第488期 – 无穷、颠覆数学的悖论、哥德尔不完备性与多重宇宙 – 乔尔·戴维·哈姆金斯

所以，如果你把这些点看作点，它们实际上就相当于火车车厢。

So those dots, if you think about them as dots, are really the same as the train cars.

如果你考虑整数格点中每一列，它都是可数无穷的。

If you think about each column of in the in that integer lattice, it's accountable infinity.

这就像是一个车厢，然后旁边是下一个车厢，再旁边是下一列，也就是下一个车厢。

It's like one train car, and then there's the next train car next to it, and then the next column next to that, the next train car.

但如果我们以这种网格方式来思考，我就可以想象一条蜿蜒的路径，沿着对角线穿过这些格点，上下穿梭。

And so but if we think about it in this grid manner, then I can imagine a kind of winding path winding through these grid points, like up and down the diagonals Mhmm.

来回蜿蜒。

Winding back and forth.

我从角落的点开始，然后向左上方走，再向右下方走，再向左上方走，再向右下方走，依此类推。

So I start at the corner point, and then I go down up into the left, and then down into the right, up into the left, down into the right, and so on.

这样，我这条路径会经过网格上的每一个点。

In such a way that I'm gonna hit every grid point in on this path.

因此，这给了我一种为这些点分配房间号的方法。

So this gives me a way of assigning room numbers to the points Mhmm.

因为每个网格点都会是这条路径上的第n个点，某个n。

Because every every grid point is gonna be the nth point on that path for some n.

这就建立了网格点与自然数本身之间的对应关系。

And that that gives a correspondence between the grid points and the natural numbers themselves.

所以这是一种不同的视角。

So it's a kind of different picture.

我的意思是，之前我们用的是3到5的s次方这种过于算术化的方式来理解它。

I mean, before we use this three to this c five times five to the s, which is a kind of, you know, overly arithmetic way to think about it.

但当你有可数个可数集合时，有一种更直接的方式理解它仍然是可数无穷，因为你只需把它们依次列出来。

But there's a kind of direct, you know, way to understand that it's still accountable infinity when you have countably many countable sets, because you can just start putting them on this list.

只要给每个无限集合机会向列表中添加一个人，你就能把所有集合中的所有人容纳进一个列表中。

And as long as you give each of the infinite collections a chance to add one more person to the list, then you're gonna accommodate everyone in any of the sets in one list.

是的。

Yeah.

这是一种非常直观的思考方式。

It's a really nice visual way to think about it.

你只需在网格上 zigzag 地走，以确保每个人都被包含。

You just zigzag your way across the grid to make sure everybody's included.

这为你提供了一种包含所有人的算法。

That gives you kind of an algorithm for including everybody.

那么你能谈谈不可数无穷大吗？

So can you speak to the uncountable infinities?

整数和实数是什么？康托尔能够找到的界限是什么？

So what are the integers and the real numbers, and what is the line that Cantor was able to find?

在做那之前，我想再补充一步，那就是有理数。

So maybe there's there's one more step I wanna insert before doing that, which is the rational numbers.

我们之前讨论了自然数的配对。

So we did we did pairs of natural numbers.

嗯。

Mhmm.

对吧？

Right?

这基本上就是火车车厢。

That that's the train car, basically.

但也许思考一下有理数——分数的集合——会更有启发性。

But maybe it's a little bit informative to think about the rational the fractions, the set of fractions or rational numbers.

因为很多人可能认为，有理数的无穷大更大，因为有理数在任意两个分数之间都是稠密排列的。

Because a lot of people maybe have an expectation that maybe this is a bigger infinity because the rational numbers are are densely ordered between any two fractions.

你总能找到另一个分数。

You can find another fraction.

对吧？

Right?

两个分数的平均值仍然是另一个分数。

The average of two fractions is another fraction.

所以有时候，人们觉得有理数和离散排列的整数性质不同。

And so so sometimes people it seems to be a different character than than the integers, which are discretely ordered.

对吧？

Right?

从任何一个整数出发，都有一个下一个和上一个，依此类推，但在有理数中却不是这样。

From every any integer, there's a next one and a previous one and so on, but that's not true in the rational numbers.

然而，有理数仍然只是可数无穷大。

And yet, the rational numbers are also still only accountable infinity.

要理解这一点，实际上就像希尔伯特的列车一样，因为每个分数都由两个整数构成：分子和分母。

And the the way to see that is actually, it's just exactly the same as Hilbert's train again, because every fraction consists of two integers, the numerator and the denominator.

所以，如果我告诉你两个自然数，你就知道我指的是哪个分数。

And so if I tell you two natural numbers, then you know what fraction I'm talking about.

我的意思是，还要考虑符号问题。

I mean, plus the sign issue.

我的意思是，它是正数还是负数。

I mean, if it's positive or negative.

但如果你只考虑正分数，那么你知道，它们的形式是 p/q，其中 q 不为零。

But if you just think about the positive fractions, then, you know, you have the numbers of the form p over q, where q is not zero.

因此，你仍然可以使用 3 的 p 次方乘以 5 的 q 次方。

So you can still do three to the p times five to the q.

同样的思路也适用于有理数。

The same idea works with the rational numbers.

所以这仍然是一个可数集。

So this is still a countable set.

你可能会想，每一个集合都是可数的，因为只有一个无穷大。

And you might think, well, every every set is gonna be countable because there's only one infinity.

我的意思是，也许你持这种观点，但这是不对的。

I mean, that's a kind of perspective maybe that you're adopting, but it's not true.

而康托尔的伟大成就正是证明了实数集不是可数无穷。

And that's the profound achievement that Cantor made is proving that the set of real numbers is not a countable infinity.

它是一个严格更大的无穷大，因此存在不止一种无穷的概念，不止一种无穷的大小。

It's a strictly larger infinity, and therefore there are there's more than one concept of infinity, more than one size of infinity.

那么我们来谈谈实数。

So let's talk about the real numbers.

什么是实数？

What are the real numbers?

为什么要把无穷大拆开？

Why did they break infinity?

可数无穷大。

The countable infinity.

对。

Right.

在Perplexity上查了一下，实数包括数轴上所有能表示的数，涵盖有理数和无理数。

Looking it up on perplexity, real numbers include all the numbers that can be represented on the number line encompassing both rational and irrational numbers.

我们已经讨论过有理数。

We've spoken about the rational numbers.

顺便说一下，有理数的定义是能表示为两个整数之比的数。

And the rational numbers, by the way, by definition, the numbers that can be represented as a fraction of two integers.

没错。

That's right.

对于实数，我们还有代数数。

So with the real numbers, we have the algebraic numbers.

当然，我们有所有的有理数。

We have, of course, all the rational numbers.

整数和有理数都属于实数系统，但此外我们还有代数数，比如根号二、五的立方根等等。

The integers and the rationals are all part of the real number system, but then also we have the algebraic numbers like the square root of two, or the cube root of five, and so on.

能够满足整数系数代数方程的数被称为代数数。

Numbers that solve an algebraic equation over the integers, those are known as algebraic numbers.

长期以来，人们一直不清楚代数数是否涵盖了所有的实数，或者是否存在所谓的超越数。

It was an open question for a long time whether that was all of the real numbers, or whether there would exist numbers that are the transcendental numbers.

