a 的后继等于零。这实际上是在说，我可以把它翻译成：不存在这样一个数，它的后继是零后继就是你每次加一时所做的操作。所以基本上这是在说，不存在这样的的东西，比如负一，是没有负数的。这一条相当于定义了数字零的性质，它说对于所有的 a，a 与零之和就是 a，零是加法单位元。这一条讲的是加法的本质，它说对于所有的 a 和 b，如果你把 a 和 b 的后继相加，那就等于取 a 与 b 之和的后继，这是一个看起来相当平凡的陈述，但它足以让加法运转起来。然后这一条是说任何数乘以零当然得到零，而这一条则是在某种意义上定义了乘法的本质，
8:37
that is if you multiply a times the successor of B it's the same thing as taking a times B in an adding a on and those statements allow you to do a lot I mean there's there other axioms are needed but I don't I don't want to try to go into it in any detail what what I want to stress though is that when I say formal what I mean is I mean today in the era of computers people are familiar with the idea that symbols can be manipulated by machines back in 1910 1913 and the years in Principia Mathematica came out and in the preceding decades and in the following decades there weren't any computers around even though uh what's his name why am i blocking on names these days uh lady Lovelace's Babbage Charles Babbage yeah Babbage had invented the concept of of computers and it sort of developed them in a theoretical way in a marvelous way so the concept of computers was sort of around not electronic of course mechanical but still but but back then the idea of symbols being manipulated formally was was a was new and uh
也就是说，如果你用 a 乘以 b 的后继，那就等于 a 乘以 b 再加上一个 a。这些陈述让你能做很多事情。我是说，还需要别的公理，但我不想试图详细展开。不过我想强调的是，当我说「形式化」的时候，我的意思是，在今天这个计算机的时代，人们已经很熟悉符号可以被机器操作这个想法。而回到 1910 年、1913 年，也就是《数学原理》问世的那些年，以及之前的几十年和之后的几十年，当时并没有计算机，尽管那个……他叫什么名字来着？我这些天怎么老想不起名字。呃，是洛夫莱斯夫人……巴贝奇，查尔斯·巴贝奇，对，巴贝奇当时已经提出了计算机的概念，并且以一种理论化的、了不起的方式把它发展了出来。所以计算机的概念可以说已经存在了，当然不是电子的，是机械的，但不管怎样。可是在那个年代，符号可以被形式化地操作这个想法是全新的，
10:04
different and and so uh it it's very important to realize that the idea of these systems was that you didn't pay any attention to the meaning of the symbols I said that this means up this upside down a means for all and that this means not but in a certain sense you're not supposed to know the meanings of these symbols when you when you perform the operations that that I said rules of inference allow you to deduce theorems from axioms those operations are not supposed to be to take into account any meaning they're supposed to ignore the meanings of the symbols the you look only at the symbols themselves and you say okay this this sequence of symbols has a certain form that allows me to proceed from from it to another sequence of symbols I don't know what they mean okay so that's the the idea of formal systems and that was the the notion that that was being pushed in mathematica so I'm going to show two proofs I'm not going to explain the rules of inference being used but I just want to show you that what a proof looks
是与众不同的。所以，非常重要的一点是要意识到，这些系统的理念在于：你完全不去关注符号的含义。我说过这个符号表示……这个倒过来的 A 表示「对于所有」，而且这意味着，在某种意义上，当你执行操作时，你是不应该知道这些符号的含义的——我说过的那些推理规则允许你从公理推导出定理，而这些操作并不应该考虑任何含义，它们应该忽略符号的含义，你只看符号本身，然后说，好，这个符号序列具有某种形式，这个形式允许我从它推进到另一个符号序列，我不知道它们是什么意思。好，这就是形式系统的思想，也是《数学原理》中所推行的观念。所以我要展示两个证明，我不打算解释所用的推理规则，我只是想让你们看看一个证明长什么样。这是一个非常短的证明，这是
04两个证明与「起伏的红线」
11:23
like this is a very short proof this is a proof of this statement that says one plus one equals two that is the successor of zero added to the successor of zero equals the successor of the successor of zero and so here is here's the proof it begins with it makes use of two axioms and then it uses some rules of inference up which I'm not going to describe but formal processes allow me to pass from this axiom to this specification of it to this to this to this to this the only thing that I want you to notice I put a red line on the right margin and I want you to notice that the line sort of goes back and forth it's these this is a long a couple of long things then they get very short and then they get a little bit it gets a little bit longer toward the end that's only a hint of things to come I just want to I just want you to notice that now the other proof that I'm going to exhibit is that one times one equals one it's a longer proof it's more complex but this is about as short as it could
这个命题的证明，它说的是一加一等于二，也就是零的后继加上零的后继等于零的后继的后继。那么这就是证明。它开头用到了两条公理，然后使用了一些推理规则——我不打算描述这些规则——但形式化的过程允许我从这条公理推到它的这个特例，再到这一步、这一步、这一步、这一步。我唯一想让你们注意的是，我在右边空白处画了一条红线，我想让你们注意到这条线大致是来回起伏的：先是很长的一两行，然后变得很短，接着到快结尾时又稍微变长了一点。这只是后面内容的一点点预示。我只是想、只是想让你们注意到这一点。而我要展示的另一个证明是一乘一等于一。这个证明更长、更复杂，但从我刚才给你们看的那组公理
12:44
get from the set of axioms that I just showed you and this and the rules of inference that I was allowing in this system and I I'm not going to go through it at all it's not important for this purpose I just again want you to look at that at this wavy shape and to see that in order to prove a this is a string of symbols that's very short I had to go through a pathway that involved some pretty long strings what's the longest one it's that one there a lot longer than the result that I was expecting and that's a very crucial fact in thinking about the nature of proofs that is that in some sense we don't know how long a proof how long the strings are going to be that we're going to have to manipulate in order to get in the end a very short string as our result nor do we know how many steps were going to have to use we can't predict that in advance so Bertrand Russell of course was very happy when he was devising he and his colleague Alfred North Whitehead were creating this theory because it was
以及我在这个系统中允许使用的推理规则出发，它差不多已经是最短的了。我完全不打算逐步讲解它，这对我们的目的来说并不重要。我只是同样想让你们看看这个波浪形状，看到为了证明——这是一个非常短的符号串——我不得不走过一条包含了相当长的符号串的路径。最长的那个是哪一行？就是那儿那一行，比我想要得到的结果长得多。而在思考证明的本质时，这是一个非常关键的事实，也就是说，在某种意义上，我们并不知道一个证明会有多长、我们必须操作的符号串会有多长，才能最终得到一个非常短的符号串作为结果；我们也不知道要用多少步，我们无法事先预测这一点。所以罗素当然非常高兴，当他在构思——他和他的同事阿尔弗雷德·诺思·怀特海在创立这套理论的时候——因为很明显，当你写下关于整数的命题时，我是说，即使是复杂的命题，比如一个
14:07
clear that when you wrote down statements about integers I mean even complicated statements like a statement that might express the pheremones Last Theorem that a to the N plus B to the N can never equal C to the N with a B and C being positive integers and n being greater than two that kind of a statement can be written down in this kind of a notation and it clearly doesn't have anything to do with the idea of paradoxical statements like saying this statement is false that as far as you can possibly get from such a thing it's talking about integers and that's all it's talking about and or a statement of the Goldbach conjecture which says any even number greater than four is the sum greater than or equal to four is the sum of two prime numbers which is still unproven but everybody believes it to be true such a statement again has only to do with prime numbers and uh and sums and such things and and how could self referential statements possibly come up in a system that is only talking about integers and their
可能表达费马大定理的命题：a 的 n 次方加 b 的 n 次方永远不可能等于 c 的 n 次方，其中 a、b、c 是正整数，n 大于二——这类命题可以用这种记号写下来，而且它显然与那种自指悖论式的命题毫无关系，比如说“这句话是假的”。它离那种东西要多远有多远，它谈的是整数，它谈的就只是整数而已。或者是哥德巴赫猜想的一个陈述，它说任何大于四的偶数都是——大于或等于四的——都是两个素数之和，这个猜想至今未被证明，但人人都相信它是对的。这样一个命题同样只涉及素数、和之类的东西，那么自指的陈述怎么可能出现在一个只讨论整数及其性质的系统里呢？所以这一点我相信让伯特兰·
05合式公式 vs 定理：可判定与不可预测
15:26
properties so that was on a very something that I'm sure made Bertrand Russell very happy that his system was clearly unable to have self-referential statements now I want to distinguish I've already talked a little bit about theorems I want to talk about something called well-formed formulas because they form a contrast to theorems a well-formed formula is really just a you could say a grammatical sentence something that expresses a statement it could be a true statement or it could be a false statement but it's just a it's just a statement so let's take a look at them here's one that says one plus one equals two that one happens to be true here what is one that says zero equals one that happens to be false but it's they're both equally good well-formed formulas they express thoughts that are either true or false and that's a that's all we need to think about here's one that expresses the commutativity of addition and this one says there is no number a whose square is two and and so forth if I took
罗素非常高兴，因为他的系统显然不可能包含自指的陈述。现在我想区分一下，我已经稍微讲了一点定理，我想讲一个叫做合式公式的东西，因为它们和定理形成对比。合式公式其实就是——你可以说是一个合乎语法的句子，是能表达一个陈述的东西。它可以是真陈述，也可以是假陈述，但它就是一个陈述。所以我们来看看，这里有一个说 1 加 1 等于 2，这个恰好是真的。这里有一个说 0 等于 1，这个恰好是假的。但它们同样都是很好的合式公式，它们表达了或真或假的想法，这就是我们在这里需要考虑的全部。这里有一个表达加法交换律的。这一个说的是不存在任何数 a，它的平方是 2，等等。如果我把
