无穷、击溃数学的悖论、哥德尔不完备性与多重宇宙——乔尔·戴维·汉姆金斯

The following is a conversation with Joel David Hamkins, a mathematician and philosopher specializing in set theory, the foundation of mathematics, and the nature of infinity.

以下是与乔尔·大卫·哈姆金斯 (Joel David Hamkins) 的对话，他是一位专门研究集合论、数学基础和无穷本质的数学家和哲学家。

00:00 He is the number one highest rated user on MathOverflow, which I think is a legendary accomplishment.

他是 MathOverflow 上评分最高的用户，我认为这是一项传奇般的成就。

MathOverflow, by the way, is like StackOverflow but for research mathematicians.

顺便说一句，MathOverflow 与 StackOverflow 类似，只不过是为研究数学家而设计的。

He is also the author of several books, including Proof in The Art of Mathematics and Lectures on the Philosophy of Mathematics.

他还是多本书的作者，包括《数学艺术的证明》和《数学哲学讲座》。

And he has a great blog, infinitelymore.xyz.

他有一个很棒的博客，infinitelymore.xyz。

This is a super technical and super fun conversation about the foundation of modern mathematics and some mind-bending ideas about infinity, nature of reality, truth, and the mathematical paradoxes that challenged some of the greatest minds of the 20th century.

这是一场超级技术性和超级有趣的对话，内容涉及现代数学的基础，以及一些关于无限、现实本质、真理的令人费解的想法，以及挑战 20 世纪一些最伟大思想的数学悖论。

01:02 I have been hiding from the world a bit, reading, thinking, writing, soul-searching, as we all do every once in a while.

我一直在躲避这个世界，阅读、思考、写作、反省，就像我们偶尔做的那样。

But mostly, just deeply focused on work and preparing mentally for some challenging travel I plan to take on in the new year.

但大多数情况下，我只是专注于工作，并为我计划在新的一年进行的一些具有挑战性的旅行做好心理准备。

Through all of it, a recurring thought comes to me, how damn lucky I am to be alive and to get to experience so much love from folks across the world.

在经历这一切的过程中，我不断地想到，我能活着并体验到来自世界各地人们如此多的爱是多么幸运。

I want to take this moment to say thank you from the bottom of my heart for everything, for your support, for the many amazing conversations I’ve had with people across the world.

我想借此机会从心底对你们所做的一切、你们的支持、以及我与世界各地的人们进行的许多精彩对话表示感谢。

I got a little bit of hate and a whole lot of love, and I wouldn’t have it any other way.

我有一点点的恨，也有很多的爱，除此之外我别无选择。

I’m grateful for all of it.

我很感激这一切。

This is the Lex Fridman Podcast.

这是莱克斯·弗里德曼播客。

02:00 To support it, please check out our sponsors in the description, where you can also find ways to contact me, ask questions, give feedback, and so on.

为了支持它，请在描述中查看我们的赞助商，您还可以在其中找到联系我、提出问题、提供反馈等的方式。

And now, dear friends, here’s Joel David Hamkins.

现在，亲爱的朋友们，这是乔尔·大卫·哈姆金斯。

02:17 Some infinities are bigger than others.

有些无穷大比其他无穷大。

This idea from Cantor at the end of the 19th century, I think it’s fair to say, broke mathematics before rebuilding it.

我认为可以公平地说，康托尔在 19 世纪末提出的这个想法在重建数学之前就打破了它。

I also read that this was a devastating and transformative discovery for several reasons.

我还读到，出于多种原因，这是一个毁灭性的、变革性的发现。

So one, it created a theological crisis, because infinity is associated with God, how could there be multiple infinities?

因此，它造成了神学危机，因为无限与上帝联系在一起，怎么可能有多个无限呢？

Also, Cantor was deeply religious himself.

此外，康托本人也非常虔诚。

Second, there’s a kind of mathematical civil war.

其次，存在一种数学内战。

The leading German mathematician, Kronecker, called Cantor a corrupter of youth and tried to block his career.

德国著名数学家克罗内克称康托是年轻人的腐蚀者，并试图阻止他的职业生涯。

02:57 Third, many fascinating paradoxes emerged from this, like Russell’s paradox, about the set of all sets that don’t contain themselves, and those threatened to make all of mathematics inconsistent.

第三，由此产生了许多令人着迷的悖论，比如罗素悖论，关于所有不包含自身的集合的集合，以及那些威胁使所有数学不一致的悖论。

Finally, on the psychological and personal side, Cantor’s own breakdown.

最后，在心理和个人方面，康托自己的崩溃。

He literally went mad, spending his final years in and out of sanatoriums, obsessed with proving the continuum hypothesis.

他真的疯了，在疗养院里进进出出度过了最后的几年，痴迷于证明连续统假说。

So laying that all out on the table, can you explain the idea of infinity, that some infinities are larger than others, and why was this so transformative to mathematics?

那么，将所有这些都摆在桌面上，您能解释一下无穷大的概念吗？某些无穷大比其他无穷大，以及为什么这对数学如此具有变革性？

Well, that’s a really great question.

嗯，这是一个很好的问题。

03:35 I would want to start talking about infinity and telling the story much earlier than Cantor actually, because, I mean, you can go all the way back to Ancient Greek times when Aristotle emphasized the potential aspect of infinity as opposed to the impossibility, according to him, of achieving an actual infinity.

我想比康托尔更早地开始谈论无限并讲述这个故事，因为，我的意思是，你可以一直追溯到古希腊时代，当时亚里士多德强调无限的潜在方面，而不是根据他的说法，实现实际无限的不可能性。

Archimedes’ method of exhaustion where he is trying to understand the area of a region by carving it into more and more triangles, say, and sort of exhausting the area and thereby understanding the total area in terms of the sum of the areas of the pieces that he put into it.

阿基米德的穷举法，他试图通过将一个区域雕刻成越来越多的三角形来了解该区域的面积，然后对面积进行穷举，从而根据他放入的各个块的面积之和来了解总面积。

And it proceeded on this kind of potential understanding of infinity for hundreds, thousands of years.

这种对无限的潜在理解持续了数百年、数千年。

04:25 Almost all mathematicians were potentialists only and thought that it was incoherent to speak of an actual infinity at all.

几乎所有数学家都只是势论者，他们认为谈论实际的无穷大是不连贯的。

Galileo is an extremely prominent exception to this, though he argued against this sort of potentialist orthodoxy in The Dialogue of Two New Sciences.

伽利略是一个极其突出的例外，尽管他在《两种新科学的对话》中反对这种势能主义正统观念。

Really lovely account there that he gave.

他给出的帐户非常可爱。

In many ways, Galileo was anticipating Cantor’s developments, except he couldn’t quite push it all the way through and ended up throwing up his hands in confusion, in a sense.

在很多方面，伽利略都在期待康托尔的发展，但从某种意义上说，他无法完全推动它，最终困惑地举起了双手。

The Galileo paradox is the idea or the observation that if you think about the natural numbers, I would start with zero, but I think maybe he would start with one.

伽利略悖论是这样一种想法或观察：如果你考虑自然数，我会从零开始，但我认为也许他会从一开始。

05:17 The numbers one, two, three, four, and so on, and you think about which of those numbers are perfect squares.

数字一、二、三、四等等，你会想这些数字中哪些是完全平方数。

So zero squared is zero and one squared is one and two squared is four, three squared is nine, 16, 25, and so on.

因此，零的平方是零，一的平方是一，二的平方是四，三的平方是九，16，25，等等。

And Galileo observed that the perfect squares can be put into a one-to-one correspondence with all of the numbers.