超越数是指那些不是代数数的实数。

The transcendental numbers are real numbers that are not algebraic.

我们甚至都不去讨论超现实数了，关于这个话题我们有一篇很棒的博客文章。

And we won't even go to the surreal numbers about which have a wonderful blog post.

我们稍后再谈这个。

We'll talk about that a little bit later.

哦，太好了。

Oh, great.

因此，是刘维尔首次证明了超越数的存在，并给出一个具体的数，即如今被称为刘维尔常数的超越数。

So it was Louisville who first proved that there are transcendental numbers, and he exhibited a very specific number that's now known as the Louisville constant, which is a transcendental number.

康托尔也著名地证明了存在无数个超越数。

Cantor also famously proved that there are many many transcendental numbers.

事实上，根据他对实数不可数性的论证，可以得出存在不可数多个超越数。

In fact, it follows from his argument on the uncountability of the real numbers that there are uncountably many transcendental numbers.

因此，大多数实数都是超越数。

So most real numbers are transcendental.

同样地，谈到困惑性，超越数是实数或复数。

And again, going to perplexity, transcendental numbers are real or complex numbers.

它们不是任何具有整数或有理系数的非零多项式的根。

They're not the root of any nonzero polynomial with integer or rational coefficients.

这意味着它们不能表示为具有整数系数的代数方程的解，从而与代数数区分开来。

This means they cannot be expressed as solutions to algebraic equations with integer coefficients, setting them apart from algebraic numbers.

没错。

That's right.

所以一些

So some of

著名的超越数包括圆周率π，也就是3.14159265等等。

the famous transcendental numbers would include the number pi, you know, the the 3.14159265 and so on.

所以这是一个超越数。

So that's a transcendental number.

还有欧拉常数e，比如e的x次方，指数函数。

Also, Euler's constant, the e, like e to the x, the exponential function.

所以你可以说，数学中最迷人的某些数字都是超越数。

So you could say that some of the sexiest numbers in mathematics are all transcendental numbers.

绝对如此。

Absolutely.

确实如此。

That's true.

没错。

Yeah.

嗯。

Yeah.

不过，我知道根号二也很

Although, you know, I know the square root of two is pretty

根号二。

Square root two.

好的。

Alright.

所以这取决于情况。

So it depends.

让我们不要认为美只能在所有不同类型的集合中找到。

Let's not beauty can be found in in all the different kinds of sets.

如果你持一种简化的观点，那么零和一看起来也不错，而且它们肯定不是。

And if you have a kind of simplicity attitude, then, you know, zero and one are looking pretty good too, so and they're definitely not.

抱歉扯远了，但你最喜欢的数字是什么？

Sorry to take that tangent, but what is your favorite number?

你有吗？

Do you have one?

天啊。

Oh, gosh.

你知道是零吗？

You know Is it zero?

你知道吗？有一个证明表明每个数都是有趣的。

Did you know there's a proof that every number is interesting?

你可以证明，因为

You can prove it because

是的。

Yeah.

那个证明是什么样的？

What's that proof look like?

是的。

Yeah.

你该怎么开始呢？

How do you even begin?

我要向你证明每一个自然数都是有趣的。

I'm gonna prove to you that every natural number is interesting.

好的。

Okay.

你知道吗？

You know?

我的意思是，零很有趣，因为它是加法单位元。

I mean, zero's interesting because, you know, it's the additive identity.

对吧？

Right?

这挺有趣的。

That's pretty interesting.

而一是乘法单位元。

And one is the multiplicative identity.

所以当你用它乘以任何其他数字时，结果就是那个数字本身。

So when you multiply it by any other number, you just get that number back.

对吧？

Right?

而二是第一个质数，非常有趣。

And two is, you know, the the first prime number that's super interesting.

对吧？

Right?

嗯。

And Mhmm.

好的。

Okay.

所以你可以这样一直继续下去，给出具体的原因，但我想从一般原理上证明每个数字都是有趣的。

So one can go on this way and and give specific reasons, but I wanna prove as a general principle that every number is interesting.

这就是证明。

And and this is the proof.

假设相反，存在一些无聊的数字。

Suppose toward contradiction that there were some boring numbers.

好的。

Okay.

但如果有一个无趣的数字，是的。

But if if there was an uninteresting number Yes.

那么必然存在一个最小的无趣数字。

Then there would have to be a smallest uninteresting number.

嗯。

Mhmm.

是的。

Yes.

但这构成了矛盾，因为最小的无趣数字本身就是一个非常有趣的性质。

But that's a contradiction because the smallest uninteresting number is a super interesting property to have.

因此，这样的数字不可能存在。这很好。

Therefore, there cannot be That's good.

不可能存在任何无聊的数字。

There cannot be any boring numbers.

我得试着找出这个证明中的漏洞，因为‘有趣’这个词里预设了很多东西。

I'm gonna have to try to find a hole in that proof because there's a lot of baked in in the word interesting.

但没错，这很美妙，真是太美了。

But, yeah, that's a beauty that's beautiful.

这并没有涉及超越数，也没有涉及实数。

That doesn't say anything about the transcendental numbers, about the real numbers.

你只是用自然数来进行证明。

You use proof from just for natural numbers.

好的。

Okay.

那我们回到康托尔的论证吗？

So should we get back to Cantor's argument?

当然。

Or Sure.

你巧妙地避开了这个问题。

You you've masterfully avoided the question.

你基本上是说，我喜欢所有的数字。

Well, you basically said, I love all numbers.

是的。

Yeah.

基本上是这样。

Basically.

是的。

Yeah.

那就是我我的

That's what I my

回到康托尔的论证。

Back to Cantor's argument.

我们开始吧。

Let's go.

好的。

Okay.

所以康托尔想证明实数的无穷大与自然数的无穷大不同，且严格更大。

So Cantor wants to prove that the infinity of the real numbers is different and strictly larger than the infinity of the natural numbers.

自然数是从零开始，逐次加一的数，即零、一、二、三，依此类推。

So the natural numbers are the numbers that start with zero and and add one successively, so zero, one, two, three, and so on.

而实数，正如我们所说，是来自数轴上的所有数，包括所有整数、有理数、代数数、超越数以及这些数的全体。

And the real numbers, as we said, are the the numbers that come from the number line, including all the integers and the rationals and the algebraic numbers and the transcendental numbers and all of those numbers altogether.

显然，由于自然数包含在实数中，我们知道实数至少与自然数一样大。

Now, obviously, since the natural numbers are included in the real numbers, we know that the real numbers are at least as large as the natural numbers.

因此，我们要证明的主张是：实数的无穷大严格更大。

And so the claim that we wanna prove is that it's strictly larger.

假设它并非严格更大，那么它们的大小就相同。

So suppose that it wasn't strictly larger, so then they would have the same size.

但要大小相同，根据定义，意味着它们之间存在一一对应关系。

But to have the same size, remember, means, by definition, that there's a one to one correspondence between them.

所以我们假设实数可以与自然数建立一一对应关系。

So we suppose that the real numbers can be put into one to one correspondence with the natural numbers.

因此，对于每一个自然数 n，都对应一个实数。

So therefore, for every natural number n, we have a real number.