16:47
away the tilde at the beginning it would say there is a number a whose square is two they're both equally good well-formed formulas now what I wanted to tell you about well-formed formulas is that it's very easy to tell if a formula is a formula meaning a string of a string of symbols it's very easy to tell if a formula is well-formed or not I mean for example there's there are some rules basically you just look at its parts and break it up into smaller parts I mean for example here I look at this and I see it begins with the right parenthesis well that tells me I better have a balancing left sorry eyes it begins with the left parenthesis I better have a balancing right parenthesis and it doesn't have to be at the end but it has to be somewhere there has to be a balance of parenthesis and indeed there that balances this balances that doesn't guarantee yet but it's well-formed but it certainly is a part of the test as to whether it's well-formed here is a this thing here stands for if then or implies and if I read this
开头的那个波浪号去掉，它就变成说存在一个数 a，它的平方是 2。它们同样都是很好的合式公式。现在我想告诉你们关于合式公式的一点是，判断一个公式——我指的是一串符号——是不是合式的，这非常容易。我的意思是，比如说，有一些规则，基本上你就是看它的各个部分，把它拆成更小的部分。比如说这里我看这个，我看到它以右括号开头——那这就告诉我最好有一个配对的左……抱歉，是我看错了，它是以左括号开头，那我最好有一个配对的右括号，而且它不一定要在末尾，但一定要在某个地方，括号必须是配平的。而确实，这个和那个配平了，这个和那个配平了。这还不能保证它是合式的，但这肯定是判断它是否合式的检验的一部分。这里的这个东西代表“如果……那么”或者“蕴含”。如果我把这个念出来，它会说：如果 1……如果 0 等于 1，那么 2 加 2 等于 4。这不是一个很
17:54
out loud it would say if 1 if 0 equals 1 then 2 plus 2 equals 4 it's not a very interesting statement kind of a silly sounding statement that's not the point but we can read it and make sense of it and this is well-formed in itself 0 equals 1 is well-formed and this is well-formed this is 2 plus 2 equals 4 and then I've combined a 1 statement on another statement with an if-then and that is also an operation that allowed that create that takes two well-formed formulas and makes another well-formed formula from them and on when I build up formulas into well-formed formulas all I do is I take smaller chunks and I build them into larger ones that's all I do so if I and and so if I want to know if a string of symbols is a well-formed formula all I need to do is look at its parts I break it down into smaller parts and I break them down into smaller parts and I keep on going and it's very predictable test of well formative well for madness I'm contrasting that with theorems it's not so easy to know if something is a
有意思的陈述，听起来有点傻，但那不是重点，重点是我们能读懂它、理解它的意思。而且它本身是合式的：0 等于 1 是合式的，这个也是合式的，这是 2 加 2 等于 4，然后我用一个“如果……那么”把一个陈述和另一个陈述组合起来，而这也是一种被允许的操作，它接受两个合式公式，并从中造出另一个合式公式。而当我把公式搭建成合式公式时，我所做的只是拿一些较小的块，把它们搭成更大的块，我做的就只有这些。所以如果我想知道一串符号是不是合式公式，我需要做的就是看它的各个部分，把它拆成更小的部分，再把那些拆成更小的部分，一直这样下去。这是一个非常可预测的、判断是否合式的检验。我把这个和定理做对比。要知道某个东西是不是定理并不那么容易，这也正是
19:08
theorem and that that's what that red line on the right margin was trying to tell you that on the the definition of theorem is something that has a demonstration something that has a proof and in order to in order to know that you have to make a search among possible proofs but you don't know how long that search is going to take and you don't know how long the formulas are that are going to be involved in in that search so there's a kind of a degree of unpredictability to it whereas again the contrast that I'm drawing is there is no unpredictability to the question of how long is it going to take me to figure out whether this formula is well-formed that's a really trivial question okay that contrast is very very crucial I guess I've already said this but so I don't need to go through it it just says test for well for madness by a recursive break down into simpler parts you break for a formula a possibly well formed possibly not well formed string down into parts and you keep on breaking
右边空白处那条红线想告诉你的：定理的定义是有推演过程的东西、有证明的东西，而为了知道这一点，你必须在可能的证明中做搜索，但你不知道那个搜索要花多长时间，也不知道搜索中会涉及的公式有多长。所以这里面有某种程度的不可预测性。而我要做的对比是，对于“我要花多长时间才能弄清楚这个公式是不是合式的”这个问题，则完全没有不可预测性，那是个非常平凡的问题。好，这个对比非常非常关键。我想我已经说过这个了，所以我不需要再讲一遍。它只是说：通过递归地拆解成更简单的部分来检验是否合式。你把一个可能合式也可能不合式的串拆成若干部分，然后不断把它们拆成各自的部分，直到你最终
20:21
them into their parts until you finally either arrive at you know the it is well-formed or that it isn't it's a very straightforward thing oh so here's here's a string this actually is an expression of the Goldbach conjecture in these formal notation i can sort of tell you what it says here it says for all a there exists a B and a C with the following properties um basically there it's saying this here is it multiplying 2 times a plus 2 so an a can be as low as 0 so a plus 2 is 2 3 4 5 6 and so forth so this is any even number from 4 onwards and it's saying for any a for any this is a this is basically an arbitrary even number and it's saying there exists a B and a C such that a is the sum of these two things and then over here it's saying that B and C are prime numbers I'm not going to decode it for you but basically this is an expression of the Goldbach conjecture as I said still today not known whether it is true or false although everyone believes it's true um so uh is it a well-formed formula just look at it you
要么得出它是合式的，要么得出它不是。这是一件非常直截了当的事。哦，这里这里有一个串，这实际上是哥德巴赫猜想在这套形式记号中的表达。我大致可以告诉你们它说的是什么。它说：对于所有的 a，存在一个 b 和一个 c，具有以下性质。基本上，这里说的是 2 乘以 (a 加 2)，而 a 可以低到 0，所以 a 加 2 就是 2、3、4、5、6，等等，所以这就是从 4 开始的任意偶数。它说的是：对于任意 a，对于任意——这是 a，这基本上是一个任意的偶数——它说存在一个 b 和一个 c，使得它是这两个东西的和，然后这边说的是 b 和 c 是素数。我不打算给你们逐字解码，但基本上这就是哥德巴赫猜想的表达。就像我说的，直到今天仍然不知道它是真是假，尽管大家都相信它是真的。那么，它是不是一个合式公式呢？你就看一看，我的意思是，你大概只要几秒钟
21:45
know I mean it'd take you a few seconds to realize that this is a well-formed formula it's nothing hard about it is it a theorem well try to prove it and hundreds of years of mathematicians working on it have not yielded that ok so you can see that although both properties being well-formed and being a theorem are formal properties that involve the properties of symbols one property is a very simple property that can be detected in a predictable amount of time the other property is one that is is much subtler and more elusive so to sum it up here on the left side it says well form formulas are built up recursively by putting smaller pieces together this idea of this picture is that you are you take small things and you put them together and you've got a big thing the build up grows you can reverse it trivially breaking well well phone formulas in two parts whereas for theorems they are arrived at by you only know something as a theorem if there exists a proof a demonstration of it that is starting with the axioms
就能意识到这是一个合式公式，这没什么难的。那它是不是定理呢？那你试着去证明它吧——几百年来数学家们的努力都没能给出结果。好，所以你可以看到，虽然这两个性质——合式和是定理——都是涉及符号性质的形式性质，但一个性质是非常简单的性质，可以在可预测的时间内判定，而另一个性质则要微妙得多、也难以捉摸得多。所以总结一下，这里左边说：合式公式是通过把较小的片段拼在一起递归地搭建起来的。这个图的意思是，你拿一些小东西把它们拼起来，就得到一个大东西，这个搭建是往上长的，而且你可以轻易地反过来，把合式公式拆成若干部分。而对于定理，它们是这样得到的：你只有在存在一个证明、一个推演过程的情况下才知道某个东西是定理——也就是从公理出发，
06斐波那契数与柯拉茨数的类比
23:03
and following rules of inference combining axioms and combining previously proven theorems into new theorems and that can be a process that involves getting longer and longer strings shorter strings longer shorter you never quite know where you're going and it can take any amount of time and you're not sure so that's a very big contrast between two kinds of structures okay now we come to two types of numbers and this is an analogy that I'm going to use to try to get you to understand I mean many of you already do but those of you who aren't familiar with girdles work to understand the the idea behind his his his construction Fibonacci numbers well you know what they are but all all I'll just mention them the Fibonacci numbers are the numbers that you get when you take sums of previous the previous two numbers 1 plus 2 is 3 2 plus 3 is 5 etc 13 plus 21 is 34 everybody these days is pretty familiar with the Fibonacci numbers now if I were to ask you about if I were to give you a number and say 672 is that a Fibonacci
遵循推理规则，把公理组合起来、把先前已证明的定理组合成新的定理。而这可能是一个会涉及串变得越来越长、又变短、又变长、又变短的过程，你永远说不准自己要走向哪里，它可能花任意长的时间，你没有把握。所以这是两类结构之间非常大的对比。好，现在我们讲两类数，这是一个类比，我打算用它来让你们理解——我是说你们很多人已经理解了，但对于那些不熟悉哥德尔工作的人来说，用它来理解他那个构造背后的思想。斐波那契数——你们都知道它们是什么，不过我还是提一下，斐波那契数就是你把前两个数相加得到的那些数：1 加 2 是 3，2 加 3 是 5，等等，13 加 21 是 34。如今大家对斐波那契数都相当熟悉了。现在，如果我要问你们关于……如果我要给你们一个数字，比如说 672，它是不是斐波那契数？你不会说，天哪，我完全不知道该怎么判断
24:21