伽利略观察到完美的平方可以与所有数字一一对应。

I mean, we just did it.

我的意思是，我们刚刚做到了。

I associated every number with its square.

我将每个数字与其平方联系起来。

And so it seems like on the basis of this one-to-one correspondence that there should be exactly the same number of squares, perfect squares, as there are numbers, and yet there are all the gaps in between the perfect squares, right?

因此，在这种一一对应的基础上，似乎应该有与数字完全相同的正方形（完美正方形）数量，但完美正方形之间却存在所有间隙，对吗？

06:03 And this suggests that there should be fewer perfect squares, more numbers than squares because the numbers include all the squares plus a lot more in between them, right?

这表明完美正方形应该更少，数字应该比正方形更多，因为数字包括所有正方形以及它们之间的更多数字，对吗？

And Galileo was quite troubled by this observation because he took it to cause a kind of incoherence in the comparison of infinite quantities, right?

伽利略对这个观察结果感到非常困扰，因为他认为它会导致无限量比较中的一种不连贯，对吧？

Another example is, if you take two line segments of different lengths, and you can imagine drawing a kind of foliation, a fan of lines that connect them.

另一个例子是，如果你选择两条不同长度的线段，你可以想象画出一种叶状结构，即连接它们的扇形线。

So the endpoints are matched from the shorter to the longer segment, and the midpoints are matched and so on.

因此端点从较短的线段到较长的线段进行匹配，并且中点进行匹配等等。

So spreading out the lines as you go.

因此，边走边分散线路。

And so every point on the shorter line would be associated with a unique, distinct point on the longer line in a one-to-one way.

因此，较短线上的每个点都将以一对一的方式与较长线上的一个独特的、不同的点相关联。

06:57 And so it seems like the two line segments have the same number of points on them because of that, even though the longer one is longer.

因此，看起来两条线段上的点数量相同，尽管较长的线段更长。

And so it makes, again, a kind of confusion over our ideas about infinity.

因此，这再次使我们对无限的想法产生了某种混乱。

Also, with two circles, if you just place them concentrically and draw the rays from the center, then every point on the smaller circle is associated with a corresponding point on the larger circle, in a one-to-one way.

另外，对于两个圆，如果将它们同心放置并从中心绘制射线，则较小圆上的每个点都与较大圆上的对应点以一对一的方式关联。

And again, that seems to show that the smaller circle has the same number of points on it as the larger one, precisely because they can be put into this one-to-one correspondence.

再次，这似乎表明较小的圆与较大的圆上具有相同数量的点，正是因为它们可以放入这种一一对应的关系中。

07:36 Now, of course, the contemporary attitude about this situation is that those two infinities are exactly the same, and that Galileo was right in those observations about the equinumerosity.

当然，现在对这种情况的当代态度是，这两个无穷大是完全相同的，伽利略对等数性的观察是正确的。

The way we would talk about it now is to appeal to what I call the Cantor-Hume principle, or some people just call it Hume’s principle, which is the idea that if you have two collections, whether they’re finite or infinite, then we want to say that those two collections have the same size, they’re equinumerous, if and only if there’s a one-to-one correspondence between those collections.

我们现在谈论它的方式是诉诸我所说的康托-休谟原理，或者有些人只是称之为休谟原理，它的想法是，如果你有两个集合，无论它们是有限的还是无限的，那么我们想说这两个集合具有相同的大小，它们是等数的，当且仅当这些集合之间存在一一对应时。

And so Galileo was observing that line segments of different lengths are equinumerous, and the perfect squares are equinumerous with the whole…

所以伽利略观察到不同长度的线段是等分的，完美的正方形是等分的……

08:17 All of the natural numbers, and any two circles are equinumerous and so on.

所有的自然数，以及任意两个圆都是等分的等等。

The tension between the Cantor-Hume principle and what could be called Euclid’s principle, which is that the whole is always greater than the part, is a principle that Euclid appealed to in the Elements.

康托-休谟原理和所谓的欧几里得原理（即整体总是大于部分）之间的紧张关系是欧几里得在《几何原本》中所诉诸的原理。

Many times when he’s calculating area and so on, he wants…

很多时候，当他计算面积等时，他想要……

It’s a kind of basic idea that if something is just a part of another thing, then the whole is greater than the part.

这是一种基本思想，如果某事物只是另一事物的一部分，那么整体大于部分。

And so what Galileo was troubled by was this tension between what we call the Cantor-Hume principle and Euclid’s principle.

因此，伽利略所困扰的是我们所说的康托-休谟原理和欧几里得原理之间的紧张关系。

08:59 It really wasn’t fully resolved, I think, until Cantor.

我认为，直到康托尔之前，这个问题确实没有完全解决。

He’s the one who really explained so clearly about these different sizes of infinity and so on in a way that was so compelling.

他是那个以一种非常引人注目的方式真正清楚地解释了这些不同大小的无穷大等的人。

So he exhibited two different infinite sets and proved that they’re not equinumerous; they can’t be put into one-to-one correspondence.

因此，他展示了两个不同的无限集，并证明它们不是等数集；它们不能一对一对应。

It’s traditional to talk about the uncountability of the real numbers.

传统上谈论实数的不可数性。

So Cantor’s big result was that the set of all real numbers is an uncountable set.

所以康托尔的重要结果是所有实数的集合是一个不可数集合。

Maybe if we’re going to talk about countable sets, then I would suggest that we talk about Hilbert’s Hotel, which really makes that idea perfectly clear.

也许如果我们要讨论可数集，那么我建议我们讨论希尔伯特旅馆，这确实使这个想法非常清晰。

Yeah, let’s talk about Hilbert’s Hotel.

是的，我们来谈谈希尔伯特酒店。

Hilbert’s Hotel is a hotel with infinitely many rooms.

希尔伯特酒店是一家拥有无限多个房间的酒店。

09:41 Each room is a full floor suite.

每个房间都是整层套房。

So there’s floor zero…

所以有零层……

I always start with zero because for me, the natural numbers start with zero, although that’s maybe a point of contention for some mathematicians.

我总是从零开始，因为对我来说，自然数从零开始，尽管这可能是一些数学家的争论点。

The other mathematicians are wrong.

其他数学家都是错的。

Like I mentioned, I’m a programmer, so starting at zero is a wonderful place to start.

就像我提到的，我是一名程序员，所以从零开始是一个很好的起点。

Exactly.

确切地。

10:01 So there’s floor zero, floor one, floor two, or room zero, one, two, three, and so on, just like the natural numbers.

所以有零楼、一楼、二楼，或者零号房间、一号房、二号房、三号房等等，就像自然数一样。

So Hilbert’s Hotel has a room for every natural number, and it’s completely full.

因此，希尔伯特酒店为每个自然数都有一个房间，而且房间已经满了。

There’s a person occupying room N for every N.

每 N 个房间就有一个人占用 N 个房间。

But meanwhile, a new guest comes up to the desk and wants a room.

但与此同时，一位新客人走到服务台前想要一间房间。

“Can I have a room, please?” And the manager says, “Hang on a second, just give me a moment.” You see, when the other guests had checked in, they had to sign an agreement with the hotel that maybe there would be some changing of the rooms during their stay.

“可以给我一个房间吗？”经理说：“等一下，给我一点时间。”你看，当其他客人入住后，他们必须与酒店签订一份协议，在他们入住期间可能会有一些房间的更换。

10:39 And so the manager sent a message up to all the current occupants and told every person, “Hey, can you move up one room, please?” So the person in room five would move to room six, and the person in room six would move to room seven and so on.