我们称它为 r_n。

Let's call it r sub n.

r_n 是列表中的第 n 个实数。

R sub n is the nth real number on the list.

基本上，我们的假设允许我们将实数视为被排列成一个列表：r₁、r₂，依此类推。

Basically, our assumption allows us to think of the real numbers as having been placed on a list, r one, r two, and so on.

好的。

Okay.

现在我要定义一个数 z，它的整数部分为零，然后加上小数点，接着我将逐位指定这个数 z 的小数位。

And now now I'm gonna define the number z, and it's gonna be the integer part is gonna be a zero, and then I'm gonna put a decimal place, and then I'm gonna start specifying the digits of this number z.

d₁、d₂、d₃，依此类推。

D one, d two, d three, and so on.

我要确保z的小数点后第n位数字与列表中第n个数的小数点后第n位数字不同。

And what I'm gonna make sure is that the nth digit after the decimal point of z is different from the nth digit of the nth number on the list.

嗯。

Mhmm.

明白吗？

Okay?

所以，要确定z的第n位数字，我要去看列表中的第n个数，也就是r_n，然后观察它小数点后第n位的数字。

So so to specify the nth digit of z, I go to the nth number on the list, r sub n, and I look at its nth digit after the decimal point.

不管那个数字是多少，我都会确保我的数字与它不同。

And whatever that digit is, I make sure that my digit is different from it.

明白吗？

Okay?

我还想再做一点改进，那就是我要确保从不使用数字0或9。

And then I wanna do something a little bit more, and that is I'm gonna make it different in a way that I 'm never using the digits zero or nine.

我只使用其他数字，从不使用0或9。

I'm just always using the the other digits and not zero or nine.

这样做的背后有一定的技术原因。

There's a certain technical reason to do that.

但主要的是，我让 z 的第 n 位数字与第 n 个数的第 n 位数字不同。

But the main thing is that I make the digits of z different in the nth place from the nth digit of the nth number.

如果你把原始列表中的数字 r1、r2、r3 等画出来，每个数占一行，然后考虑第 n 个数的第 n 位数字，就会形成一条向右下方延伸的对角线。

If you had if you had drawn out the numbers on the original list, r one, r two, r three, and so on, and you made it, you know they were each filling a whole row, and you thought about the nth digit of the nth number, it would form a kind of diagonal going down into the right.

因此，这个论证被称为对角线论证，因为我们关注的是第 n 个数的第 n 位数字，而这些数字正好位于一条向下的对角线上。

And that for that reason, this argument is called the diagonal argument, because we're looking at the nth digit of the nth number, and those exist on a kind of diagonal going down.

我们构造的数字 z，使得 z 的第 n 位数字与第 n 个数的第 n 位数字不同。

And we've made our number z so that the nth digit of z is different from the nth digit of the nth number.

但由此可以推出，z 不在列表中，因为 z 与 r1 不同——毕竟，z 的小数点后第一位数字与 r1 的小数点后第一位数字不同。

But now it follows that z is not on the list because z is different from r one because well, the the first digit after the decimal point of z is different from the first digit of r one after the decimal point.

这正是我们构造它的方法。

That's exactly how we built it.

z 的第二位数字与 r2 的第二位数字不同，以此类推。

And the second digit of z is different from the second digit of r two, and so on.

对于每一个n，z的第n位数字都与r_n的第n位数字不同。

The nth digit of z is different from the nth digit of r sub n for every n.

因此，z不等于这些r_n中的任何一个数。

So therefore, z is not equal to any of these numbers r sub n.

但这构成了一个矛盾，因为我们原本假设列表中包含了所有的实数，然而这里却出现了一个不在列表中的实数z。

And but that's a contradiction because we had assumed that we had every real number on the list, but yet here is a real number z that's not on the list.

明白吗？

Okay?

所以这就是主要的矛盾。

And so that's the main contradiction.

因此，这是一种构造性的证明。

And so it's a kind of proof by construction.

没错。

Exactly.

给定一个数字列表，证明这一点很有趣，因为你提到这一点时，实际上引发了一种哲学上的争议，即关于康托尔的构造是否具有构造性。

So given a list of numbers, proving it's interesting that you say that actually because there's a kind of philosophical controversy that occurs as a in connection with this observation about whether Cantor's construction is constructive or not.

给定一个数字列表，康托尔提供了一种具体的方法来构造一个不在列表中的实数，这是一种思考方式。

Given a list of numbers, Cantor gives us a specific means of constructing a real number that's not on the list is a way of thinking about it.

我之前提到过一个方面，即某些实数具有多个十进制表示，这会给论证带来一点问题。

There's this one aspect which I alluded to earlier, but some real numbers have more than one decimal representation, and it causes this slight problem in the argument.

例如，数字1可以写成1.0000...无限下去，但也可以写成0.999...无限下去。

For example, the number one, you can write it as one point zero zero zero zero forever, but you can also write it as 0.999 forever.

这两个是同一个数的两种不同的十进制表示。

Those two those are two different decimal representations of exactly the same number.

你巧妙地避开了零和九。

You beautifully got rid of the zeros and the nines.

因此，我们甚至不需要考虑这个问题，证明仍然成立。

Therefore, we don't need to even consider that, and the proof still works.

没错。

Exactly.

因为这种现象只会在数字最终变为零或最终变为九的情况下发生。

Because the only kind of case where that phenomenon occurs is when the number is eventually zero or eventually nine.

因此，我们的数 z 从未包含任何零或九，它不属于那些数。

And so since our number z never had any zeros or nines in it, it wasn't one of those numbers.

所以实际上，在这些情况下，我们根本不需要做任何特殊处理。

And so, actually, in those cases, we didn't need to do anything special to diagnose.

仅仅因为我们的数具有唯一的表示形式，就已经意味着它不等于那些数。

Just the mere fact that our number has a unique representation already means that it's not equal to those numbers.

也许在一百多年前康托尔的时代，这还颇具争议，但如今人们普遍认为，这是集合论的最初主要成果之一，它深刻、惊人且富有洞察力，是许多后续论证的起点。

So maybe it was controversial in Cantor's day more than a hundred years ago, but I think it's most commonly looked at today as, you know, one of the initial main results in set theory, and it's profound and amazing and insightful and the beginning point of so many later arguments.

这种对角化思想已被证明是一种极其富有成效的证明方法。

And this diagonalization idea has proved to be an extremely fruitful proof method.

数学逻辑中的几乎所有重大结果，都在抽象意义上运用了对角化思想。

And almost every major result in mathematical logic is using in an abstract way the idea of diagonalization.

它真正开启了众多其他发现的先河，包括罗素悖论、停机问题、递归定理，以及许多其他原理，其核心都使用了对角化。

It was really the start of so many other observations that were made, including Russell's paradox and the halting problem and the recursion theorem and so many other principles are using diagonalization at their core.

我们能不能稍微退一步？

Can we can we just step back a little bit?

这种无穷危机导致了数学的重建。

This infinity crisis led to a kind of rebuilding of mathematics.

所以如果你能梳理一下由此产生的成果，那就很好了。

So it would be nice if you lay out the things that resulted in.