number or not you wouldn't say oh my god I have no idea how to figure that out what you would do is you would you you would generate the Fibonacci numbers up to that size what did I say 672 up to that size or close to it and you would say well it looks like it's not because the next one is going to be above its going to be about nine hundred and something nine hundred eighty seven and that's all I've already gone and I'm never going to get smaller the Fibonacci numbers always get bigger and if I passed 692 well or 670 whatever it was that I said then I guess it's not but if I had said instead three hundred and seventy seven and you hit it you'd say yes it is so it's very easy to know if number is or is not a Fibonacci number it's a very simple test similarly you could say that for primes do i you know is 691 prime or not you divide it by a series of numbers and eventually when you hit the square root of 691 and you haven't found a divisor you know you're done and you know it's it's a prime or
你会做的是，把斐波那契数一直生成到那个大小，我刚说的是 672 吧，生成到那个大小或者接近它，然后你就会说，看起来它不是，因为下一个会超过它，下一个大概是九百多少来着，九百八十七，而我已经越过去了，而且我再也不会变小了，斐波那契数总是越来越大的，如果我已经越过了 692，或者 670，反正我刚说的那个数，那我猜它就不是了；但如果我说的是三百七十七，而你正好命中了，你就会说是的，它就是。所以要知道一个数是不是斐波那契数非常容易，这是个非常简单的检验。类似地，你也可以对素数这么说，比如691 是不是素数？你用一系列数去除它，最终当你除到 691 的平方根时，如果你还没找到一个因数，你就知道结束了，你知道它是素数；或者如果你在那个范围内找到了因数，
25:29
if you found a divisor in that period then it's it's it's not a prime and so there are lots of properties of numbers that can be tested very quickly and very predictably in contrast I'm going to talk about I'm not even going to show that you get the idea I'm going to talk about a famous conjecture made by Co lots lotta Co lots in the 1930s he was a mathematician who was specialty with differential equations but he probably is going to be known more for his conjecture than for anything else that he did although he was a good mathematician so I'm going to phrase it in a way that is a little bit different from the way he phrased it but it's equivalent it's the same thing it's just sort of looking at it from the backwards perspective start with N equals one we always start with one and then we have two things we can do we can always jump from n 2 to n we can always double anything we have we can double and if the number has the form 3 K plus 1 such as for example 7 or 22 then we are allowed to jump from this big number
那它就不是素数。所以数的很多性质都可以很快、很可预测地被检验。作为对比，我要讲的是——我甚至不打算展示，你们已经明白意思了——我要讲的是柯拉茨（Collatz）在 1930 年代提出的一个著名猜想。他是一位数学家，专长是微分方程，但他大概会因为这个猜想而出名，而不是因为他做的其他任何事情，尽管他是个不错的数学家。我要用一种和他当初表述稍微不同的方式来陈述它，但它是等价的，是同一回事，只是从反过来的角度去看它。我们从 N 等于 1 开始，我们总是从 1 开始，然后我们有两件事可以做：我们总是可以从 n 跳到 2n，我们总是可以把手上的任何数翻倍。我们可以翻倍；另外，如果这个数具有 3K 加 1 的形式，比如说 7 或者 22，那么我们就可以从这个大数——比如说 22——跳到 7，因为 22 具有 3k 加 1 的形式，它是 3 乘以 7 加 1，所以我可以
26:41
let's say 22 down to 7 because 22 is of the four 3k plus 1 is 3 times 7 plus 1 so I can jump from 22 downwards because it's of the form 3k plus 1 I can't do that with 23 because it's not of that form I can't do it with 24 because it's not of that form but with 25 I can do it I can jump from 25 down to 8 because it's of that form all right now so I have these two rules that allow me to do things to the numbers that I have I can go 1 double it get to double it get 4 double it get 8 double it get 16 now 16 is of the form 3k plus 1 so I can go down now I can go to it under 5 if I wish to I could also just double it but I decided that I would use the other rule and I would go down to 5 then I can go to 10 then I can go to 20 and then I can go to 40 oh look 40 is of the form 3n plus or 3k plus 1 so I can go down now I can go down to 13 and I can go up to 26 and so forth I can take pathways in the set of numbers where what what destinations can I reach here is actually the the shortest pathway to 7 it's very very kind of
从 22 往下跳，因为它是 3k 加 1 的形式。我不能对 23 这么做，因为它不是那个形式；我也不能对 24 这么做，因为它不是那个形式；但 25 可以，我可以从 25 跳到 8，因为它是那个形式。好，现在我有了这两条规则，让我可以对手上的数做操作。我可以从 1 开始，翻倍，得到 2，翻倍得到 4，翻倍得到 8，翻倍得到 16。现在 16 是 3k 加 1 的形式，所以我可以往下走，我可以走到 5，如果我愿意的话。我也可以就继续翻倍，但我决定用另一条规则，我要往下走到 5，然后我可以到 10，再到 20，再到 40。哦你看，40 是 3n 加 1、哦是 3k 加 1 的形式，所以我现在可以往下走，我可以下到 13，然后我可以上到 26，等等。我可以在整数集合里走出各种路径，那么我能到达哪些目的地呢？这其实是到 7 的最短路径，这是一条非常非常有意思的路径：我先变大，然后变
28:00
funny pathway I I get bigger then I get smaller then I get bigger then I get smaller then I got bigger and I get smaller then I get bigger than I got smaller then I get bigger then finally I wind up at 7 it's in a kind of a chaotic pathway and it's the shortest pathway to reach 7 now you can see that if I say a is a given number does I'll call these call-outs numbers is a given number a call that's number is not so trivial as as is determining whether it's a Fibonacci number because I don't know how long the pathway is going to be to get there can you get to 3 so here is checking it out and in fact I can get to 3 but I have to you know if I take all the possible routes starting with one I I start going all sorts of different ways it turns out this is the promising route but we didn't know in advance that this was the route to take all right what about 27 this is a famous case that is the shortest route to get to 27 and it goes all the way up I put it I put this number in green and made it large it
小，然后变大，然后变小，然后变大，再变小，再变大，再变小，再变大，最后我落到了 7。这是一条相当混乱的路径，而它是到达 7 的最短路径。现在你可以看到，如果我问某个给定的数——我把这些叫做柯拉茨数——某个给定的数是不是柯拉茨数，这可就不像判断它是不是斐波那契数那么平凡了，因为我不知道到达它的路径会有多长。你能到达 3 吗？这里是验算过程，事实上我确实能到 3，但你得——你知道，如果我把从 1 开始的所有可能路线都走一遍，我会往各种不同的方向走，结果发现这条是可行的路线，但我们事先并不知道这就是该走的路线。好，那 27 呢？这是个著名的例子，这是到达 27 的最短路线，它一路往上走。我把这个数标成绿色并且放大了，它一路涨到 9232。
29:20
goes all the way up to 9,000 232 it had you can only go that's the shortest route to get to 27 you have to go all the way up to 9,000 in order to come back down to get to 27 that is really quite astonishing so here's a sort of a picture of it I don't know why it's not it's cut off a little bit at the bottom but I guess that's okay can you see the graph or what I wear those think can you see the graph that's that idea is that it goes kind of chaotic jumps then it goes way up to that then it comes down then it goes way up then it comes down it jumps and finally we get to 27 that's that's an interesting picture and um so I said all of this it's not a test into by recursive break down to the simpler smaller cases and it's not guaranteed to succeed we do we don't know if I say what about getting to 542 using this rule well we don't know how long it's going to take or even if we can get there in advance so now Google although he wasn't familiar with Cole outs numbers because they were only invented
你只能这样走，这是到达 27 的最短路线，你必须一路上到九千多，才能再降回来到达 27，这真是相当令人吃惊。这是它的一个图示，我不知道为什么它底部被截掉了一点，不过我想没关系。你们能看到这个图吗？我想应该能看到吧。这个图的意思是，它先是有些混乱的跳动，然后一路冲到那个高点，然后掉下来，然后又冲上去，然后又掉下来，来回跳动，最后我们到达 27。这是一幅很有意思的图。所以我说了这么多，是想说这不是那种可以通过递归地分解成更简单、更小的情形来完成的检验，而且它不保证一定成功。我们不知道，如果我问用这条规则能不能到达 542，我们事先并不知道要花多长时间，甚至不知道能不能到得了。现在，哥德尔虽然并不了解柯拉茨数，因为这些是在他的工作完成之后才提出来的，但他明白这个道理。是的，
30:37
after his work was done but he understood yeah yeah
07哥德尔的洞见：证明本身是数学对象
30:51
the conjecture states that every possible intent positive integer can be reached in this fashion every positive integer can be reached by following this process and it's been tested of all the way out to many billions and it's always been it's always been found but every number works but it's not it's never been proven so um I want to come back to a girdle now gödel had a deep insight I mean in the in the preceding centuries people use numbers to stand for all sorts of things on and I mean everybody knew for example that you could model the physical world I don't mean everybody but people in science knew that you could model the physical world by using equations that that the coordinates of the say the center of gravity of planets could be thought of as numbers and that there were equations that governed their evolution in time and so it was understood that numbers could it could simulate the world but what goodwill saw was that numbers could also integers in fact not real numbers could very easily
这个猜想说的是，每一个正整数都可以用这种方式到达，每一个正整数都可以通过这个过程到达。它已经被验证到了很多很多亿，而且一直都成立，一直都发现每个数都行，但它从来没有被证明过。所以，我想回到哥德尔。哥德尔有一个深刻的洞见。我的意思是，在此前的几个世纪里，人们用数来代表各种各样的东西。我是说，比如大家都知道你可以对物理世界建模——我不是说所有人，而是搞科学的人知道，你可以用方程来给物理世界建模，比如说行星重心的坐标可以被看作数，而且有方程支配它们随时间的演化。所以人们理解数可以模拟这个世界。但哥德尔看到的是，数——事实上是整数，而不是实数——也可以非常容易地模拟一些人们从来没想过是数学
32:20