于是经理向所有现有住户发送了一条消息，并告诉每个人：“嘿，请问您可以换一间房间吗？”因此，五号房间的人会搬到六号房间，六号房间的人会搬到七号房间，依此类推。

And everyone moved at the same time.

而所有人同时动了。

Of course, we never want to be placing two different guests in the same room, and we want everyone to have their own private room.

当然，我们绝不希望将两位不同的客人安排在同一个房间，我们希望每个人都有自己的私人房间。

But when you move everyone up one room, then the bottom room, room zero, becomes available, of course.

但是，当您将每个人都移到一个房间时，那么底部的房间（零号房间）当然就变得可用。

So he can put the new guest in that room.

这样他就可以把新客人安排在那个房间里。

So even when you have infinitely many things, then the new guest can be accommodated.

因此，即使你有无限多的东西，也可以容纳新客人。

And that’s a way of showing how the particular infinity of the occupants of Hilbert’s Hotel, it violates Euclid’s principle.

这是展示希尔伯特旅馆的居住者的无限性如何违反欧几里得原理的一种方式。

11:25 It exactly illustrates this idea because adding one more element to a set didn’t make it larger, because we can still have a one-to-one correspondence between the total new guests and the old guests by the room number, right?

它恰好说明了这个想法，因为在集合中添加一个元素并不会使其变得更大，因为我们仍然可以通过房间号在新客人总数和老客人之间进行一一对应，对吧？

So to just say one more time, the hotel is full.

再说一遍，酒店已经满了。

The hotel is full.

酒店已满。

And then you could still squeeze in one more, and that breaks the traditional notion of mathematics and breaks people’s brains about when they try to think about infinity, I suppose.

然后你仍然可以再挤进去一个，这打破了传统的数学概念，并且在人们试图思考无穷大时打破了他们的大脑，我想。

11:46 This is a property of infinity.

这是无穷大的性质。

It’s a property of infinity that sometimes when you add an element to a set, it doesn’t get larger.

这是无穷大的一个属性，有时当你向集合中添加一个元素时，它不会变得更大。

11:59 That’s what this example shows.

这就是这个例子所显示的内容。

But one can go on with Hilbert’s Hotel, for example.

但我们可以继续以希尔伯特酒店为例。

I mean, maybe the next day, 20 people show up all at once.

我的意思是，也许第二天，就会有 20 个人同时出现。

We can easily do the same trick again, just move everybody up 20 rooms.

我们可以轻松地再次执行同样的操作，只需将每个人向上移动 20 个房间即可。

Then we would have 20 empty rooms at the bottom, and those new 20 guests could go in.

那么底层就会有 20 个空房间，新来的 20 个客人就可以进去了。

But on the following weekend, a giant bus pulled up, Hilbert’s bus.

但在接下来的周末，一辆巨型巴士停了下来，那是希尔伯特的巴士。

And Hilbert’s bus has, of course, infinitely many seats.

当然，希尔伯特的巴士有无限多个座位。

There’s Seat Zero, Seat One, Seat Two, Seat Three, and so on.

有零号座位、一号座位、二号座位、三号座位等等。

So one wants to…

所以有人想要……

You know, all the people on the bus want to check into the hotel, but the hotel is completely full.

要知道，车上的人都想入住酒店，但酒店却完全住满了。

So what is the manager going to do?

那么经理要做什么呢？

12:50 And when I talk about Hilbert’s Hotel in class, I always demand that the students provide the explanation of how to do it.

当我在课堂上谈论希尔伯特旅馆时，我总是要求学生解释如何做。

So maybe I’ll ask you.

所以也许我会问你。

Can you tell me, yeah, what is your idea about how to fit them all in the hotel, everyone on the bus, and also the current occupants?

您能告诉我，是的，您对如何将他们全部安置在酒店、公共汽车上的每个人以及当前的乘客有什么想法吗？

You separate the hotel into even and odd rooms, and you squeeze in the new Hilbert bus people into the odd rooms, and the previous occupants go into the even rooms.

你把酒店分成偶数和奇数房间，然后把新的希尔伯特巴士人员塞进奇数房间，而以前的乘客则挤进偶数房间。

That’s exactly right.

完全正确。

13:20 So, I mean, that’s a very easy way to do it.

所以，我的意思是，这是一个非常简单的方法。

If you just tell all the current guests to double their room number, so in Room N, you move to Room 2 times N.

如果您只是告诉所有当前客人将其房间号加倍，那么在 N 号房间中，您将移动到 2 倍 N 的房间。

So they’re all going to get their own private room, the new room, and it will always be an even number because 2 times N is always an even number.

所以他们都会有自己的私人房间，新房间，而且它总是偶数，因为 2 乘以 N 总是偶数。

And so all the odd rooms become empty that way.

这样所有奇怪的房间就都空了。

And now we can put the bus occupants into the odd-numbered rooms.

现在我们可以将公交车上的乘客安排到奇数房间。

And by doing so, you have now shoved an infinity into another infinity.

通过这样做，你现在已经将一个无穷大推入了另一个无穷大。

That’s right.

这是正确的。

13:47 So what it really shows, I mean, another way of thinking about it is that, well, we can define that a set is countable if it is equinumerous with a set of natural numbers.

所以它真正表明的，我的意思是，另一种思考方式是，我们可以定义一个集合是可数的，如果它与一组自然数相等。

And a kind of easy way to understand what that’s saying in terms of Hilbert’s Hotel is that a set is countable if it fits into Hilbert’s Hotel, because Hilbert’s Hotel basically is the set of natural numbers in terms of the room numbers.

理解希尔伯特旅馆的一种简单方法是，如果一个集合适合希尔伯特旅馆，那么它就是可数的，因为希尔伯特旅馆基本上是房间号的自然数集合。

So to be equinumerous with a set of natural numbers is just the same thing as to fit into Hilbert’s Hotel.

因此，用一组自然数进行等分与适应希尔伯特旅馆是一样的。

And so what we’ve shown is that if you have two countably infinite sets, then their union is also countably infinite.

所以我们已经证明，如果有两个可数无限集，那么它们的并集也是可数无限的。

If you put them together and form a new set with all of the elements of either of them, then that union set is still only countably infinite.

如果将它们放在一起并使用其中任何一个的所有元素形成一个新集合，那么该并集仍然只是可数无限的。

14:34 It didn’t get bigger.

它并没有变得更大。

And that’s a remarkable property for a- a notion of infinity to have, I suppose.

我想，对于无穷大的概念来说，这是一个非凡的特性。

But if you thought that there was only one kind of infinity, then it wouldn’t be surprising at all, because if you take two infinite sets and put them together, then it’s still infinite.

但如果你认为只有一种无穷大，那就一点也不奇怪了，因为如果你把两个无穷集放在一起，那么它仍然是无穷大。

And so if there were only one kind of infinity, then it shouldn’t be surprising- …

因此，如果只有一种无限，那么也就不足为奇了——……

that the union of two countable sets is countable.

两个可数集合的并集是可数的。

So there’s another way to push this a bit harder, and that is when when Hilbert’s train arrives, and Hilbert’s train has infinitely many train cars- …

所以还有另一种方法可以让这个问题变得更困难，那就是当希尔伯特的火车到达时，希尔伯特的火车有无限多的车厢——……

and each train car has infinitely many seats.

每节火车车厢有无限多个座位。

15:13 And so we have an infinity of infinities of the train passengers together with the current occupants of the hotel, and everybody on the train wants to check in to Hilbert’s Hotel.

因此，我们有无数的火车乘客以及酒店的当前入住者，火车上的每个人都想入住希尔伯特酒店。

So the manager can, again, of course, send a message up to all the rooms telling every person to double their room number again.