一个是集合论成为了数学的基础。

So one is set theory became the foundation of mathematics.

所有的数学都可以通过集合来构建，为数学提供了第一个真正严谨的基础。

All mathematics could not be built from sets, giving math its first truly rigorous foundation.

数学的公理化、悖论迫使数学家们发展出ZFC及其他公理系统，数学逻辑由此诞生。

The axiomatization of mathematics, the paradoxes forced mathematicians to develop ZFC and other axiomatic systems, and mathematical logic emerged.

哥德尔、图灵等人开创了全新的领域。

Gurudo Turing and others created entire new fields.

你能解释一下什么是集合论，以及它如何作为现代数学的基础，甚至作为真理的基础吗？

So can you explain what set theory is, and how does it serve as a foundation of modern mathematics and maybe even the foundation of truth?

这是个很好的问题。

That's a great question.

集合论实际上承担着两种角色。

Set theory really has two roles that it's serving.

集合论的出现有两种方式。

There's kind of two ways that set theory emerges.

一方面，集合论本身是一个独立的数学领域，有其独特的问题、疑问、答案和证明方法。

On the one hand, set theory is the is its own subject of mathematics, which with its own problems and questions and answers and proof methods.

因此，从这个角度来看，集合论关注的是超限递归构造或良基定义与构造。

And so, really, from this point of view, set theory is about the transplanet recursive constructions or well founded definitions and constructions.

这些思想极为富有成果，集合论者深入研究了它们，并由此发展出众多理念。

And those ideas have been enormously fruitful, and set theorists have looked into them and developed so many ideas coming out of that.

但集合论也恰好发挥了另一种基础性作用。

But set theory has also happened to serve in this other foundational role.

人们经常听到一些关于集合论的说法，却没有意识到它所承担的这两种角色之间的区别。

It's very common to hear things said about set theory that really aren't taking account of this distinction between the two roles that it's serving.

它既是一个独立的学科，同时也作为数学的基础。

It's its own subject, but it's also serving as a foundation of mathematics.

因此，在其基础作用中，集合论提供了一种将事物的集合视为一个整体的方式。

So in its foundational role, set theory provides a way to think of a collection of things as one thing.

这是集合论的核心思想。

That's the the central idea of set theory.

集合是一些事物的集合，但你要将集合本身视为一个抽象的单一实体。

A set is a collection of things, but you think of the set itself as one abstract thing.

所以当你形成实数的集合时，它就是一个集合。

So when you form the set of real numbers, then that is a set.

它是一个整体。

It's one thing.

它是一个集合，并且内部包含元素。

It's a set, and it has elements inside of it.

所以它有点像一个装满对象的袋子。

So it's sort of like a bag of objects.

集合就像一个装满对象的袋子，因此我们有许多不同的公理，用以描述将事物的集合视为一个整体、一个抽象实体这一概念的性质。

A set is kind of like a bag of objects, and so we have a lot of different axioms that describe the nature of this idea of thinking of a collection of things as as one thing itself, one abstract thing.

公理是我们假设为真的事实，基于这些事实，我们构建了数学的各个概念。

And axioms are, I guess, facts that we assume are true based on which we then build the ideas of mathematics.

因此，关于集合有一些公理，我们可以将它们组合起来，如果它们足够强大，就可以在此基础上构建出许多非常有趣的数学理论。

So there's a bunch of facts, axioms about sets that we can put together, and if they're sufficiently powerful, we can then build on top of that a lot of really interesting mathematics.

是的。

Yeah.

我觉得这是对的。

I think that's right.

也就是说，当前被称为策梅洛-弗兰克尔公理的集合论公理体系，是在二十世纪初由策梅洛的思想发展而来的。

So, I mean, the history of how of the current set theory axioms known as the Zermelo Frankel axioms came out in the early twentieth century with with Zermelo's idea.

这个历史非常有趣，因为策梅洛在1904年提出了一项证明，表明选择公理蕴含良序原理。

I mean, the history is quite fascinating because Zermelo in nineteen o four offered a proof that the what's called the axiom of choice implies the well order principle.

他描述了自己的证明，这在当时引起了极大的争议。

So he described his proof, and that was extremely controversial at the time.

当时还没有一套完整的理论。

And there was no theory.

那时候没有任何公理。

There weren't any axioms there.

康托尔并没有在公理化框架下工作。

Cantor was not working in an axiomatic framework.

他没有像我们现在对集合论那样列出一套公理，策梅洛也没有。

He didn't have a list of axioms in the way that we have for set theory now, And Zermelo didn't either.

他的关于良序定理的想法受到了很多质疑。

And his ideas were challenged so much with regard to the well ordered theorem

嗯。

Mhmm.

因此他被迫提出一种理论，使他的论证能够被形式化，这便是后来被称为策梅洛集合论的起源。

That he was pressed to produce the theory that in which his argument could be formalized, and that was the origin of what's known as Dermelo set theory.

至于困惑，选择公理是

And going to perplexity, the axiom of choice is

集合论中的一个基本原理，它指出：对于任何非空集合的集合，即使没有给出明确的选择规则，也有可能从每个集合中恰好选出一个元素。

a fundamental principle in set theory, which states that for any collection of non empty sets, it is possible to select exactly one element from each set even if no explicit rule to make the choice is given.

这个公理允许构造一个新集合，其中包含来自每个原始集合的一个元素，即使在集合是无限的或没有自然选择规则的情况下也是如此。

This axiom allows the construction of a new set containing one element from each original set even in cases where the collection is infinite or where there is no natural way to specify a selection rule.

因此，这引发了争议，而当时甚至还没有形成公理系统的语言。

So this was controversial, and this was described before there's even a language for axiomatic systems.

没错。

That's right.

一方面，公理选择原则是显而易见的，我们希望它成立，它确实成立。

So on the one hand, I mean, the axiomatic choice principle is completely obvious that we want this to be true, that it is true.

也就是说，许多人把它视为逻辑法则。

I mean, a lot of people take it as a law of logic.

如果你有一堆集合，那么就存在一种方法，可以从每个集合中选出一个元素。

If you have a bunch of sets, then there's a way of picking an element from each of them.

存在一个函数。

There's a function.

如果我有一堆集合，那么就存在一个函数，当你将其应用于其中任何一个集合时，都会得到该集合中的一个元素。

If I have a bunch of sets, then there's a function that when you apply it to any one of those sets gives you an element of that set.

这是一个非常自然的原则。

It's it's a completely natural principle.

我的意思是，它被称为选择公理，这是一种将数学概念拟人化的方式。

I mean, it's called the axiom choice, which is a way of sort of anthropomorphizing the mathematical idea.

并不是说这个函数在做出选择。

It's not like the function is choosing something.

我的意思是，如果你真的要做这样的选择，就会存在一个由你所做选择构成的函数。

I mean, it's just that if you were to make such choices, there would be a function that consisted of the choices that you made.

困难在于，当你无法指定选择的规则或程序时，就很难说明你断言存在的那个函数是什么。

And the difficulty is that when you when you can't specify a rule or a procedure by which you're making choices, then it's difficult to say what the function is that you're asserting exists.

你知道吗？

You know?

你希望认为，确实有一种选择的方式。

You wanna have the view that well, there is a way of choosing.