simulate some things that people had never thought of as being mathematical objects it seemed may be clear after Kepler and Newton that planets moving in orbits were sort of mathematical object and that they could be simulated by real numbers following differential equations but the idea that that formal systems of of the sort that Russell and Whitehead had invented was itself a mathematical system was not something that anybody had thought about they they didn't think of the process of creating a proof as being a mathematical process it was somehow not thought of in terms of in terms of numbers what Google realized was that essentially the manipulation of symbols was a mathematical operation and therefore could be modeled using integers so well here is a picture of a piece of music of piece by Bach and you can imagine encoding this piece into one very large integer the method by which you encode it into a very large integer is not relevant to this discussion but you can imagine that you could encode
对象的东西。在开普勒和牛顿之后，行星沿轨道运动似乎显然算是某种数学对象，而且可以用满足微分方程的实数来模拟。但是，罗素和怀特海所发明的那种形式系统本身也是一个数学系统——这个想法是没有人想过的。他们没有把构造一个证明的过程看作一个数学过程，它不知怎么就没有被从数的角度来思考。哥德尔意识到的是，本质上，符号的操作就是一种数学运算，因此可以用整数来建模。好，这里是一段乐谱的图片，是巴赫的一首曲子，你可以想象把这首曲子编码成一个非常大的整数。至于用什么方法把它编码成一个非常大的整数，跟我们现在讨论的无关，但你可以想象你能把它编码进一个非常大的整数里，然后
33:48
this in a very large integer and then you could ask questions about that integer which would amount to asking questions about the piece of music even though it seemed to be asking questions about the about the integer it would be asking questions about the piece of music also at the same time so for example question is how many notes are in the piece that's a mathematical question which you could answer by looking at this big integer that stood for it what is the key signature of the piece is it by Bach or not that's a little bit harder to imagine a mathematical operation that will take a big integer and tell you whether or not the piece that it stands for was written by BA but you can imagine maybe maybe it's a mathematical question what nationality was the composer that gets a little bit tricky uh so I I just wanted to raise some of the kinds of issues that you could imagine asking such questions some of them seem trivial how many notes are in the piece and some of them seem a little bit harder ah so if
你可以问关于那个整数的问题，而这些问题实际上等同于在问关于这首乐曲的问题。尽管看起来像是在问关于那个整数的问题，但它同时也是在问关于这首乐曲的问题。比如说，问题是：这首曲子里有多少个音符？这是一个数学问题，你可以通过考察代表它的那个大整数来回答。这首曲子的调号是什么？它是不是巴赫写的？这就稍微难想象一点了——要有一种数学运算，接受一个大整数，然后告诉你它所代表的那首曲子是不是巴赫写的。但你可以想象，也许这也是个数学问题。作曲家是哪国人？这就有点棘手了。所以我只是想提出一些你可以想象去问的这类问题。有些看起来很平凡，比如曲子里有多少个音符；有些看起来就难一些。那么，如果
34:55
you did the same thing to us strings of symbols like the Goldbach conjecture is a string of symbols I don't know how many it contained it the way I wrote it out I guess it probably contained about 30 symbols and you could imagine now that you could encode it in some fashion as a large integer and I'll explain how good did that how he encoded him as integers but imagine that you have a big integer that in some sense stands for this string then you could ask how many symbols are in this string and that should be something you could be able to figure out is this a fact from the times table that is is it a mere multiplication like two times two equals four or nine times eight equals 72 is that what this string is expressing or is it expressing something else how about whether it's true or false is this a true statement or a false statement can you apply a mathematical operation to the large integer and and know whether it's a true or false statement does it follow directly from another statement is it a theorem of
你对符号串做同样的事情——比如哥德巴赫猜想就是一串符号，我不知道它包含多少个符号，按我写出来的方式，我猜大概包含 30 个符号——你现在可以想象，你能以某种方式把它编码成一个大整数。我等会儿会解释哥德尔是怎么做的，他是怎么把它们编码成整数的，但先想象你有一个大整数，在某种意义上代表这个符号串。那么你可以问：这个串里有多少个符号？这应该是你能算出来的。这是不是乘法表里的一个事实？也就是说，它是不是仅仅是一个乘法，比如二乘二等于四，或者九乘八等于 72？这个串表达的是这个吗？还是在表达别的什么？那么它是真是假呢？这是一个真命题还是假命题？你能不能对这个大整数施加某种数学运算，就知道它是真命题还是假命题？它是不是直接由另一个命题推出？它是不是《数学原理》中的一个定理？等等
08哥德尔配数：素因数分解编码符号串
36:05
principia mathematica or not and so forth and so forth so on um alright so let's see how did he do oh I have a picture of gödel with an unidentified peasant I think so so good all mapped strings into large integers by taking advantage of the fact that you can on the fact that all all any any integer can be factored in a unique way so here 6 is 2 to the 1 times 3 to the 125 is 5 squared 37 is a prime so it's already factored into its own thing now ah here is a large integer fairly large 1700 we factor it into 2 squared times 5 squared times 17 that is its prime factorization and on the exponents here of the primes that are involved well up to the biggest prime the biggest prime involved is 17 the intermediate primes are 2 3 5 7 11 13 and 17 those are all the primes up to 17 each of them in this factorization has an exponent most of them here are zeros but uh 2 squared 3 2 0 5 squared 7 0 11 etc and so I have a sequence of integers being coded by this large integer sequence of small integers being coded
等等，诸如此类。好，那我们来看看他是怎么做的。哦，我这里有一张哥德尔和一位身份不明的农民的合影，我想是吧。哥德尔把符号串映射成大整数，靠的是利用这样一个事实：任何整数都可以以唯一的方式做因数分解。比如这里 6 是 2 的 1 次方乘以 3 的 1 次方，25 是 5 的平方，37 是素数，所以它本身就已经是分解好的了。啊，这里有个相当大的整数，1700，我们把它分解成 2 的平方乘以 5 的平方乘以 17，这就是它的素因数分解。再看这里涉及到的那些素数的指数，一直到最大的那个素数——涉及的最大素数是 17，中间的素数是 2、3、5、7、11、13 和 17，这些就是17 以内的所有素数。在这个分解里每一个都有一个指数，这里大多数是 0，但是嗯，2 的平方、3 的 0 次、5 的平方、7 的 0 次、11 等等，于是我就有了一列整数被这个大整数编码起来，一列小整数被这个大整数编码起来。这就是
37:45
by this large integer this is what goodwill saw he said a large integer can stand for a sequence of small integers an arbitrarily long sequence of small integers and so he then recast all of what was being done by Whitehead and Russell in principia mathematica as a sequence of operations on large integers instead of seeing it as an operations on strings it was operations on large integers so let's take an example that let's look at just one string so see how girdle could encode it as a large integer the string I've taken is that axiom that was the first axiom for all a it is not the case that the successor of a equals 0 which basically is saying there are no negative numbers ok that's the sequence of how many symbols 1 2 3 4 5 6 7 8 symbols so we'll take the first 8 prime numbers and we'll put them to various powers what I have written here is I've actually put 2 to the upside-down capital a but I'm that that may not look like it's an integer but if you think of upside-down capital a as being a digit in in Martian this is
哥德尔看到的。他说，一个大整数可以代表一列小整数，可以代表任意长的一列小整数。于是他就把怀特海和罗素在《数学原理》中所做的一切，重新表述为对大整数的一系列操作，不再把它看作对符号串的操作，而是对大整数的操作。那我们来看个例子，我们只看一个符号串，看看哥德尔怎么把它编码成一个大整数。我取的这个串就是那条公理，第一条公理：对所有 a，a 的后继等于 0 这件事不成立——基本上就是在说没有负数。好，这个序列有多少个符号呢？1、2、3、4、5、6、7、8，八个符号。所以我们取前 8 个素数，把它们各自取不同的幂。我在这里写的其实是 2 的倒过来的大写 A 次方。你可能觉得那看起来不像个整数，但如果你把倒写的大写A 想成火星文里的一个数字，这就只是火星记数法，倒写的大写 A 在火星记数法里也许就是 7，
39:14
just Martian notation and upside-down capital a might be 7 in Martian notation and lowercase a might be 9 and each of these symbols is really just an integer in Martian notation so so when I do do this I I'm just creating a large integer that's all I'm doing and if I want to know what formula I was I was doing it for all I need to do is take that large energy factor it and then look at the sequence of exponents and I can spell out the exact string that I was encoding in the large integer so goodwill had this way of encoding strings into large integers now um it does creates a kind of an ambiguity is this large integer is it a number well yes is it a formula well in a sense yes it's a formula that is a formula to in compute mathematically yes it is it's it's a formula in the sense that it stands for this formula and if we are dealing with it in the context of these formal operations that allow us to pass from one string to another or from one number to another according to certain rules then it can be thought of
小写 a 也许是 9，这些符号中的每一个其实都只是火星记数法里的一个整数。所以当我这么做的时候，我只是在造一个大整数，仅此而已。而如果我想知道我当初编的是哪个公式，我只需要把那个大整数拿来分解，然后看指数的序列，我就能一字不差地拼出我编码进这个大整数里的那个串。所以哥德尔有了这套把符号串编码成大整数的办法。现在，嗯，这确实造成了某种含混：这个大整数，它是一个数吗？是的。它是一个公式吗？在某种意义上是的，它是一个公式。它是可以用来做数学计算的公式吗？是的，它是。它是公式，是在这个意义上说的：它代表着那个公式。而如果我们是在这些形式操作的语境下处理它——那些让我们按照某些规则从一个串过渡到另一个串、或者从一个数过渡到另一个数的操作——那么它就可以被看作
40:34