因此，经理当然可以再次向所有房间发送一条消息，告诉每个人再次将其房间号加倍。

And so that will occupy all the even-numbered rooms again and free up again the odd-numbered rooms.

这样就会再次占用所有偶数房间，并再次释放奇数房间。

So somehow, we want to put the train passengers into the odd-numbered rooms.

所以不知何故，我们想把火车乘客安排到奇数房间。

And so while every train passenger is on some car, let’s say Car C and Seat S, somehow, we have to take these two coordinates, you know, C, S, the car number and the seat number, and produce from it an odd number in a one-to-one way.

因此，当每个火车乘客都乘坐某节车厢时，比如说 C 车厢和 S 座位，我们必须以某种方式获取这两个坐标，即 C、S、车厢编号和座位编号，并以一对一的方式从中生成一个奇数。

And that’s actually not very difficult.

这实际上并不是很困难。

16:10 In fact, one can just use, say…

事实上，人们可以直接使用，比如说……

An easy way to do it is to just use the number 3 to the C times 5 to the S.

一种简单的方法是使用 C 的数字 3 乘以 S 的 5。

3 to the C, 3 to the car number, so 3 x 3 x 3, you know, the number of the car.

3 到 C，3 到汽车号码，所以 3 x 3 x 3，你知道，汽车的号码。

You multiply 3 by itself the number of the train car, and then you multiply 5 by itself the seat number of times, and then you multiply those two numbers together.

将火车车厢的编号乘以 3，然后将座位数乘以 5，然后将这两个数字相乘。

So 3 to the C times 5 to the S.

所以 3 乘以 5 到 S。

That’s always an odd number, because the prime factorization has only 3s and 5s in it.

这始终是奇数，因为素因数分解中只有 3 和 5。

There’s no 2 there.

那里没有2。

So therefore, it’s definitely an odd number, and it’s always different because of the uniqueness of prime factorization.

因此，它绝对是一个奇数，并且由于素数分解的唯一性，它总是不同的。

So every number can be factored uniquely into primes.

所以每个数字都可以唯一地因式分解为素数。

So if you have a number of that form, then you can just factor it, and that tells you the exponent on 3 and the exponent on 5.

因此，如果您有这种形式的数字，那么您只需对它进行因式分解即可，这样您就可以得到 3 的指数和 5 的指数。

17:06 And so you know exactly which person it was, which car they came from, and which seat they came from.

这样您就可以准确地知道那是谁、他们来自哪辆车以及他们来自哪个座位。

And prime factorization is every single number can be decomposed into the atoms of mathematics, which is the prime numbers.

而素因数分解就是每一个单独的数字都可以分解为数学原子，这就是素数。

17:10 You can multiply them together to achieve that number.

您可以将它们相乘以获得该数字。

That’s, uh-

那就是，呃-

And that’s prime factorization.

这就是质因数分解。

17:23 You’re showing 3 and 5 are both prime numbers, odd.

您显示 3 和 5 都是质数，都是奇数。

So through this magical formula, you can deal with this train, an infinite number of cars, with each car having an infinite number of seats.

所以通过这个神奇的公式，你可以处理这列火车，无限多的车厢，每节车厢有无限多个座位。

Exactly right.

完全正确。

17:41 We’ve proved that if you have countably many countable sets, then the union of those sets, putting all those sets together into one giant set, is still countable.

我们已经证明，如果你有可数的多个可数集合，那么这些集合的并集，将所有这些集合组合成一个巨大的集合，仍然是可数的。

You know, because the train cars are each countable, plus the current hotel.

要知道，因为火车车厢每节都是可数的，加上现在的酒店。

It’s sort of like another train car, if you want to think about it that way.

如果你想这样想的话，它有点像另一节火车车厢。

The current occupants of the hotel could, you know, have the same number as any of the train cars.

你知道，酒店目前的住户人数可能与任何火车车厢的人数相同。

So putting countably many countable sets together to make one big union set is still countable.

因此，将可数个可数集合放在一起形成一个大并集仍然是可数的。

It’s quite remarkable, I think.

我认为这非常了不起。

When I first learned this many, many years ago, I was completely shocked by it and transfixed by it.

很多很多年前，当我第一次了解到这一点时，我完全被它震惊了，被它惊呆了。

18:20 It was quite amazing to me that this notion of countable infinity could be closed under this process of infinitely many infinities adding up still to the very same infinity, which is a strong instance, a strong violation of Euclid’s principle once again, right?

让我感到非常惊讶的是，可数无穷大的概念可以在无限多个无穷大加起来仍然是同一个无穷大的过程下得到封闭，这是一个强有力的例子，再次严重违反了欧几里得原理，对吗？

So, the new set that we built is…

所以，我们构建的新套装是……

has many more elements than the old set in the sense that there’s additional elements, but it doesn’t have many more elements in terms of its size because it’s still just a countable infinity and it fits into Hilbert’s Hotel.

从有附加元素的意义上来说，它比旧集合有更多的元素，但就其大小而言，它没有更多的元素，因为它仍然只是可数无穷大，并且适合希尔伯特酒店。

Have you been able to sort of internalize a good intuition about countable infinity?

您是否能够内化关于可数无穷的良好直觉？

18:53 ‘Cause that is a pretty weird thing.

因为这是一件非常奇怪的事情。

You can have a countably infinite set of countably infinite sets, and you can shove it all in and it still is a countable infinite set.

你可以有一个由可数无限集组成的可数无限集，你可以把它全部塞进去，它仍然是一个可数无限集。

Yeah, that’s exactly right.

是的，完全正确。

19:11 I mean, I guess, of course when you work with these notions that the argument of Hilbert’s Hotel becomes kind of clear, there are many, many other ways to talk about it too.

我的意思是，我想，当然，当你处理这些概念时，希尔伯特旅馆的论点变得有点清晰，还有很多很多其他方式来谈论它。

For example, let’s think about, say, the integer lattice, the grid of points that you get by taking pairs of natural numbers, say, so the upper right quadrant of the integer lattice, yeah?

例如，让我们考虑一下整数格，即通过取自然数对得到的点网格，比如说，整数格的右上象限，是吗？

So there’s the, you know, row zero, row one, row two and so on, column zero, column one, column two and so on, and each row and column has a countable infinity of points on it, right?

所以，你知道，第零行、第一行、第二行等等，零列、第一列、第二列等等，每一行和每一列上都有可数无穷个点，对吗？

So those dots, if you think about them as dots, are really the same as the train cars if you think about each column in the integer lattice, it’s a countable infinity.

因此，如果你将这些点视为点，那么它们实际上与火车车厢相同，如果你考虑整数格中的每一列，它们都是可数无穷大。

20:01 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.

这就像一节火车车厢，然后旁边有下一节车厢，然后是旁边的下一列，下一节车厢。

And so…

所以……

But if we think about it in this grid manner, then I can imagine a- a kind of winding path winding through these grid points, like up and down the diagonals- …

但如果我们以这种网格方式来思考，那么我可以想象一种——一种蜿蜒穿过这些网格点的路径，就像对角线上下——……

winding back and forth.

来回缠绕。

So I start at the corner point and then I go down, up and to the left, and then down and to the right, up and to the left, down and to the right, and so on, in such a way that I’m gonna hit every grid point in- on this path.

所以我从角点开始，然后向下，向上和向左，然后向下和向右，向上和向左，向下和向右，依此类推，这样我就能击中这条路径上的每个网格点。

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

因此，这为我提供了一种为点分配房间号的方法。

20:38 Because every grid point is going to be the Nth point on that path for some N.