我无法轻易说出这个函数是什么，但它肯定存在。

I don't have an easy way to say what the function is, but there definitely is one.

是的。

Yeah.

这就是看待选择公理的方式。

This is the way of thinking about the axiom of choice.

所以在这次对话中，我们可能会频繁提到ZFC这三个字母。

So we're gonna say the the the three letters of ZFC maybe a lot in this conversation.

你已经提到过策梅洛-弗兰克尔集合论了。

You already mentioned Zermela Frankel set theory.

那就是Z和F，而C就来自这个选择公理。

That's the z and the f, and the c in that is this comes from this axiom of choice.

没错。

That's right.

所以ZFC听起来像是一个非常技术性的术语，但它实际上是现代数学的基础公理集合。

So ZFC sounds like a super technical thing, but it is the set of axioms that's the foundation of modern mathematics.

是的。

Yeah.

当然。

Absolutely.

因此，人们还应该意识到，数学中有很大一部分领域并不关注是否使用了选择公理，它们甚至不愿意使用选择公理，而是研究在不使用选择公理或使用弱化形式的策梅洛-弗兰克尔集合论等情况下可能得出的结论。

So one should be aware also that there's huge parts of mathematics that don't that pay attention to whether the X amount of choice is being used, and they don't want to use the X amount of choice, so they work out the consequences that's that are possible without the X amount of choice or with weakened forms of of Simonelle Frankel's theory and so on.

在这一领域，确实有相当活跃的研究工作。

And that's quite a there's quite a vibrant amount of work in that area.

我的意思是，但让我们稍微回到选择公理上，或许有趣的是，介绍一下罗素对选择公理的描述方式。

I mean, but going back to the axiom of choice for a bit, it's maybe interesting to to give Russell's description of how to think about the axiom of choice.

罗素描述了一个富人，他拥有一个无限大的衣橱。

So Russell describes this rich person who has a an infinite closet.

在这个衣橱里，他有无穷多双鞋子。

And in that closet, he has infinitely many pairs of shoes.

他告诉他的管家，请从每双鞋中拿一只给我。

And he tells his butler to please go and give me one shoe from each pair.

而管家可以轻松完成这个任务，因为对于任何一双鞋，他都可以始终选择左脚的那只。

And and the butler can do this easily because he can for any pair of shoes, he can just always pick the left shoe.

我的意思是，有一种我们可以描述的挑选方式。

I mean, there's a way of picking that we can describe.

我们总是选左边的，或者总是选右边的，或者如果鞋子是红色的就选左边，如果是棕色的就选右边，或者我们可以发明一些规则来产生这类选择函数。

We always take the left one, or always take the right one, or take the left one if it's a red shoe, and the right one if it's a brown shoe, or, you know, we can invent rules that would result in these kind of choice functions.

因此，我们可以描述明确的选择函数。

So we can describe explicit choice functions.

对于这些情况，你不需要选择公理就知道存在一个选择函数。

And for those cases, you don't need the axiom of choice to know that there's a choice function.

当你能描述一种具体的挑选方式时，就不需要诉诸选择公理来确认存在选择函数。

When you can describe a specific way of choosing, then you don't need to appeal to the axiom to know that there's a choice function.

但问题出现在你考虑这个人衣橱里那无限多双袜子的时候。

But the problematic case occurs when you think about the infinite collection of socks that the person has in their closet.

如果我们假设每双袜子内部是无法区分的，也就是说它们彼此匹配，但又完全不可辨别，那么管家就没有任何规则来决定从每双袜子中选哪一只。

And if we assume that socks are sort of indistinguishable within each pair, you know, they match each other, but they're sort of, you know, indiscernible, then the butler wouldn't have any kind of rule for which sock in each pair to pick.

因此，他是否真的有一种方法能从每双袜子中选出一只，就不那么明确了，对吧？

And so it's not so clear that he has a way of of producing one sock from each pair because right?

因此，关键问题在于：你能否指定一个规则，使得选择函数遵循该规则并定义该函数，还是说其中存在某种任意选择的成分。

So that's what's at stake, is the question of whether you can specify a rule by which the choice function, you know, a rule that it obeys that defines the choice function, or whether there's sort of this arbitrary choosing aspect to it.

这时你就需要选择公理来确认这样的函数确实存在。

That's when you need the axiom of choice to know that there is such a function.

但当然，从数学本体论的角度来看，我们可能会觉得以下观点很有吸引力：你看。

But, of course, as a matter of mathematical ontology, we might find attractive the idea that well, look.

我的意思是，并非每一种选袜子的方式都必须由一个规则来定义。

I mean, I don't I don't Not every way of choosing the socks has to be defined by a rule.

为什么数学现实中所有存在的事物都必须遵循某种规则或程序呢？

Why should everything that exists in mathematical reality follow a rule or a procedure of that sort?

如果我认为我的数学本体论包含丰富的对象，那么我会认为存在着各种各样的函数和选择方式。

If I have the idea that my mathematical ontology is rich with objects, then I think that that there are all kinds of functions and ways of choosing.

这些全部都是我想讨论的数学现实的一部分。

Those are all part of the mathematical reality that I wanna be talking about.

因此，我对断言选择公理没有任何问题。

And so I don't have any problem asserting the axiom choice.

是的。

Yes.

确实存在一种选择方式，但我无法明确告诉你它是什么。

There is a way of choosing, but I can't tell necessarily tell you what it is.

但在数学论证中，我可以假设我固定了选择函数，因为我确信它存在。

But in a mathematical argument, I can assume that I fix the choice function because I know that there is one.

因此，在有选择公理和没有选择公理的情况下进行工作，其哲学差异在于论证的构造性本质。

So it's a the philosophical difference between working when you have the axiom of choice and when you don't is the question of this constructive nature of the argument.

所以，如果你的论证依赖于选择公理，那么你可能是在承认，你在证明中产生的对象并非构造性的。

So if you make an argument and you appeal to the axiom of choice, then maybe you're admitting that the objects that you're producing in the proof are not gonna be constructive.

你无法确切地说明它们的具体性质。

You're not gonna be able to necessarily say specific things about them.

但如果你只是声称做出一个存在性断言，那完全没问题。

But if you're just claiming to make an existence claim, that's totally fine.

而如果你持构造主义的数学观，认为数学断言只有在你能提供明确程序来生成所讨论的数学对象时才成立，那么你可能会拒绝选择公理，甚至更多。

Whereas if you have a constructive attitude about the nature of mathematics, and you think that mathematical claims maybe are only warranted when you can provide an explicit procedure for producing the mathematical objects that you're dealing with, then you're probably gonna wanna deny the axiom of choice and maybe much more.

我们能谈谈支撑ZSC的公理吗？

Can we maybe speak to the axioms that underlie ZSC?

所以，正如我们提到的，ZSC或策梅洛-弗兰克尔集合论加上选择公理，是现代数学最标准的基础。

So going to Perplexity ZSC or Simmelo Frenkel set theory with the axiom of choice, as we mentioned, is the standard foundation for most modern mathematics.