as a formula it's just a formula written in a funny notation this actually is on the integer that stands for one of the axioms in principia mathematica it's a big integer because the axiom is a complicated statement and and it requires a lot of symbols and using there are many ways of doing what is called girdle numbering this is using a girdles actual system I mean there are many other ways of doing it which would make it simpler but I just thought it's amusing to see that this one large integer is standing for one of the axioms of principia mathematica and you could figure out that it was by just factoring it it might take you a while to factor it but you once you've factored it you could you could you could exactly map out symbol by symbol exactly what this axiom says however it's another thing to figure out if it's a theorem well if you recognize that it's an axiom you're done you know but anyway does a given integer not that one in particular but just to give an integer stand for a well-formed formula
一个公式，只不过是用一种奇怪的记法写出来的公式。这个其实就是代表《数学原理》中一条公理的整数。它是个很大的整数，因为这条公理是个复杂的陈述，需要很多符号。而且做所谓的哥德尔编号有很多种方式，这里用的是哥德尔本人的系统。我是说，还有很多别的做法会更简单，但我只是觉得有意思：看到这一个大整数代表着《数学原理》中的一条公理，而你只要把它分解就能弄清楚它是哪一条。分解它可能要花你一阵子，但一旦你分解完了，你就可以一个符号一个符号地精确还原出这条公理说的是什么。然而，要弄清楚它是不是一个定理，那就是另一回事了。当然，如果你认出它是一条公理，那你就完事了。不过总之，一个给定的整数——不是特指刚才那个，而是随便给一个整数——它是不是代表一个合式公式？这是个容易的问题，非常容易的问题，因为你只要，你知道，只要
09定理数像柯拉茨数：可证性进入算术
41:48
that's easy question very easy question because you just you know you can just look at the exponents and then you look at that that's formula and you know how to break it down you can do this with it but you can bypass the translation into this into the the notational system of principia mathematica you can just operate on the integer itself you can have rules that will tell you how to break that integer down into smaller pieces and to this yet smaller pieces and they will tell you whether that integer stands for a well-formed formula or not it's just as mechanical and isomorphic to the process of breaking down a string and asking whether it is a well formed string so it's it's it's a the same question just asked in a different notation does the an integer stand a given integer stand for a theorem well there's no obvious way to know that because proofs are like zigzagging Collatz pathways and this is sort of one of the crucial slides of this lecture I did was I took the proof that this is a proof of a beginning with
看指数，然后你看那个公式，你知道怎么把它拆解开，你可以这么做。但你也可以绕过翻译回《数学原理》记号系统这一步，你可以直接对整数本身做操作。你可以有一些规则，告诉你怎么把那个整数拆成更小的部分，再拆成更小的部分，这些规则会告诉你那个整数是不是代表一个合式公式，还是不是。这跟拆解一个符号串、问它是不是一个合式串的过程一样机械，而且是同构的。所以这是同一个问题，只是用不同的记法来问。那么一个整数是不是代表一个定理呢？这就没有明显的办法可以知道了，因为证明就像曲折跳跃的柯拉茨路径。这算是这次讲座最关键的幻灯片之一。我做的是，我取了一个证明，这是一个从
43:04
the axioms let's see yeah this is a proof of the commutativity of addition it says at the end for all D and C C plus D equals D plus C so that's the commutativity of addition and it begins with axioms and uses rules of inference it's a fairly long proof I mean it's actually very short but for our purposes it seems fairly long in it and it requires getting some fairly long statements and and there as well and it's a bumpy root in order to derive the commutativity of addition but in the end we do derive it but notice that we don't know how long this proof is going to be in advance and we don't know how long those strings involved the intermediate stages are going to be there are two completely unknown things and and notice how similar this feels to the Collatz numbers and if we think of these things not as strings now but as integers because remember we can encode as integers were basically doing the very same thing we're taking some smallish integers mathematically manipulating them in various ways
公理开始的证明。我们看看，对，这是加法交换律的证明，它在最后说：对所有 D 和 C，C加 D 等于 D 加 C，这就是加法交换律。它从公理开始，使用推理规则，这是个相当长的证明。我是说，其实它非常短，但就我们的目的而言它显得相当长，而且它需要写出一些相当长的语句，还有那边也是，而且这是一条崎岖的路径，才能推导出加法的交换律，但最终我们确实推导出来了。不过注意，我们事先并不知道这个证明会有多长，也不知道中间各个阶段涉及的那些字符串会有多长，这里有两个完全未知的东西。而且注意，这感觉和考拉兹数列多么相似。如果我们把这些东西现在不看作字符串，而看作整数——因为记住，我们可以把它们编码成整数——我们基本上是在做完全相同的事情：我们拿一些不太大的整数，用各种方式对它们做数学操作，得到一些非常大的整数，类似于 9232 那样，然后我们继续
44:22
getting some very big integers the analog of 9000 232 then we keep you lations we get some other big integers and finally at the end we wind up with the thing that we were hoping for so this is what goodwill saw that the question about whether something is a theorem is a potentially a very complex question unlike whether it's a well-formed formula which is not a very complex question ok so well formed formula numbers the numbers that stand through gödel numbering before well-formed formulas they're like Fibonacci numbers you can test very easily theorem numbers numbers that stand for theorems in the pen Capilla Mathematica are not so simple they're more like call-outs numbers so a way of putting it is well-formed formulas there comes the numbers that correspond to them to make such numbers you just increase the pathway just it goes from smaller integers to larger ones and that's all there is it's just a monotonically increasing pathway this kind of thing theorem numbers is not a monotonically
操作，得到另外一些大整数，最后在结尾处我们得到了我们期望的那个东西。所以这就是哥德尔看到的：某个东西是不是定理，这个问题有可能是一个非常复杂的问题，不像“它是不是合式公式”那样——那不是一个很复杂的问题。好，那么合式公式的数，也就是通过哥德尔配数来代表合式公式的那些数，它们就像斐波那契数一样，你可以很容易地检验。而定理数，也就是代表《数学原理》中定理的那些数，就没那么简单了，它们更像考拉兹数。所以换个说法就是：合式公式，以及对应于它们的那些数，要造出这样的数，你只需要一路递增，它就是从较小的整数走向较大的整数，就这么回事，它只是一条单调递增的路径，就是这么一类东西。而定理数就不是一条单调递增的路径，它会以混沌的方式来回
45:35
increasing pathway it jumps back and forth in chaotic ways alright um well formed formula numbers are isomorphic to well-formed formulas through girdle's mapping and the pathways that get you to well formed formula numbers are monotonic and thus monotonous nothing very interesting theorem numbers are isomorphic to theorems ah but again since theorem the pathways that get you to theorems are not predictable the same holds for theorem numbers so that's just reiterating what I told you earlier um theorem number of pathways follow unpredictable zigzags ok and so girdle was able to show that the notion of a theorem number was as mathematically precise as the notion of let's say prime number in other words a prime number is a is clearly a mathematical notion right it just involves asking you know when I factored this number do I get anything in between 1 and itself as a factor and that's a very mathematical notion what gödel showed was that being a theorem number is also a perfectly mathematical notion
跳动。好，合式公式数通过哥德尔的映射与合式公式是同构的，而通往合式公式数的路径是单调的，因此也是单调乏味的，没什么特别有意思的。而定理数与定理是同构的，但同样，既然通往定理的路径是不可预测的，那么同样的情况也适用于定理数。所以这只是在重复我前面告诉你们的东西。定理数的路径遵循的是不可预测的曲折。好，于是哥德尔能够证明，“定理数”这个概念在数学上和比方说“素数”这个概念一样精确。换句话说，素数显然是一个数学概念，对吧？它无非就是问：当我把这个数做因式分解时，我能不能得到介于 1 和它自身之间的因子？这是一个非常数学化的概念。而哥德尔所证明的是，“成为一个定理数”同样是一个完全数学化的概念。也就是说，你首先
47:05
that is you begin with that you begin with by stating that a certain set of numbers are theorem numbers which ones are those the ones that stand for the axioms the axioms are all by definition theorem numbers that the axiom numbers are all theorem numbers and so you're just given those out right here's you're told these five or these ten whatever however many axioms there are these numbers are all theorem numbers and then you're given some formal rules that allow you to manipulate these numbers and from those numbers you can create new numbers that are big sometimes bigger and sometimes smaller but you will get new numbers those are called the theorem numbers and those numbers are just as mathematical as the ko lots numbers just as mathematical as the prime numbers just as mathematical as the Fibonacci numbers so what girdle was doing was bringing a discussion of proved ability into the domain of discourse that principia mathematica dealt with what is principia mathematica dealing with its dealing
声明某一组数是定理数——那是哪些呢？就是代表公理的那些数。按定义，所有公理都是定理数，也就是公理的哥德尔数都是定理数，所以这些是直接给你的：有人告诉你，不管有五条还是十条，或者不管有多少条公理，这些数全都是定理数。然后再给你一些形式规则，允许你对这些数进行操作，从这些数出发你可以造出新的数，有时更大，有时更小，但你会得到新的数，那些就叫做定理数。而这些数和考拉兹数一样是数学的，和素数一样是数学的，和斐波那契数一样是数学的。所以哥德尔所做的，是把关于“可证明性”的讨论带入了《数学原理》所处理的论域之中。《数学原理》处理的是什么？它处理的
48:18