因为对于某个 N，每个网格点都将是该路径上的第 N 个点。

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

这就给出了网格点和自然数本身之间的对应关系。

So it’s a kind of different picture.

所以这是一种不同的景象。

I mean, before we used this 3 to the C, 5 times 5 to the S, which is a kind of, you know, overly arithmetic way to think about it.

我的意思是，在我们对 C 使用 3，对 S 使用 5 乘以 5 之前，这是一种过于算术的思考方式。

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

但是，有一种直接的方法可以理解，当您有可数个可数集合时，它仍然是可数无穷大，因为您可以开始将它们放入此列表中。

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

只要你给每个无限集合一个机会在列表中再添加一个人，那么你就可以容纳一个列表中任何集合中的每个人。

Yeah, it’s a really nice visual way to think about it.

是的，这是一种非常好的视觉思考方式。

21:20 You just zigzag your way across the grid to make sure everybody’s included, that gives you kind of an algorithm for including everybody.

你只需在网格中蜿蜒前行，以确保每个人都被包括在内，这为你提供了一种包含每个人的算法。

So can you speak to the uncountable infinities?

那么你能说出不可数的无穷大吗？

Yeah, absolutely.

是的，绝对是。

What are the integers and the real numbers-

什么是整数和实数-

Correct

正确

…

……

21:36 and what is the line that Cantor was able to find?

康托找到的线是什么？

Maybe there’s one more step I want to insert before doing that.

也许在此之前我还想插入一个步骤。

Right

右

which is the rational numbers.

这是有理数。

21:43 So we did pairs of natural numbers.

所以我们做了自然数对。

Right?

正确的？

That’s the train car, basically.

基本上，这就是火车车厢。

But maybe it’s a little bit informative to think about the rational, the fractions, the set of fractions, or rational numbers, because a lot of people maybe have an expectation that maybe this is a bigger infinity because the rational numbers are densely ordered; between any two fractions you can find another fraction, right?

但考虑有理数、分数、分数集或有理数可能会提供一些信息，因为很多人可能会期望这是一个更大的无穷大，因为有理数是密集排序的；在任意两个分数之间你可以找到另一个分数，对吧？

The average of two fractions is another fraction.

两个分数的平均值是另一个分数。

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

所以有时人们，它似乎是与离散排序的整数不同的字符，对吧？

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

对于任何整数，都有下一个和上一个，依此类推，但在有理数中却不是这样。

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

然而，有理数仍然只是可数无穷大。

22:35 And 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.

事实上，它与希尔伯特火车完全相同，因为每个分数都由两个整数组成，即分子和分母。

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

因此，如果我告诉你两个自然数，那么你就知道我在说什么分数。

I mean, plus the sign issue, I mean if it’s positive or negative.

我的意思是，加上符号问题，我的意思是它是正数还是负数。

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.

但如果你只考虑正分数，那么你就会知道，你有 P 超过 Q 形式的数字，其中 Q 不为零。

So you can still do 3 to the P times 5 to the Q; the same idea works with the rational numbers.

所以你仍然可以对 P 进行 3 次，对 Q 进行 5 次；同样的想法也适用于有理数。

So this is still a countable set.

所以这仍然是一个可数集。

And you might think, well, every set is going to be countable because there’s only one infinity.

你可能会想，好吧，每一组都是可数的，因为只有一个无穷大。

23:21 I mean if that’s a kind of perspective maybe that you’re adopting, but it’s not true, and that’s the profound achievement that Cantor made is proving that the set of real numbers is not a countable infinity.

我的意思是，如果这是一种你可能会采用的观点，但事实并非如此，康托所取得的深远成就就是证明实数集不是可数无穷大。

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

它是一个严格意义上更大的无穷大，因此有不止一种无穷大的概念，不止一种无穷大的大小。

So let’s talk about the real numbers.

那么我们来谈谈真实的数字。

23:40 What are the real numbers?

真实数字是多少？

Why do they break infinity?

为什么他们会打破无穷大？

Right.

正确的。

The countable infinity.

可数无穷大。

Right.

正确的。

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.

在 Perplexity 上查找，实数包括可以在数轴上表示的所有数字，包括有理数和无理数。

23:46 We’ve spoken about the rational numbers, and the rational numbers, by the way, are, by definition, the numbers that can be represented as a fraction of two integers.

我们已经讨论过有理数，顺便说一下，根据定义，有理数是可以表示为两个整数的分数的数字。

That’s right.

这是正确的。

24:05 So with the real numbers, we have the algebraic numbers.

因此有了实数，我们就有了代数数。

We have of course all the rational numbers.

我们当然有所有的有理数。

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 2 or the cube root of 5 and so on.

整数和有理数都是实数系的一部分，但我们也有代数数，例如 2 的平方根或 5 的立方根等。

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

求解整数代数方程的数称为代数数。

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.

长期以来，这是一个悬而未决的问题，这是否是所有实数，或者是否存在超越数。

The transcendental numbers are real numbers that are not algebraic.

超越数是非代数的实数。

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

我们甚至不会讨论超现实的数字，您有一篇精彩的博客文章介绍了这些数字。

24:38 We’ll talk about that a little bit later.

我们稍后会讨论这个问题。

Oh, great.

哦，太好了。

24:43 So it was Liouville who first proved that there are transcendental numbers, and he exhibited a very specific number that’s now known as the Liouville constant, which is a transcendental number.

所以是刘维尔第一个证明了超越数的存在，他展示了一个非常具体的数字，现在被称为刘维尔常数，这是一个超越数。

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

康托尔还证明了存在很多很多超越数。

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

事实上，从他关于实数不可数的论证可以得出，存在不可数的超越数。

So most real numbers are transcendental.

所以大多数实数都是超越数。

And again, going to Perplexity, “Transcendental numbers are real or complex numbers; they are not the root of any non-zero polynomial with integer or rational coefficients.

再次回到困惑，“超越数是实数或复数；它们不是任何具有整数或有理系数的非零多项式的根。

25:12 This means they cannot be expressed as solutions to algebraic equations with integer coefficients, setting them apart from algebraic numbers.”

这意味着它们不能表示为具有整数系数的代数方程的解，从而将它们与代数数区分开来。”

That’s right.

这是正确的。

25:29 So some of the famous transcendental numbers would include the number pi, you know, the 3.14159265 and so on.

因此，一些著名的超越数包括 pi，你知道，3.14159265 等等。

So that’s a transcendental number.

所以这是一个超越数。

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

此外，欧拉常数 e，就像 e 与 x（指数函数）一样。

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

所以你可以说数学中一些最性感的数字都是超越数？

Absolutely.

绝对地。

25:51 That’s true.

这是真的。

Yeah, yeah.

是啊是啊。

Although, you know, I don’t know, square root of two is pretty.

虽然，你知道，我不知道，二的平方根很漂亮。

Square root.

平方根。

25:56 All right.

好的。

So it depends.

所以这取决于。

Let’s not.

我们不要。

Beauty can be found in-

美可以发现在——

That’s right

没错

…in all the different kinds of sets, but yeah.

……在所有不同类型的场景中，但是是的。

That’s right.

这是正确的。

26:02 And if you have a kind of simplicity attitude, then, you know, zero and one are looking pretty good too, so…

如果你有一种简单的态度，那么，你知道，零和一看起来也不错，所以……

And they’re definitely not.

但他们绝对不是。

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

很抱歉这么切线，但是你最喜欢的数字是多少？

26:07 Do you have one?

你有吗？

Oh, gosh.