它包括以下主要公理：外延公理、空集公理、配对公理、并集公理、幂集公理、无穷公理、分离公理、替换公理、正则性公理和选择公理。

It consists of the following main axioms, axiom of extensionality, axiom of empty set, axiom of pairing, axiom of union, axiom of power set, axiom of infinity, axiom of separation, axiom of replacement, axiom of regularity, and axiom of choice.

其中一些公理相当基础，但最好能让人对什么是公理有一个大致的理解——我们能摆在桌面上作为基础、进而构建优美数学的那些基本事实是什么？

Some of these are quite basic, but it would be nice to kinda give people a sense Sure.

什么是公理？比如，我们能奠定哪些基本事实，作为构建优美数学的基础？

Of what it means to be an axiom, like, what kind of basic facts we can lay on the table on which we can build some beautiful mathematics?

是的。

Yeah.

它的历史其实非常引人入胜。

So the history of it is really quite fascinating.

策梅洛提出了这些公理中的大部分，也就是说，他提出了现在被称为策梅洛集合论的一部分，目的是形式化他从选择公理推导出良序原理的证明，而这一结果在当时极具争议。

So Zermelo introduced most of these axioms, I mean, part of what's now called Zermelo set theory to formalize his proof from the axiom of choice to the well order principle, which was an extremely controversial result.

因此，在1904年，他给出了一个没有理论基础的证明，随后被要求提供理论依据。

So in nineteen o four, he gave the proof without the theory, and then he was challenged to provide the theory.

于是在1908年，他提出了策梅洛集合论，并证明了在该理论中，每个集合都可以被良序化。

And so in nineteen o eight, he produced the Zermelo set theory and gave the proof that in that theory, you can prove that every set admits a well ordering.

因此，列表中的这些公理，比如外延性，表达了他希望讨论集合时所依据的最根本原则。

And so the axioms on the list, these things like extensionality express the the most fundamental principles of the understanding of sets that he wanted to be talking about.

例如，外延性指出：如果两个集合具有相同的元素，那么它们就是相等的。

So for example, extensionality says, if two sets have the same members, then they're equal.

这意味着集合仅仅是由其元素构成的集合，仅此而已。

So it's this idea that the sets consists of the collection of their members, and that's it.

集合中不再包含其他任何东西。

There's nothing else that's going on in the set.

所以，如果两个集合的元素完全相同，那么它们就是同一个集合。

So it's just if if two sets have the same members, then they are the same set.

因此，从某种意义上说，这可能是最原始的公理。

So it's maybe the most primitive axiom in some respect.

此外，为了让大家有个直观感受，还有一个不含任何元素的集合，叫做空集。

Well, there's also just to give a flavor, there's just a set with no elements called the empty set.

对于任意两个集合，存在一个集合，其元素恰好就是这两个集合。

For any two sets, there's a set that contains exactly those two sets as elements.

对于任意一个集合，存在一个集合，其元素恰好是该集合各元素的元素，也就是并集。

For any set, there's a set that contains exactly the elements of the elements of that set, so the union set.

然后是幂集。

And then there's the power set.

对于任意一个集合，存在一个集合，其元素恰好是原集合的所有子集，这就是幂集。

For any set, there's a set whose elements are exactly the subsets of the original set, the power set.

在无穷公理中，存在一个无限集合，通常是一个包含空集且在添加一个元素的操作下封闭的集合。

In the axiom of infinity, there exists an infinite set, typically, a set that contains the empty set and is closed under the operation of adding one more element

回到我们之前的旅馆例子。

back to our hotel example.

没错。

That's right.

还有更多，但这确实很迷人。

And there's there's more, but this it's kind of fascinating.

我曾经试着让自己置身于那些最初试图形式化集合论的人的心态中。

I used to put put yourself in the mindset of people at the beginning of this of trying to formalize set theory.

人类能够做到这一点，真是令人着迷。

It's it's fascinating that humans can do that.

我读过一些历史学家关于那个时代的记载，特别是关于策梅洛公理和他的良序定理的证明。

I read some historical accounts by historians about that time period, specifically about Zermelo's axioms and his proof of the well ordered theorem.

历史学家们说，在数学史上，从未有过任何数学定理像策梅洛的这个定理那样被如此公开、如此激烈地争论过。

And the historians were saying, never before in the history of mathematics has a mathematical theorem been argued about so publicly and so vociferously as that theorem of Cermallo's.

这也很迷人，因为选择公理最初被广泛视为一种基本原理，但人们却对良序定理非常怀疑，因为没人能想象出实数集的良序排列。

And it's fascinating also because the axiom of choice was widely regarded as a kind of, you know, basic principle at first, but then when but people were very suspicious of the well ordered theorem because no one could imagine a well ordering, say, of the real numbers.

因此，这成了一个案例：策梅洛似乎从一些看似合理的公理出发，证明了一个显然错误的结论。

And so this was a case when Zermelo seemed to be, from principles that seemed quite reasonable, proving this obvious untruth.

因此，数学家们提出了反对。

And so people were mathematicians were objecting.

但后来策梅洛和其他人仔细研究了那些激烈反对者的数学论文，发现许多情况下，他们其实暗中在自己的论证中使用了选择公理，尽管他们公开反对它。

But then Zermelo and others actually looked into the mathematical papers and so on of some of the people who had been objecting so vociferously and found in many cases that they were implicitly using the axiom of choice in their own arguments, even though they would argue publicly against it.

因为使用它太自然了，从某种意义上说，它是一个如此显而易见的原则。

Because it's so natural to use it because it's such an obvious principle in a way.

我的意思是，如果你不够严谨，很容易无意中使用它，甚至根本意识不到自己正在使用选择公理。

I mean, it's easy to just use it by accident with if you're not critical enough, and you don't even realize that you're using the axiom of choice.

这种情况现在依然如此。

That that's true now even.

人们喜欢关注数学论证中是否使用了选择公理。

People like to pay attention to when the axiom of choice is used or not used in mathematical arguments.

我的意思是，直到今天，它曾经更重要。

I mean, up until this day, it used to be more important.

在二十世纪初。

In the early twentieth century.

当时它非常重要，因为人们不知道它是否是一个一致的理论，而且当时出现了许多悖论。

It was very important because people didn't know if it was a consistent theory or not, and there were these antinomies arising.

因此，人们担心公理的一致性。

And so there was a worry about consistency of the axioms.

但后来，随着哥德尔和科恩等人的结果，关于选择公理的一致性问题逐渐消失了。

But then, of course, eventually, with the result of of Godel and Cohen and so on, this consistency question specifically about the axiom of choice sort of falls away.

我们知道，选择公理本身永远不会成为集合论不一致的根源。

We know that the axiom of choice itself will never be the source of inconsistency in set theory.

如果选择公理导致不一致，那么即使没有它，系统也早已不一致了。

If there's inconsistency with the axonal choice, then it's it's already inconsistent without the axonal choice.

所以它并不是不一致的原因。

So it's not the cause of inconsistency.

因此，从这个角度来看，从一致性角度关注是否使用选择公理的重要性某种程度上降低了。

And so in that from that point of view, the need to pay attention to whether you're using it or not from a consistency point of view is somehow less important.

但仍然有理由根据我之前提到的构造主义观点来关注它。

But still, there's this reason to pay attention to it on the grounds of these constructivists ideas that I had mentioned earlier.