with statements about integers does this is this integer prime or not is this integer eco lots number or not is this integer a Fibonacci number is this integer a well-formed formula number all of these are mathematical questions is this integer a theorem number another mathematical question and yet in code it is saying is this particular theorem is this particular string a theorem or not if you ask if is this a theorem number you're saying effectively is this string a theorem of print capilla Mathematica or not and so despite the fact that Russell and Whitehead were under the illusion that self reference was not possible in their system girdle had snuck it in by the Trojan horse of gödel numbering because once you have numbers that are acting isomorphic Li to the structures in principia mathematica then you're able to talk about all the structures that bring capilla Mathematica numerically and so he was I'm not going to explain how he did this exactly but he was not only able to import the notion of theorem number into
是关于整数的陈述：这个整数是不是素数？这个整数是不是考拉兹数？这个整数是不是斐波那契数？这个整数是不是合式公式数？所有这些都是数学问题。而“这个整数是不是定理数”，又是另一个数学问题；然而与此同时它其实在说：这个特定的字符串是不是一个定理？如果你问“这是不是一个定理数”，你实际上是在问：这个字符串是不是《数学原理》的一个定理。所以，尽管罗素和怀特海抱着一种幻觉，以为他们的系统里不可能出现自指，哥德尔却通过哥德尔配数这个特洛伊木马把它偷偷带了进来。因为一旦你有了与《数学原理》中的结构同构地运作的那些数，你就能够用数值的方式谈论《数学原理》中所有的结构了。所以他——我不打算精确解释他是怎么做到的——但他不仅能够把“定理数”这个概念引入《数学原理》，引入这个系统本身，而且他还能写
10哥德尔句 G：真而不可证
49:34
rinkeby mathematic into the system itself but also he was able to write down a very special string that expressed this following statement it was a particular particular integer G it said the integer G is not the number of a theorem is not the number is not a theorem number where a theorem number remember is just a mathematical notion be like saying the integer 641 is not prime which incidentally is false it is prime but that's not irrelevant it's just a mathematical statement the integer G is not the number of a theorem is not a theorem number however it turned out that the integer G was actually the exact number that correspond 'add to this string that expressed this statement so the G this is girdle's formula says this thing in the notation of principia mathematica it says G this lower case number this lower case letter G is not a theorem number so it says this number has a certain number theoretical property and it also says that by doing that it it says that the formula that it stands for is not a theorem but
出一个非常特殊的字符串，它表达了下面这个陈述：有一个特定的整数 G，它说“整数 G 不是某个定理的编号”，也就是说它不是一个定理数。而记住，“定理数”只是一个数学概念，就好比说“整数 641 不是素数”——顺便说一句这是假的，它是素数，但这并不相关——它只是一个数学陈述。整数 G 不是某个定理的编号，即不是定理数。然而结果发现，整数 G恰恰正是那个表达了这个陈述的字符串所对应的编号。所以这个 G，这就是哥德尔的公式，它用《数学原理》的记号说了这件事：它说小写字母 G 这个数不是定理数。所以它说这个数具有某种数论性质，同时它这么说的时候，也就是在说：它所代表的那个公式不是定理。可偏偏这个公式恰恰就是那个陈述本身。所以他实际上
50:56
it happens that this formula is actually that exact statement so he actually created a statement that said about itself that it was not a theorem and that's what Marcus put up he said this theory this statement is not it's not provable so um I guess I just added this little thing these are the same assertion just understood in two different ways a certain number has a certain number theoretical property or a certain formula is not a theorem of principia mathematica identical statements I guess all I I see I've already done in effect well I didn't really because this is my last slide well let me just conclude then by mentioning the consequences are not the but some of the consequences of this construction because the construction creates a statement that speaks of itself which is the the thing that inspired such terror in Bertrand Russell's mind self reference to his mind was the source of all paradox and he had striven so hard to make sure that it was not a part of pink hippy Mathematica and girdle had actually
造出了一个陈述，这个陈述关于它自己说：它不是定理。这就是马库斯放上去的那句话，他说：这个陈述是不可证明的。所以，我想我只是加了这么一小点：这两者是同一个断言，只是用两种不同的方式来理解：某个数具有某种数论性质，或者某个公式不是《数学原理》的定理——它们是同一个陈述。我想我实际上已经把该讲的都讲了……其实也没有，因为这是我最后一张幻灯片。那我就以提一下这个构造的一些后果来做个结尾吧——不是全部后果，而是其中一些。因为这个构造造出了一个谈论自身的陈述，而这正是让伯特兰·罗素心生恐惧的东西。在他心目中，自指是一切悖论的根源，他曾如此努力地要确保自指不会成为《数学原理》的一部分，而哥德尔却通过哥德尔配数这个特洛伊木马，硬是把它带进了《数学原理》内部。
52:17
brought it inside principia mathematica by this Trojan horse of gödel numbering now the statement that girdle created said of itself that it was not provable well it if it was uh let's let's think about that supposing that it was provable then it would be a false statement because it states that it is not provable but if it were provable then a false statement would be provable oh that seems horrible we don't want there to be any false statements provable that would be that would be the destruction of principle Mathematica nobody believed that or where any false statements will be provable so we have to go back and read Shakhtar assumption which was that it was provable so it's not a provable thing but that's what it says that's exactly what it says it says that it's not provable so it turns out that what it says is true and B precisely for that reason it is not provable it's like here's a pair of real paradox it's the exact reason that it is not provable is that it is true that's the exact
现在，哥德尔造出的这个陈述说自己是不可证明的。那么，如果它是可证的……我们来想想这个：假设它是可证的，那么它就是一个假陈述，因为它声称自己不可证；但如果它是可证的，那就意味着一个假陈述是可证的。哦，这看起来很可怕，我们不希望有任何假陈述是可证的，那将会是《数学原理》的毁灭，没有人相信会有任何假陈述可证。所以我们必须回过头去否定我们的假设，也就是“它是可证的”这个假设。所以它是不可证的——可这正是它自己说的，这正是它所说的：它说自己不可证。于是结果就是，它所说的是真的，而且恰恰因为这个原因，它是不可证的。这就像是一个真正的悖论：它之所以不可证，恰恰是因为它是真的。这与数学家们所喜欢的完全相反，他们希望可证性和真理是同义的。而这个字符串之所以
53:34
opposite of what mathematicians like they think that they want prove ability and truth to be synonymous the exact reason that this string is not provable is that it is true very very strange and this construction that showed that there was a statement that was true but not provable in principia mathematica didn't apply only to principia mathematica it applied to any system of this structure it didn't depend on any of the details of principia mathematica zwey of the particular axioms that he used or the particular rules of inference that it used all it depended on was the idea that there was some axioms and some rules of inference and um and it showed that in any system that was rich enough to get the truths of number theory in it there were statements that were true and not provable and this would in you could then throw this in as a new axiom but that wouldn't hurt that wouldn't help anything because that's a new system that has a new set of axioms and you just apply the the godel prop procedure
不可证，恰恰就是因为它是真的。非常非常奇怪。而这个构造表明存在一个在《数学原理》中为真但不可证的陈述，它并不只适用于《数学原理》，它适用于任何具有这种结构的系统。它不依赖于《数学原理》的任何细节，不依赖于他所用的那些特定公理，也不依赖于它所用的特定推理规则。它所依赖的全部，只是这样一个想法：存在一些公理和一些推理规则。而且它表明，在任何丰富到足以容纳数论真理的系统中，都存在为真但不可证的陈述。而你可以把这个陈述当作一条新公理加进去，但那没有用，那不会有任何帮助，因为那是一个新的系统，它有一组新的公理，于是你只要把哥德尔的那套程序应用到这个稍微大一点的系统上，
11不可判定性侵入常规数学
54:43
to that system which is a little larger and you've got another string which is which is not provable but true and you can keep on going forever and so there are an infinite number of holes and they can't be systematically all filled I guess the last thing I want to say is that it has been shown in the last few decades that although these number theoretical statements I mean girdle's statement says certain number has certain kind of property but it it seems like a very artificial kind of a property that is its it's constructed using this mapping that Goodall created and and and so forth and it might seem like it's such a very very esoteric kind of formula that even though that such things exist that they are very remote from anything in a mathematician would ever actually talk about but it's been shown that that's not the case at all and in fact John Conway and colleagues were able to show that there are if you consider the Co Lots problem as one of a family of problems that are all of that sort that
你就又得到另一个字符串，同样是不可证但为真的。你可以这样一直进行下去，所以存在无穷多个漏洞，而它们不可能被系统地全部填补。我想最后我还想说的是，最近几十年人们已经证明：虽然这些数论陈述——我是说哥德尔的那个陈述说某个数具有某种性质——但它看起来像是一种非常人为的性质，也就是说它是用哥德尔造出的那个映射之类的东西构造出来的，因此它可能看起来是一种非常非常玄奥的公式，以至于即便这样的东西存在，它们也离数学家真正会谈论的任何东西非常遥远。但人们已经证明情况完全不是这样。事实上，约翰·康威和同事们能够证明：如果你把考拉兹问题看作一族同类问题中的一个——这类问题基本上涉及把整数乘或者除，就这样上上
56:03
basically involve multiplying or dividing integers and just going up and down in that way I'm not going to try to define the exact family but you can imagine it I mean the Cole that's the problem is often called the three n plus one problem and you can imagine a five n plus one problem or a five n plus two problem or something like that if you just consider all of those problems together it's been shown that there are problems of that sort that are undecidable for good alien reasons that is that the the girdle the girdle construction includes it you can you can prove that the girdle construction is equivalent to a Collatz problem of of some sort and so in fact that's a very very normal number theoretical kind of question it's not it's not something that seems obscure or strange or non mathematical or artificial it's a very natural mathematical kind of question it's also been shown that the Daiya fantine equations are there exist unsolvable Daiya undecidable diet die of fantine equations and those are