哦，天哪。

26:10 You know-

你知道-

Is it zero?

是零吗？

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

您知道有证据表明每个数字都很有趣吗？

26:13 You can prove it, because…

你可以证明这一点，因为……

Yeah?

是吗？

26:22 What’s that proof look like?

那个证明是什么样的？

Yeah, okay.

是啊，好吧。

How do you even begin?

你如何开始呢？

I’m gonna prove to you-

我要向你证明——

Okay

好的

…

……

26:26 that every natural number is interesting.

每个自然数都是有趣的。

Okay.

好的。

Yeah.

是的。

26:29 I mean, zero’s interesting because, you know, it’s the additive identity, right?

我的意思是，零很有趣，因为，你知道，它是加法恒等式，对吧？

That’s pretty interesting.

这很有趣。

And one is the multiplicative identity, so when you multiply it by any other number, you just get that number back, right?

其中一个是乘法恒等式，所以当你将它乘以任何其他数字时，你就会得到那个数字，对吗？

And two is, you know, the f- the first prime number.

你知道，二是 f——第一个素数。

That’s super interesting, right?

这非常有趣，对吧？

Okay.

好的。

So…

所以……

…One can go on this way and give specific reasons, but I want to prove as a general principle that every number is interesting.

……人们可以继续这种方式并给出具体原因，但我想证明作为一般原则，每个数字都很有趣。

And this is the proof.

这就是证据。

Suppose, toward contradiction, that there were some boring numbers.

假设存在一些无聊的数字，这可能是矛盾的。

Okay?

好的？

Okay.

好的。

But if, if there was an uninteresting number- …

但如果，如果有一个无趣的数字——……

27:04 then there would have to be a smallest uninteresting number.

那么就必须有一个最小的无趣数。

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

但这是一个矛盾，因为最小的无趣数是一个超级有趣的属性。

So therefore-

所以因此——

Ah, that’s good

啊，那就好

…there cannot be any boring numbers.

……不能有任何无聊的数字。

I’m going to have to try to find a hole in that proof, because there’s a lot of baked in in the word interesting, but yeah, that’s beautiful.

我将不得不尝试在该证明中找到一个漏洞，因为“有趣”这个词中有很多内容，但是，是的，这很漂亮。

Right.

正确的。

That doesn’t say anything about the transcendental numbers, about the real numbers that you just…

这并没有说明超越数，也没有说明你只是……的实数。

That’s right

没错

…

……

27:38 proved from just-

刚刚证明——

That’s right

没错

…

……

27:39 four natural numbers.

四个自然数。

Yeah.

是的。

27:40 Okay, so should we get back to Cantor’s argument, or?

好吧，那么我们应该回到康托的论点吗？

Sure.

当然。

Okay.

好的。

You’ve masterfully avoided the question.

你巧妙地回避了这个问题。

27:43 Well, you basically said, “I love all numbers.”

嗯，你基本上是说，“我喜欢所有的数字。”

Yeah, basically.

是的，基本上。

Is that what you said?

是你说的吗？

27:48 Okay.

好的。

All right.

好的。

That was my intention.

这就是我的意图。

Back to Cantor’s argument.

回到康托尔的论证。

27:49 Let’s go.

我们走吧。

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

好吧，康托尔想证明实数的无穷大是不同的，并且严格大于自然数的无穷大。

27:51 So the natural numbers are the numbers that start with zero and add one successively, so zero, one, two, three, and so on.

所以自然数就是从零开始依次加一的数字，所以零、一、二、三等等。

And the real numbers, as we said, are 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.

正如我们所说，实数是来自数轴的数字，包括所有整数、有理数、代数数、超越数以及所有这些数字。

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.

现在显然，由于自然数包含在实数中，所以我们知道实数至少与自然数一样大。

And so the claim that we want to prove is that it’s strictly larger.

因此，我们想要证明的是，它确实更大。

28:36 So suppose that it wasn’t strictly larger, then they would have the same size.

因此，假设它不是严格更大，那么它们将具有相同的大小。

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

但请记住，具有相同的大小，根据定义意味着它们之间存在一一对应的关系。

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

所以我们假设实数可以与自然数一一对应。

So therefore, for every natural number N, we have a real number, let’s call it R sub N.

因此，对于每个自然数 N，我们都有一个实数，我们称其为 R sub N。

R sub N is the Nth real number on the list.

R sub N 是列表中的第 N 个实数。

Basically, our assumption allows us to think of the real numbers as having been placed on a list, R1, R2, and so on.

基本上，我们的假设允许我们将实数视为已放置在列表中的 R1、R2 等。

Okay, and now I’m going to define the number Z, and it’s going to be…

好的，现在我要定义数字 Z，它将是……

The integer part is going to be a zero, and then I’m going to put a decimal place, and then I’m going to start specifying the digits of this number Z, D1, D2, D3, and so on.

整数部分将为零，然后我将放置一个小数位，然后我将开始指定该数字 Z、D1、D2、D3 等的数字。

29:28 And what I’m going to 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.

我要确定的是，Z 小数点后的第 N 位与列表中第 N 个数字的第 N 位不同。

29:40 Okay?

好的？

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.

因此，为了指定 Z 的第 N 位，我转到列表中的第 N 个数字，R sub N，然后查看其小数点后的第 N 位。

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

无论那个数字是什么，我都会确保我的数字与它不同。

Okay?

好的？

And then I want to do something a little bit more, and that is I’m going to make it different in a way that I’m never using the digits zero or nine.

然后我想做更多的事情，那就是我将以一种我从不使用数字零或九的方式让它变得不同。

I’m just always using the other digits and not zero and.

我只是总是使用其他数字而不是零和。

It would form a kind of diagonal going down and to the right, and 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.

它会形成一种向下和向右的对角线，因此，这个参数被称为对角线参数，因为我们正在查看第 n 个数字的第 N 位数字，而这些数字存在于一种向下的对角线上。

And we’ve made our number Z so that the Nth digit of Z is different from the Nth digit of the nth number.

我们将数字设为 Z，使得 Z 的第 N 位数字与第 n 个数字的第 N 位数字不同。

30:58 But now it follows that Z is not on the list because Z is different from R1 because, well, the first digit after the decimal point of Z is different from the first digit of R1 after the decimal point.

但现在 Z 不在列表中，因为 Z 与 R1 不同，因为 Z 小数点后的第一位数字与 R1 小数点后的第一位数字不同。

That’s exactly how we built it.

这正是我们构建它的方式。

And the second digit of Z is different from the second digit of R2 and so on.

Z的第二位数字与R2的第二位数字不同，依此类推。

The Nth digit of Z is different from the Nth digit of R7 for every end.

Z的第N位与R7的第N位每一端都不同。

So therefore, Z is not equal to any of these numbers R7.

因此，Z 不等于这些数字 R7 中的任何一个。

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, okay?

但这是一个矛盾，因为我们假设我们拥有列表中的每个实数，但这里有一个实数 Z 不在列表中，好吗？

And so that’s the main contradiction.

这就是主要矛盾。

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

所以这是一种构造证明。

Exactly.

确切地。

31:45 So, given a list of numbers, Cantor is proving…

所以，给定一个数字列表，康托正在证明……

It’s interesting that you say that, actually, because there’s a kind of philosophical controversy that occurs in connection with this observation about whether Cantor’s construction is constructive or not.

有趣的是，你这么说实际上是因为与关于康托尔的构造是否具有建设性的观察有关，发生了一种哲学争议。

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.

给定一个数字列表，康托尔为我们提供了一种构造列表中没有的实数的特定方法，这是一种思考它的方式。

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.