我们还应该说明，在集合论中，一致性意味着不可能从该理论的公理中推导出矛盾。

And we should say, in set theory, consistency means that it is impossible to derive a contradiction from the axioms of the theory.

这意味着没有矛盾。

So means that there's no contradictions.

一致的公理系统就是没有矛盾的。

That's that's a That's consistent a axiomatic systems that there's no contradictions.

一个一致的理论是指无法从该理论中推导出矛盾的理论。

A consistent theory is one for which you cannot prove a contradiction from that theory.

也许我们稍作休息，短暂放松一下，去趟洗手间。

Maybe a quick pause, quick break, quick bathroom break.

你之前私下跟我提到过罗素悖论，还说有一个很巧妙的、可以拟人化的不可数性证明。

You mentioned to me offline, we were talking about Russell's paradox, and that there's a nice another kind of anthropomorphizable proof of uncountability.

我想知道你能不能详细讲讲这个证明。

I was wondering if you can lay that out.

哦，当然。

Oh, yeah.

当然可以。

Sure.

当然。

Absolutely.

罗素悖论和这个证明都是。

Both Russell's paradox and the proof.

对。

Right.

所以，我们之前讨论过康托尔关于实数集是不可数无穷大的证明。

So let's so we we talked about Cantor's proof that the real numbers, the set of real numbers, is an uncountable infinity.

它是一个比自然数集严格更大的无穷大。

It's a strictly larger infinity than the natural numbers.

但康托尔实际上证明了一个更普遍的结论，即对于任意集合，其幂集都严格更大。

But Cantor actually proved a much more general fact, namely that for any set whatsoever, the power set of that set is a strictly larger set.

幂集是指包含原集合所有子集的集合。

So the power set is the set containing all the subsets of the original set.

所以，如果你有一个集合，并考察它的所有子集构成的集合，康托尔证明了这个幂集比原集合更大。

So if you have a set and you look at the collection of all of its subsets, then Cantor proves that this is this is a bigger set.

它们不是等势的。

They're not equinumerous.

当然，子集的数量总是至少和元素数量一样多，因为对于任何一个元素，你都可以构造一个只包含该元素的单元素子集。

Of course, there's always at least as many subsets as elements because for any element, you can make the the singleton subset that has only that guy as a member.

对吧？

Right?

所以子集的数量总是至少和元素数量一样多。

So there's always at least as many subsets as elements.

但问题是，子集的数量是否严格更多。

But the question is whether they whether it's strictly more or not.

于是康托尔这样推理。

And so Kantor reasoned like this.

这非常简单。

It's very simple.

这是一种提炼抽象诊断思想的方法，避开了实数复杂性的干扰。

It's a kind of distilling the abstract diagnosis idea without encumbered by the complexity of the real numbers.

所以我们有一个集合 x，我们正在考察它的所有子集。

So we have a set x, and we're looking at all of its subsets.

这就是 x 的幂集。

That's the power set of x.

假设 x 和 x 的幂集大小相同。

Suppose that x and the power set of x have the same size.

假设相反，它们的大小相同。

Suppose towards contradiction, they have the same size.

这意味着我们可以将 x 中的每一个元素与一个子集对应起来。

So that means we can associate to every individual of x a subset.

现在，让我定义一个新的集合。

And so now let me define a new set.

我的意思是，另一个集合。

I mean, another set.

我要来定义它。

I'm gonna define it.

我们称它为 d。

Let's call it d.

d 是 x 的一个子集，包含所有不属于其对应集合的元素。

And d is the subset of x that contains all the individuals that are not in their set.

每个元素都与 x 的一个子集相关联，而我关注的是那些不属于其对应集合的元素。

Every individual was associated with a subset of x, and I'm looking at the individuals that are not in that their set.

也许根本不存在这样的元素。

Maybe nobody's like that.

也许 x 中没有任何元素是这样的，也许所有元素都是这样的，也许一部分是，一部分不是。

Maybe there's no element of x that's like that, or maybe they're all like that, or maybe some of them are and some of them aren't.

对于这个论证来说，这其实并不重要。

I don't it doesn't really matter for the argument.

我定义了一个子集 d，它由那些不属于其对应集合的元素组成。

I defined a subset d consisting of the individuals that are not in the set that's attached to them.

但这是一个完全有效的子集，因此由于基数相等，它必须对应于某个特定的元素。

But that's a perfectly good subset, and so because of the equinumerosity, it would have to be attached to it to a particular individual.

你知道吗？

You know?

嗯。

And Mhmm.

但让我们称那个人为以D开头的名字，比如黛安娜。

But that let let's call that person it should be a name starting with d, so Diana.

嗯。

Mhmm.

现在我们问，黛安娜是d的元素吗？

Now we ask, is Diana an element of d or not?

但如果黛安娜是d的元素，那么她就在自己的集合中。

But if Diana is an element of d, then she is in her set.

但她不应该是，因为集合d是由那些不在自己集合中的个体组成的。

So she shouldn't be because the set d was the the set of individuals that are not in their set.

所以如果黛安娜在d中，那她就不应该在。

So if Diana is in d, then she shouldn't be.

但如果她不在d中，那么她就不在自己的集合里，因此她应该在d中。

But if she isn't in d, then she wouldn't be in her set, and so she should be in d.

嗯。

Mhmm.

这真是个矛盾。

That that's a contradiction.

因此，对于任何集合，其子集的数量总是大于元素的数量。

So therefore, the number of subsets is always greater than the number of elements for any set.

而拟人化的想法如下。

And the anthropomorphizing idea is the following.

我喜欢这样来解释它。

I like to talk about it this way.

对于任何一群人，你可以从中组建出比人数更多的委员会。

For any collection of people, you can you can form more committees from them than there are people.

即使你有无限多的人。

Even if you have infinitely many people.

嗯。

Mhmm.

假设你有一个无限多的人的集合。

Suppose you have an infinite set of people.

那么什么是委员会呢？

And what's a committee?

委员会就是列出委员会的成员，基本上就是委员会的成员名单。

Well, a committee is just the list of who's on the committee, basically, the members of the committee.

所以有所有两人委员会，所有一人委员会，还有那个最差的委员会——每个人都加入的那个。

So there's there's all the two person committees, and there's all the one person committees, and there's the universal the worst committee, the one that everyone is on.

好的。

Okay.

最好的委员会是空委员会，没有成员，从不开会，等等。

The best committee is the empty committee with no members and never meets and so on.

或者空委员会一直在开会吗？

Or is the empty committee meeting all the time?

我不确定。

I'm not sure.

嗯。

Yeah.

那是个深刻的问题。

That's well, that's a profound question.

只有一个成员的委员会也会开会吗？

And does a a committee with just one member meet also?

嗯。

Yeah.

也许它一直在开会。

Maybe it's always in session.

我不知道。

I don't know.

嗯。

Yeah.

所以，这个说法是委员会的数量比人还多。

So so the the the claim is that there's more committees than people.

嗯。

Mhmm.

好的。

Okay.

假设不是这样。

Suppose not.

那么，我们可以在人和委员会之间建立一种对应关系。

Well, then we could make an association between the people and the committees.