下下——我不打算给出这一族的确切定义，但你们可以想象。我是说，考拉兹问题常被称为 3n+1 问题，你可以想象一个 5n+1 问题，或者一个 5n+2问题之类的。如果你把所有这些问题放在一起考虑，人们已经证明，存在这一类问题是不可判定的，而且是出于哥德尔式的原因。也就是说，哥德尔的构造包含在其中；你可以证明哥德尔构造等价于某种形式的考拉兹问题。所以事实上，那是一个非常常规的数论问题，它并不是什么看起来晦涩、古怪、非数学或人为的东西，它是一种非常自然的数学问题。人们还证明了，丢番图方程中存在不可解的、不可判定的丢番图方程。那类方程基本上是这样的形式：一些
57:20
equations of the basically the form some integers to some powers equal a set of sum of integers two powers equals another set of integers to sum of integers to powers it's it's a very very simple basic algebraic kind of equation and those two there exist ones that are undecidable so it turns out that as a result of as a consequence of girdles work that there are questions a very standard mathematical form that have been shown to be undecidable and so the undecidability doesn't just apply to very very weird remote regions but to very standard kinds of constructions and that's a really an astounding somewhat scary thought and with that I think I'll conclude
整数的若干次幂之和等于另一组整数的若干次幂之和。这是一种非常非常简单基础的代数方程，而其中确实存在不可判定的。所以结果就是，作为哥德尔工作的一个后果，存在一些具有非常标准数学形式的问题，已经被证明是不可判定的。因此不可判定性并不只适用于非常非常古怪偏远的角落，而是适用于非常标准的那类构造。这真是一个惊人的、甚至有点吓人的想法。就讲到这里，我想我该做个总结了。
视频总结 · 一句话概括与核心要点
一句话概括
Douglas Hofstadter 概述了哥德尔不完备定理的核心机制：通过"哥德尔编码"把形式系统的字符串和证明变成整数运算，让本应被《数学原理》严密排除的自指偷渡进系统，从而构造出一个"真却不可证"的命题，并说明这种不可判定性已延伸到 Collatz 类问题和丢番图方程等极其普通的数学问题。
核心要点
- 形式化运动源于对悖论的恐惧：数学在四五百年前几乎没有符号（等号、用字母表示常量/变量都是后来才发明），17–19 世纪逐步形式化后，Euler 等人发现的种种悖论令人担忧数学推理的可靠性。Whitehead 与 Russell 在 1910–1913 年出版三卷《数学原理》，试图把全部数学奠基于逻辑与集合论；Russell 认为一切悖论的根源是自指，专门设计"类型论"作为防止自指的堡垒。
- 形式系统的本质是"无视意义地操作符号"：系统由公理和推理规则组成，推导时只看符号串的形状，不看含义。讲者展示了数论公理（如 ∀a: ~(Sa=0) 表示不存在负数、a+0=a、a+Sb=S(a+b) 等）以及 1+1=2 和 1×1=1 的形式证明，后者虽结论极短，中间却要经过很长的字符串。
- 良构公式与定理是两类根本不同的性质：判断一个字符串是否良构（语法正确）只需递归拆分成更小部分，耗时可预测；判断是否是定理则要在证明空间中搜索，既不知道要多少步，也不知道中间公式会有多长——证明中字符串长度"忽长忽短、曲折不定"。哥德巴赫猜想的形式表达几秒就能确认是良构公式，但数百年无人能证明它是定理。
- 斐波那契数 vs Collatz 数的类比：判断 672 是否为斐波那契数或 691 是否为素数都有单调、可预测的算法；而 Collatz 过程（从 1 出发，可加倍，或在数为 3k+1 时跳到 k）到达一个数的路径是混沌的——到达 27 的最短路径要先冲到 9232 再降下来。Collatz 猜想断言所有正整数都可达，已验证到数十亿却从未被证明。
- 哥德尔的关键洞见是"符号操作本身就是数学运算"：人们早知道实数可以模拟行星轨道，但没人想到形式系统本身也是可以用整数模拟的数学对象。就像可以把一首巴赫乐曲编码成一个大整数，再用数学问题问"有多少音符"，字符串也能编码成整数，然后问"它是良构公式吗""它是定理吗"。
- 哥德尔编码利用素因数分解的唯一性：把 n 个符号的字符串映射为 2^s₁·3^s₂·5^s₃·…（每个符号视作一个"火星数字"），对大整数做因数分解、读出指数序列即可还原字符串。例如 8 个符号的公理 ∀a:~(Sa=0) 用前 8 个素数编码；《数学原理》的一条公理对应一个巨大的整数。
- "定理数"和"素数"一样是纯粹的数论概念：良构公式数像斐波那契数一样易于检测；定理数则像 Collatz 数——从公理数出发，按形式规则生成新数，路径忽大忽小、不可预测。但它的定义完全是数学的，所以"是否为定理"的可证性问题被引入了《数学原理》自己的讨论域（整数性质）。
- 自指经由"特洛伊木马"混入：Russell 以为只谈整数的系统不可能出现"这句话是假的"这类语句，但哥德尔构造出一条特殊公式 G，它说"整数 G 不是定理数"，而 G 恰好就是这条公式本身的编码——于是这条语句在说"我不可证"。
- 真与可证由此分道扬镳：若 G 可证，则它为假，意味着系统能证明假命题（灾难）；所以 G 不可证——而这正是它所断言的，故 G 为真。"它不可证的确切原因恰恰是它为真"，与数学家希望真理和可证等同的愿望截然相反。把 G 加为新公理也无济于事，对扩大后的系统再做一次哥德尔构造又得到新的漏洞，无穷无尽，无法系统性填补。
- 定理适用于任何足够丰富的系统，且不可判定性并不"偏僻"：论证不依赖《数学原理》的具体公理或规则，只依赖"有公理和推理规则"的结构，凡能表达数论真理的系统皆然。近几十年 John Conway 等人证明 Collatz 型问题（3n+1、5n+1 等广义家族）中存在哥德尔式不可判定的实例；也存在不可判定的丢番图方程——都是极为标准、自然的数学问题。
结论与值得注意的细节
- 结论：可证性不等于真理；任何包含算术的一致形式系统都有真却不可证的命题，且这种"漏洞"无法通过增补公理彻底修复。更令人不安的是，不可判定性不只藏在人造的怪异命题里，还出现在 Collatz 类问题和丢番图方程这样的日常数论问题中——讲者称之为"惊人且有些可怕的想法"。
- 讲者刻意回避哲学后果，只聚焦于机制本身，留待讨论环节。
- 讲者反复强调"良构 vs 定理"和"斐波那契 vs Collatz"两组对比，用红色边线标记证明中字符串长度的起伏，作为理解哥德尔构造的直觉支架。
- 历史细节：Babbage 已在理论上构想计算机，但 1910 年代"机械操作符号"的观念仍属新奇；Collatz 猜想提出于 1930 年代，晚于哥德尔 1931 年的工作，讲者的类比是事后视角。
- 讲者顺带提到 641 是素数、哥德巴赫猜想"人人相信为真却未证明"等小例子，用以说明"定理数"陈述在形式上与普通数论断言并无二致。
核心句型 · 9
1. It's kind of hard to imagine X without Y at all, but that tells us that …
“It's kind of hard to imagine mathematics without any symbols at all but that tells us that we've come a long way”
先用「难以想象」制造反差，再用 that tells us 引出结论。适合演讲中从历史事实过渡到评价。仿写：It's hard to imagine coding without version control, but that tells us how far tooling has come.
2. What I want to stress, though, is that …
“What I want to stress though is that when I say formal what I mean is …”
口语演讲中标记「重点来了」的信号句，though 放句中缓和语气。可接 when I say X, what I mean is 来澄清术语定义。
3. The only thing I want you to notice is …
“The only thing that I want you to notice I put a red line on the right margin and I want you to notice that …”
引导听众聚焦一个细节而忽略其他复杂内容，演示图表时极常用。仿写：The only thing I want you to notice is how the curve flattens after week three.
4. X is as far as you can possibly get from Y
“That as far as you can possibly get from such a thing”
强调两者毫无关联的夸张表达，比 completely different 更生动。仿写：A spreadsheet is as far as you can possibly get from poetry.
5. It's not so trivial as (doing) …, because I don't know how …
“Is not so trivial as is determining whether it's a Fibonacci number because I don't know how long the pathway is going to be”
通过对比说明难度来源，because 后给出「不知道……」的具体不确定性。适合解释为何某任务比表面看起来难。
6. Despite the fact that A were under the illusion that …, B had snuck it in by …
“Despite the fact that Russell and Whitehead were under the illusion that self reference was not possible in their system girdle had snuck it in by the Trojan horse of gödel numbering”
「尽管某人以为……，某人却以……方式偷偷做到」——叙述反转的高级句式。under the illusion 和 snuck in 都带轻微戏剧色彩。
7. Supposing that it was X, then it would be Y … so we have to go back and reject our assumption
“Supposing that it was provable then it would be a false statement … so we have to go back and reject our assumption which was that it was provable”
反证法的口语化表述：supposing 引出假设，then it would be 推出后果，最后 reject our assumption 收尾。写论证段落时可直接套用。
8. The exact reason that X is that Y
“The exact reason that this string is not provable is that it is true”
用 the exact reason … is that 强调唯一且出人意料的原因，比 because 更有力。适合陈述反直觉结论。
9. X doesn't just apply to …, but to …
“The undecidability doesn't just apply to very weird remote regions but to very standard kinds of constructions”
not just … but … 的变体，用于扩展某结论的适用范围，从边缘推向核心。仿写：This bias doesn't just apply to edge cases but to everyday decisions.
生词精讲 · 96 · 按出现顺序
unknowable
/ʌnˈnoʊəbl/
adj.
0:05
不可知的
paradoxical
/ˌpærəˈdɑːksɪkl/
adj.
0:05
悖论的，自相矛盾的
rigorous
/ˈrɪɡərəs/
adj.
1:35
严格的，严密的（数学语境指论证严谨）
gave rise to
phr.
2:47
引起，导致
pin it down
phr.
2:47
把……明确固定下来，钉死
for once and for all
phr.
2:47
一劳永逸地
opus
/ˈoʊpəs/
n.
2:47
（大部头的）著作，作品
ground
/ɡraʊnd/
v.
2:47
使建立在……基础之上（ground A in B）
self-reference
/ˌself ˈrefrəns/
n.
4:18
自指
elaborate
/ɪˈlæbərət/
adj.
4:18
精心设计的，复杂精巧的
Bastion
/ˈbæstʃən/
n.
4:18
堡垒；捍卫某事物的据点
dubious
/ˈduːbiəs/
adj.
5:47
怀疑的，半信半疑的（be dubious of/about）
axioms
/ˈæksiəmz/
n.
5:47
公理
rules of inference
phr.
5:47
推理规则
decode
/ˌdiːˈkoʊd/
v.
5:47
解读，译解
negation
/nɪˈɡeɪʃn/
n.
7:21
否定（逻辑）
successor
/səkˈsesər/
n.
7:21
后继（数）；继任者
additive identity
phr.
7:21
加法单位元
get off the ground
phr.
7:21
起步，开始运转
manipulated
/məˈnɪpjuleɪtɪd/
v.
8:37
操作，处理（符号、数据）
blocking on
phr.
8:37
（口语）一时想不起（名字等）
deduce
/dɪˈduːs/
v.
10:04
推导，演绎
take into account
phr.
10:04
考虑到，把……计入
specification
/ˌspesɪfɪˈkeɪʃn/
n.
11:23
（逻辑）特例化，将全称量词实例化
a hint of things to come
phr.
11:23
后事的预兆，伏笔
pathway
/ˈpæθweɪ/
n.
12:44
路径，途径
crucial
/ˈkruːʃl/
adj.
12:44
至关重要的
conjecture
/kənˈdʒektʃər/
n.
14:07
猜想（数学中未证明的命题）
as far as you can possibly get from
phr.
14:07
与……相去甚远，离……要多远有多远
well-formed formulas
phr.
15:26
合式公式（语法正确的逻辑公式）
commutativity
/kəˌmjuːtəˈtɪvəti/
n.
15:26
交换律
tilde
/ˈtɪldə/
n.
16:47
波浪号 ~
balancing
/ˈbælənsɪŋ/
adj.
16:47
配对的，使之平衡的（括号）
implies
/ɪmˈplaɪz/
v.
16:47
蕴含（逻辑）
chunks
/tʃʌŋks/
n.
17:54
块，片段
demonstration
/ˌdemənˈstreɪʃn/
n.
19:08
（数学）推演，证明
unpredictability
/ˌʌnprɪˌdɪktəˈbɪləti/
n.
19:08
不可预测性
recursive
/rɪˈkɜːrsɪv/
adj.
19:08
递归的
straightforward
/ˌstreɪtˈfɔːrwərd/
adj.
20:21
直截了当的，简单明了的
yielded
/ˈjiːldɪd/
v.
21:45
产出，给出（结果）
subtler
/ˈsʌtlər/
adj.
21:45
更微妙的
elusive
/iˈluːsɪv/
adj.
21:45
难以捉摸的
trivially
/ˈtrɪviəli/
adv.
21:45
轻而易举地，平凡地
analogy
/əˈnælədʒi/
n.
23:03
类比
differential equations
phr.
25:29
微分方程
phrase
/freɪz/
v.
25:29
表述，措辞
equivalent
/ɪˈkwɪvələnt/
adj.
25:29
等价的
chaotic
/keɪˈɑːtɪk/
adj.
28:00
混乱的，混沌的
promising
/ˈprɑːmɪsɪŋ/
adj.
28:00
有希望的，可行的
wind up at
phr.
28:00
最终到达/落在
astonishing
/əˈstɑːnɪʃɪŋ/
adj.
29:20
令人震惊的
center of gravity
phr.
30:51
重心
govern
/ˈɡʌvərn/
v.
30:51
支配，决定（规律）
encoding
/ɪnˈkoʊdɪŋ/
v./n.
32:20
编码
amount to
phr.
33:48
等同于，相当于
key signature
phr.
33:48
（音乐）调号
tricky
/ˈtrɪki/
adj.
33:48
棘手的
in some fashion
phr.
34:55
以某种方式
follow directly from
phr.