我之前提到过这一方面，但有些实数有不止一种十进制表示形式，这会导致论证中出现这个小问题。

For example, the number one, you can write it as 1.0000 forever, but you can also write it as 0.999 forever.

比如第一，你可以永远写成1.0000，但也可以永远写成0.999。

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

这是完全相同的数字的两种不同的十进制表示形式。

You beautifully got rid of the zeros and the nines.

你漂亮地摆脱了零和九。

32:38 Therefore, we don’t need to even consider that, and the proof still works.

因此，我们甚至不需要考虑这一点，证明仍然有效。

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

确切地说，因为发生这种现象的唯一情况是当数字最终为零或最终为九时。

32:44 Since our number Z never had any zeros or nines in it, it wasn’t one of those numbers.

由于我们的数字 Z 中从来没有任何零或九，所以它不是那些数字之一。

So actually, in those cases, we didn’t need to do anything special to diagonalize.

所以实际上，在这些情况下，我们不需要做任何特殊的事情来对角化。

The mere fact that our number has a unique representation already means that it’s not equal to those numbers.

我们的数字具有唯一的表示形式，这一事实本身就意味着它不等于那些数字。

So maybe it was controversial in Cantor’s day, more than 100 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.

所以也许它在 100 多年前的康托时代是有争议的，但我认为今天它最常被视为集合论的最初主要结果之一，它是深刻的、令人惊奇的、富有洞察力的，也是后来许多争论的起点。

33:24 And this diagonalization idea has proved to be an extremely fruitful proof method, and almost every major result in mathematical logic is using in an abstract way the idea of diagonalization.

而这种对角化思想已被证明是一种极其卓有成效的证明方法，几乎​​数理逻辑中的每一个重大成果都以抽象的方式使用了对角化思想。

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.

这实际上是许多其他观察结果的开始，包括罗素悖论、停止问题和递归定理。

So many other principles are using diagonalization at their core.

许多其他原则都以对角化为核心。

Can we just step back a little bit?

我们可以退后一点吗？

Sure.

当然。

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

这种无穷危机导致了数学的某种重建。

33:58 So it’d be nice if you lay out the things it resulted in.

所以如果你能列出它所产生的结果那就太好了。

One is set theory became the foundation of mathematics.

一是集合论成为数学的基础。

All mathematics could now be built from sets, giving math its first truly rigorous foundation.

现在所有数学都可以由集合构建，为数学提供了第一个真正严格的基础。

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

数学的公理化、悖论迫使数学家发展ZFC和其他公理系统，数理逻辑出现。

Gödel, Turing, and others created entirely new fields.

哥德尔、图灵和其他人创造了全新的领域。

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?

那么，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，甚至是真理的基础？

That’s a great question.

这是一个很好的问题。

34:43 Set theory really has two roles that it’s serving.

集合论确实有两个作用。

There are two ways that set theory emerges.

集合论有两种出现的方式。

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

一方面，集合论是它自己的数学学科，有它自己的问题、问答和证明方法。

So really, from this point of view, set theory is about the transfinite recursive constructions or well-founded definitions and constructions.

所以实际上，从这个角度来看，集合论是关于超限递归构造或有充分根据的定义和构造的。

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

这些想法非常富有成效，集合论学家对它们进行了研究，并从中发展出了许多想法。

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

但集合论也恰好发挥了另一个基础作用。

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.

人们经常听到有关集合论的说法，但实际上并没有考虑到它所服务的两个角色之间的区别。

35:38 It’s its own subject, but it’s also serving as a foundation of mathematics.

它是它自己的学科，但它也是数学的基础。

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

因此，集合论的基础作用是提供一种将一组事物视为一个事物的方法。

That’s the central idea of set theory.

这就是集合论的中心思想。

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

集合是事物的集合，但您将集合本身​​视为一个抽象事物。

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

所以当你形成一组实数时，那就是一个集合。

It’s one thing.

这是一回事。

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

它是一个集合，里面有元素。

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

所以它有点像一袋物体。

A set is kind of like a bag of objects.

集合有点像一袋物体。

So we have a lot of different axioms that describe the nature of this idea of thinking of a collection of things as one thing itself, one abstract thing.

因此，我们有很多不同的公理来描述将一组事物视为一个事物本身、一个抽象事物的想法的本质。

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

我想，公理是我们假设为真的事实，然后我们在此基础上建立数学思想。

36:20 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.

因此，我们可以将大量事实、关于集合的公理放在一起，如果它们足够强大，我们就可以在此基础上构建许多非常有趣的数学。

Yeah, I think that’s right.

是的，我认为这是对的。

36:42 So, the history of the current set theory axioms, known as the Zermelo-Fraenkel axioms, came out in the early 20th century with Zermelo’s idea.

因此，当前集合论公理的历史，即所谓的策梅洛-弗兰克尔公理，是随着策梅洛的想法于20世纪初问世的。

The history is quite fascinating because Zermelo in 1904 offered a proof that what’s called the axiom of choice implies the well-order principle.

这段历史非常有趣，因为策梅洛在 1904 年证明了所谓的选择公理蕴含着良序原则。

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

所以他描述了他的证明，这在当时引起了极大的争议。

There was no theory, there weren’t any axioms there.

那里没有理论，没有任何公理。

Cantor was not working in an axiomatic framework.

康托并没有在公理框架下工作。

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

他没有像我们现在的集合论那样列出公理列表，策梅洛也没有。

And his ideas were challenged so much with regard to the well-order theorem—

他的想法在良序定理方面受到了很大的挑战——

37:29 that he was pressed to produce the theory in which his argument could be formalized, and that was the origin of what’s known as Zermelo set theory.

他被迫提出一种理论，使他的论证得以形式化，这就是所谓的策梅洛集合论的起源。

And going to Perplexity, the axiom of choice is 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 choices is given.

说到困惑，选择公理是集合论中的一项基本原则，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，也可以从每个集合中精确地选择一个元素。

37:39 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.

该公理允许构造一个包含每个原始集合中的一个元素的新集合，即使在集合是无限的或没有自然方式来指定选择规则的情况下也是如此。

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

所以这是有争议的，而且在公理系统语言出现之前就已经描述过这一点。

That’s right.

这是正确的。

38:14 So on the one hand, the axiom of choice principle is completely obvious that we want this to be true, that it is true.

因此，一方面，选择原理公理是完全明显的，我们希望这是真的，它是真的。

A lot of people take it as a law of logic.

很多人将其视为逻辑法则。

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

如果你有一堆集合，那么有一种方法可以从每个集合中挑选一个元素。

There’s a function.

有一个功能。

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.

如果我有一堆集合，那么有一个函数，当您将其应用于其中任何一个集合时，都会为您提供该集合的一个元素。

It’s a completely natural principle.

这是一个完全自然的原则。

It’s called the axiom of choice, which is a way of anthropomorphizing the mathematical idea.

它被称为选择公理，是一种将数学思想拟人化的方式。

It’s not like the function is choosing something.

这不像函数正在选择某些东西。

It’s just that if you were to make such choices, there would be a function that consisted of the choices that you made.

只是如果你要做出这样的选择，就会有一个函数包含你所做的选择。

39:01 And the difficulty is that 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.

困难在于，当您无法指定做出选择的规则或程序时，就很难说出您断言存在的功能是什么。

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

你想要这样的观点：嗯，有一种选择的方式。

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

我没有一个简单的方法来说明这个功能是什么，但肯定有一个。

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

这就是思考选择公理的方式。

So we’re going to say the three letters of ZFC may be a lot in this conversation.