因此，每个委员会都可以以一个人的名字来命名，一一对应。

So we would have a kind of every committee could be named after a person in a one to one way.

我并没有说名字被用来命名委员会的那个人是否在该委员会中，不管怎样。

And I'm not saying that the person is on the committee that's named after them or not on it, whatever.

有时候会发生这种情况，有时候不会。

Maybe sometimes that happens, sometimes it doesn't.

我不知道。

I don't know.

这无关紧要。

It doesn't matter.

但让我们来构造我所谓的委员会D，它由所有那些不在以自己名字命名的委员会中的人组成。

But let's form what I call committee d, which consists of all the people that are not on the committee that's named after them.

嗯。

Mhmm.

好的。

Okay.

也许这是所有人。

Maybe that's everyone.

也许一个人都没有。

Maybe it's no one.

也许是半数人。

Maybe it's half the people.

这不重要。

It doesn't matter.

这是一个委员会。

That's a committee.

它是一群人。

It's a set of people.

所以它必须以某个人的名字命名。

And so it has to be named after someone.

我们叫这个人达尼埃拉。

Let's call that person Daniella.

那么现在我们问，达尼埃拉是否在以她命名的委员会里？

So now we ask, is Daniella on the committee that's named after her?

如果她在，那她就不应该在，因为这个委员会是由那些不在自己委员会里的人组成的。

Well, if she is, then she shouldn't be because it was the committee of people who aren't on their on their own committee.

如果她不在，那她就应该在。

And if she isn't, then she should be.

所以，这又是一个矛盾。

So, again, it's a contradiction.

当我还在牛津大学任教时，我的一个学生提出了对坎托论证的另一种拟人化解释。

So When I was teaching at Oxford, one of my students came up with the following different anthropomorphization of Kanto's argument.

让我们考虑所有可能的水果沙拉。

Let's consider all possible fruit salads.

我们有一组给定的水果。

We have a given collection of fruits.

嗯。

Mhmm.

比如苹果、橙子、葡萄之类的。

You know, apples and oranges and grapes, whatever.

水果沙拉由这些水果中的一些组合而成。

And a fruit salad consists of some collection of those fruits.

比如香蕉、梨、葡萄沙拉等等。

So there's the banana, pear, grape salad, and so on.

沙拉的种类很多。

There's a lot of different kinds of salad.

每一种水果组合都构成一种沙拉，一种水果沙拉。

Every set of fruits makes a salad, a fruit salad.

好的。

Okay.

我们想证明，对于任何水果集合，即使有无限多种不同的水果，可能的水果沙拉数量也总是多于水果的数量。

And we want to prove that for any collection of fruits, even if there are infinitely many different kinds of fruit, for any collection of fruits, there are more possible fruit salads than there are fruits.

如果不是这样，那么水果和水果沙拉之间就可以建立一一对应关系。

So if not, then you can put a one to one correspondence between the fruits and the fruit salads.

你可以用每种水果来命名每一种水果沙拉。

So you could name every fruit salad after a fruit.

那个水果本身可能并不在那个沙拉里。

Might not be that fruit might not be in that salad.

这没关系。

It doesn't matter.

这只是一个命名，一种一一对应的关系。

We're just it's a naming, a one to one correspondence.

然后，我们构造出对角沙拉，它由所有那些未包含在以自己命名的沙拉中的水果组成。

And then, of course, we form the diagonal salad, which consists of all the fruits that are not in the salad that's named after them.

嗯。

Mhmm.

而且这确实是一个完全合理的沙拉。

And and that that's a perfectly good salad.

它可能是那种清淡沙拉，如果是空沙拉的话；也可能是全能沙拉，如果包含了所有水果的话。

It might be the kind of diet salad if it was the empty salad, or it might be the universal salad, which had all fruits in it if all the fruits are in it.

或者它可能只包含部分水果，而不是全部。

Or it might have just some and not all.

因此，这个对角沙拉必须被某种水果命名。

So that diagonal salad would have to be named after some fruit.

所以我们假设它被命名为榴莲，意味着在一一对应关系中它与榴莲相关联。

So let's suppose it's named after durian, meaning that it was associated with durian in the one to one correspondence.

然后我们问：榴莲是否在以它命名的沙拉中？

And then we ask, well, is durian in the salad that it's named after?

如果它在，那它就不应该在。

And if it is, then it shouldn't be.

如果它不在，那它就应该在。

And if it isn't, then it should be.

因此，这又是一个同样的矛盾。

And so it's again the same contradiction.

所以，所有这些论证都与康托尔证明任何集合的幂集都大于该集合的证明完全相同。

So so all of those arguments are just the same as Cantor's proof that the power set of any set is bigger than the set.

这正是罗素悖论中出现的逻辑，因为罗素认为所有集合的类不能是一个集合。

And this is exactly the same logic that comes up in Russell's paradox, because Russell is arguing that the class of all sets can't be a set.

因为如果它是，那么我们就可以构造出所有不包含自身的集合的集合。

Because if it were, then we could form the the set of all sets that are not elements of themselves.

基本上，罗素所证明的是：集合的集合比元素本身更多。

So basically, he's what Russell is proving is that there are more collections of sets than than elements.

因为我们能构造出对角类，也就是所有不属于自身的集合的类。

Because we can form the diagonal class, you know, the class of all sets that are not elements of themselves.

如果这是一个集合，那么它当且仅当它不属于自身时，才属于自身。

If that were a set, then it would be an element of itself if and only if it was not an element of itself.

这四个论点中的逻辑完全相同。

It's exactly the same logic in all four of those arguments.

是的。

Yeah.

因此，不可能存在一个包含所有集合的类，因为如果存在，那就必须存在一个包含所有不属于自身的集合的类。

So there can't be a class of all sets, because if there were, then there would have to be a class of all sets that aren't elements of themselves.

但这个集合当且仅当它不属于自身时，才属于自身，这就构成了矛盾。

But but that set would be an element of itself if and only if it's not an element of itself, which is a contradiction.

这就是罗素悖论的本质。

So this is the essence of the Russell paradox.

我不称它为罗素悖论。

I don't call it the Russell paradox.

实际上，当我讲授时，我称它为罗素定理。

Actually, when I teach it, I call it Russell's theorem.

不存在全集。

There's no universal set.

现在这已经不再令人困惑了。

And it's not really confusing anymore.

当时，这非常令人困惑。

At the time, it was very confusing.

但如今，我们已经将集合论的这种特性融入了我们对集合本质的基本理解中，因此它不再令人困惑。

But now, we've absorbed this nature of set theory into our fundamental understanding of of how sets are, and it's not confusing anymore.

我的意思是，关于罗素悖论的历史非常引人入胜，因为在那之前，弗雷格正在致力于他的宏伟著作，试图实现逻辑主义哲学，即试图将所有数学还原为逻辑。

I mean, the history is fascinating, about the Russell paradox, because before that time, Frege was working on his monumental work undertaking implementing the philosophy of logicism, which is the attempt to reduce all of mathematics to logic.

因此，弗雷格希望用逻辑概念来解释整个数学。

So Frege wanted to give an account of all of mathematics in terms of logical notions.

他正在撰写这部宏伟著作，并已确立了其基本原理。

And he was writing this monumental work and had formulated his basic principles.