34:55
直接由……推出
factored
/ˈfæktərd/
v.
36:05
做因数分解
prime factorization
phr.
36:05
素因数分解
exponents
/ɪkˈspoʊnənts/
n.
36:05
指数，幂次
arbitrarily
/ˌɑːrbəˈtrerəli/
adv.
37:45
任意地
recast
/ˌriːˈkæst/
v.
37:45
重新表述，改写
spell out
phr.
39:14
逐字拼出；详细说明
ambiguity
/ˌæmbɪˈɡjuːəti/
n.
39:14
歧义，含混
amusing
/əˈmjuːzɪŋ/
adj.
40:34
有趣的，好玩的
map out
phr.
40:34
详细勾画，逐一还原
bypass
/ˈbaɪpæs/
v.
41:48
绕过
isomorphic
/ˌaɪsəˈmɔːrfɪk/
adj.
41:48
同构的
zigzagging
/ˈzɪɡzæɡɪŋ/
adj.
41:48
曲折的，之字形的
bumpy
/ˈbʌmpi/
adj.
43:04
颠簸的，崎岖的
derive
/dɪˈraɪv/
v.
43:04
推导出
analog
/ˈænəlɔːɡ/
n.
44:22
类似物，对应物
monotonically
/ˌmɑːnəˈtɑːnɪkli/
adv.
44:22
单调地（数学：始终递增或递减）
monotonous
/məˈnɑːtənəs/
adj.
45:35
单调乏味的（此处与 monotonic 双关）
reiterating
/riˈɪtəreɪtɪŋ/
v.
45:35
重申
by definition
phr.
47:05
按定义
domain of discourse
phr.
47:05
论域（逻辑术语）
under the illusion that
phr.
48:18
误以为，抱有……的幻觉
snuck
/snʌk/
v.
48:18
偷偷带入（sneak 的过去式）
Trojan horse
phr.
48:18
特洛伊木马（暗中带入的东西）
incidentally
/ˌɪnsɪˈdentli/
adv.
49:34
顺便说一句
number theoretical
phr.
49:34
数论的
assertion
/əˈsɜːrʃn/
n.
50:56
断言
striven
/ˈstrɪvn/
v.
50:56
努力，奋力（strive 的过去分词）
provable
/ˈpruːvəbl/
adj.
52:17
可证明的
supposing
/səˈpoʊzɪŋ/
conj.
52:17
假设
synonymous
/sɪˈnɑːnɪməs/
adj.
53:34
同义的，等同的
rich enough
phr.
53:34
（系统）表达力足够强
systematically
/ˌsɪstəˈmætɪkli/
adv.
54:43
系统地
esoteric
/ˌesəˈterɪk/
adj.
54:43
玄奥的，晦涩难懂的
remote from
phr.
54:43
与……相去甚远
undecidable
/ˌʌndɪˈsaɪdəbl/
adj.
56:03
不可判定的
obscure
/əbˈskjʊr/
adj.
56:03
晦涩的，冷僻的
astounding
/əˈstaʊndɪŋ/
adj.
57:20
令人震惊的
理解自测 · 11 题 · 是真懂了，还是以为自己懂
1. 讲者说数学在四五百年前「没有符号」，具体提到了哪两项发明和发明者？
讲者提到两项：一是用字母表示数——用字母表开头的字母（a、b、c）表示常量、末尾的字母（x、y、z）表示变量，他归功于笛卡尔；二是等号「=」，由一位苏格兰数学家在四五百年前发明，讲者只记得他名叫 Robert（实际是 Robert Recorde，1557 年）。这段出现在开场的历史铺垫部分，用意是说明「形式化」并非自古就有，而是一步步建立起来的，为后文《数学原理》的极端形式化做背景。
2. 《数学原理》是谁写的、何时出版，其核心目标是什么？罗素认为悖论的根源在哪？
《数学原理》由阿尔弗雷德·诺思·怀特海与伯特兰·罗素合著，1910–1913 年分三卷出版。目标是把全部数学形式化，并把数学推理奠基于逻辑与集合论之上，一劳永逸地确定什么是数学真理。罗素认为一切悖论的根源是「自指」——能谈论自身的句子（如「这句话是假的」）或包含自身的集合，因此他发展了类型论，作为一道精巧的堡垒阻止自指进入系统。这是第 2–3 段的内容，也是全讲最终被「特洛伊木马」颠覆的前提。
3. 讲者展示了到达 27 的柯拉茨路径，其中提到了什么关键数字？这个例子想说明什么？
到达 27 的最短路径必须一路升到 9232 才能回落到 27，讲者把这个数标成绿色放大展示。它想说明：与斐波那契数不同，柯拉茨路径的中间值可以远大于目标值，且路径长度事先不可知，所以「某数是否为柯拉茨数」无法通过有界搜索判定。这个例子出现在第 21–22 段，后文（第 33–34 段）把它与证明过程中「中间公式远长于结论」直接对应，9232 被称为长中间公式的「类似物」。
4. 什么是「合式公式」？讲者用哪个例子说明合式与真假无关？
合式公式（well-formed formula）就是语法正确、能表达一个陈述的符号串，它可以为真也可以为假。讲者举了「1+1=2」（合式且真）和「0=1」（合式但假），说两者「同样都是很好的合式公式」；还举了「若 0=1 则 2+2=4」，说它听起来傻但语法完全合式。这在第 11–13 段。区分「合式」与「真/可证」是全讲论证骨架的第一步：合式性是可以通过递归拆解在可预测时间内判定的语法性质。
5. 讲者为何反复强调形式系统「不看符号的意义」？这一点对哥德尔的构造有何作用？
讲者强调（第 7 段）推理规则只根据符号序列的「形状」操作，不考虑含义，这是形式系统的定义特征。这一点是后文的关键前提：既然操作纯粹是语法性的、机械的，那它就可以被翻译成对整数的算术运算——哥德尔配数正是把符号换成素数指数，把推理规则换成算术变换。如果推理依赖「理解意义」，就无法在数论内部模拟它。所以「无意义操作」看似是形式主义的局限，实际正是让「证明本身成为数学对象」得以成立的条件。
6. 请重述讲者的核心类比：合式公式数、定理数分别对应哪种数？对应的依据是什么？
合式公式数对应斐波那契数（或素数），定理数对应柯拉茨数。依据是「判定所需的搜索是否有可预知边界」：判断某数是否斐波那契数，只需生成到超过它即可，因为序列单调递增；合式性检查同样是从大到小递归拆解、必然终止。而判断某数是否柯拉茨数，路径可能先飞到 9232 再回落，长度不可预知；同样，一个公式是否为定理取决于是否存在证明，而证明的步数与中间公式长度都无法事先知道。这个类比在第 34–35 段被明确点出。
7. 哥德尔配数如何工作？为什么「唯一因数分解」对它至关重要？
每个符号被指派一个整数（讲者戏称「火星数字」），符号串的第 k 个符号作为第 k 个素数的指数，把这些幂相乘得到一个大整数；如 8 个符号就用 2,3,5,…,19 的幂相乘（第 29 段）。要还原原串，只需对该大整数做素因数分解，读出指数序列。这依赖算术基本定理：每个整数的素因数分解是唯一的（第 28 段），否则同一个数可能解码成不同字符串，编码就不可逆。由此，「关于公式的问题」都成了「关于整数的问题」，比如「这串有几个符号」变成「分解后有几个非零指数」。
8. 请完整复述哥德尔句 G 为何「真而不可证」的推理链。
G 的内容是「编号为 G 的公式不是定理数」，而 G 恰好就是这个公式自身的编号，因此 G 在说「我不可证」（第 38 段）。推理（第 40 段）：假设 G 可证，那么 G 所说的「我不可证」为假，于是系统证明了一个假命题；没人接受《数学原理》能证明假命题，所以必须否定假设——G 不可证。但「G 不可证」正是 G 所断言的，所以 G 为真。结论：G 为真，且正因为它为真才不可证。讲者说这与数学家的期望恰好相反——他们希望「可证」与「真」是同义词。
9. 为什么把 G 加为新公理不能修复不完备性？讲者对此的结论是什么？
讲者在第 41–42 段说明：哥德尔的构造不依赖《数学原理》的具体公理或推理规则，只依赖「存在一组公理和一组推理规则」这一结构。因此把 G 加进去得到的新系统同样满足这一结构，对它再做一次哥德尔构造，就得到新的 G'，同样真而不可证；如此可以无限进行下去。结论是「漏洞有无穷多个，而且无法被系统地全部填补」——任何足以表达数论真理的形式系统必然不完备，这不是《数学原理》的缺陷，而是这类系统的普遍性质。
10. 有人反驳：「哥德尔句是人为构造的怪物，与真正的数学无关。」讲者会如何回应？
讲者在第 42–44 段正面回应了这一反驳。他承认哥德尔句表面上看是通过映射构造出来的、「玄奥」的性质，但指出近几十年的结果表明并非如此：John Conway 及同事证明，把 3n+1 问题推广为一族类似问题（5n+1、5n+2 等），其中存在因哥德尔式理由而不可判定的问题，且哥德尔构造可被证明等价于某种柯拉茨问题；此外还存在不可判定的丢番图方程——形式极其基本的代数方程。因此不可判定性「不只出现在偏远角落，而是出现在非常标准的构造中」，他称之为「惊人甚至有点吓人的想法」。
11. 把这一论点迁移到 AI：如果一个 AI 系统是一台按固定规则运行的形式机器，哥德尔定理是否意味着它「必然有人类才能看到的盲点」？按讲者的框架该如何谨慎判断？
按讲者框架，哥德尔定理只断言：任何一致、足以表达算术的固定公理系统，都存在它无法证明的真命题——但它并没有说「人类」能证明这些命题。讲者本人在第 40 段展示的推理也是在系统之外、假设系统一致的前提下进行的，这一「元层」推理同样可以被另一个更大的形式系统执行（第 41 段：加公理后再做哥德尔构造）。因此从定理推出「人脑必然超越机器」需要额外假设：人类心智是一致的、且不等价于任何形式系统，而这两点定理并未提供（这正是 Lucas–Penrose 论证的争议所在）。讲者作为《GEB》作者，更倾向于把自指看作心智与机器共有的结构，而非划分二者的界线。所以谨慎的结论是：定理限制的是「固定系统的完备性」，对「人 vs 机」的比较保持开放。
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