所以我们要说ZFC这三个字母在这次谈话中可能会很多。

39:28 You already mentioned—

你已经提到过——

Right

右

Zermelo-Fraenkel set theory.

策梅洛-弗兰克尔集合论。

39:33 The Z and the F and the C in that come from this axiom of choice.

其中的 Z、F 和 C 来自这个选择公理。

That’s right.

这是正确的。

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

所以 ZFC 听起来像是一个超级技术性的东西，但它是一组公理，是现代数学的基础。

Yeah, absolutely.

是的，绝对是。

39:49 So one should be aware also that there are huge parts of mathematics that pay attention to whether the axiom of choice is being used, and they don’t want to use the axiom of choice, so they work out the consequences that are possible without the axiom of choice or with weakened forms of Zermelo-Fraenkel set theory, and so on.

因此，人们还应该意识到，数学中有很大一部分关注是否使用选择公理，并且他们不想使用选择公理，因此他们计算出在没有选择公理或弱化形式的策梅洛-弗兰克尔集合论的情况下可能出现的后果，等等。

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

该领域有大量充满活力的工作。

But going back to the axiom of choice for a bit, it’s maybe interesting to give Russell’s description of how to think about the axiom of choice.

但稍微回到选择公理，给出罗素关于如何思考选择公理的描述可能会很有趣。

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

所以拉塞尔描述了这个拥有无限衣柜的富人。

40:31 In that closet, he has infinitely many pairs of shoes, and he tells his butler, “Please go and give me one shoe from each pair.” And the butler can do this easily because for any pair of shoes, he can just always pick the left shoe.

在那个衣柜里，他有无数双鞋子，他对他的管家说：“请去给我每双鞋子中的一只。”管家可以轻松做到这一点，因为对于任何一双鞋子，他总是可以选择左边的鞋子。

There’s a way of picking that we can describe.

我们可以描述一种选择方法。

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, you know.

我们总是走左边的，或者总是走右边的，或者如果是红色鞋子就走左边，如果是棕色鞋子就走右边，你知道。

We can invent rules that would result in these kinds of choice functions so we can describe explicit choice functions.

我们可以发明产生这些选择函数的规则，这样我们就可以描述显式的选择函数。

For those cases, you don’t need the axiom of choice to know that there’s a choice function.

对于这些情况，您不需要选择公理就知道存在选择函数。

41:14 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.

当你可以描述一种特定的选择方式时，你就不需要诉诸公理来知道存在选择函数。

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

但当你想到这个人衣柜里有无数的袜子时，就会出现有问题的情况。

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

如果我们假设每双袜子都无法区分，你知道，它们彼此匹配，但难以辨别，那么管家就没有任何规则来选择每双袜子中的哪只。

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

所以目前还不清楚他是否有办法从每双袜子中生产出一只袜子，对吧？

41:56 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.

所以这就是关键的问题，你是否可以指定一个选择函数所遵循的规则，你知道，一个它所遵循的定义选择函数的规则，或者是否存在某种任意选择的方面。

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

这时你需要选择公理来知道存在这样一个函数。

But of course, as a matter of mathematical ontology, we might find attractive the idea that, well, look, I mean, not every way of choosing the socks has to be defined by a rule.

但是，当然，作为数学本体论的问题，我们可能会发现这样一个想法很有吸引力：嗯，看，我的意思是，并不是每种选择袜子的方式都必须由规则定义。

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

为什么数学现实中存在的一切都应该遵循某种规则或程序？

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

如果我认为我的数学本体是丰富的对象，那么我认为有各种各样的函数和选择方式。

42:45 Those are all part of the mathematical reality that I want to be talking about, and so I don’t have any problem asserting the axiom of choice.

这些都是我想讨论的数学现实的一部分，所以我在断言选择公理方面没有任何问题。

Yes, there is a way of choosing, but I can’t necessarily tell you what it is.

是的，有一种选择的方式，但我不一定告诉你它是什么。

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

但在数学论证中，我可以假设我修复了选择函数，因为我知道存在一个。

So it’s a…

所以这是一个……

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.

当你有选择公理时和没有选择公理时工作之间的哲学差异在于论证的这种建设性性质的问题。

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 going to be constructive.

因此，如果你提出一个论点并诉诸选择公理，那么也许你承认你在证明中产生的对象不会具有建设性。

You’re not going to be able to necessarily say specific things about them.

你不一定能说出关于他们的具体事情。

43:30 But if you’re just claiming to make an existence claim, that’s totally fine.

但如果你只是声称存在，那完全没问题。

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 going to want to deny the axiom of choice and maybe much more.

然而，如果你对数学的本质抱有建设性的态度，并且你认为只有当你能够提供一个明确的过程来产生你正在处理的数学对象时，数学主张才可能是有道理的，那么你可能会想要否认选择公理，甚至更多。

Can we maybe speak to the axioms that underlie ZFC?

我们可以谈谈 ZFC 背后的公理吗？

43:51 So ZFC, or Zermelo-Fraenkel set theory with the axiom of choice, as we mentioned, is the standard foundation for most modern mathematics.

因此，正如我们提到的，带有选择公理的 ZFC 或 Zermelo-Fraenkel 集合论是大多数现代数学的标准基础。

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.

它由以下主要公理组成：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、正则公理和选择公理。

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

其中一些是非常基本的，但如果能给人们一种感觉那就太好了……

Sure

当然可以

of what it means to be an axiom.

成为公理意味着什么。

44:28 Like, what kind of basic facts we can lay on the table on which we can build some beautiful mathematics.

比如，我们可以将什么样的基本事实摆在桌面上，并在此基础上构建一些美丽的数学。

Yeah, so the history of it is really quite fascinating.

是的，所以它的历史真的很有趣。

44:37 So, Zermelo introduced most of these axioms, as 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，策梅洛引入了大部分公理，作为现在所谓的策梅洛集合论的一部分，以形式化他从选择公理到良序原理的证明，这是一个极具争议的结果。

So in 1904, he gave the proof without the theory, and then he was challenged to provide the theory.

所以在1904年，他在没有理论的情况下给出了证明，然后他就被要求提供理论。

And so in 1908, 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，在 1908 年，他提出了 Zermelo 集合论，并证明了在该理论中，您可以证明每个集合都具有良序性。

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

因此，列表中的公理，例如外延性，表达了他想要讨论的理解集合的最基本原则。

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

例如，外延性表示如果两个集合具有相同的成员，那么它们是相等的。

45:26 So it’s this idea that the sets consist of the collection of their members, and that’s it.

所以这个想法是集合由其成员的集合组成，就是这样。

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

剧组中没有其他事情发生。

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

因此，如果两个集合具有相同的成员，那么它们就是同一个集合。

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

所以从某些方面来说，这可能是最原始的公理。

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

好吧，为了给大家带来一点趣味，存在一个没有元素的集合，称为空集。

45:44 For any two sets, there’s a set that contains exactly those two sets as elements.

对于任何两个集合，都有一个集合恰好包含这两个集合作为元素。

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

对于任何集合，都有一个集合完全包含该集合的元素的元素，因此称为并集。

And then there’s the power set.

然后是电源组。

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

对于任何集合，都有一个集合，其元素恰好是原始集合（即幂集）的子集。

And 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.

而无穷公理，存在一个无限集合，通常是包含空集合并且在添加一个元素的操作下封闭的集合。

Back to our hotel example.

回到我们酒店的例子。

That’s right.

这是正确的。

And there’s more, but it’s kind of fascinating to put yourself in the mindset of people at the beginning of this, of trying to formalize set theory.

还有更多，但在一开始就将自己置于人们的思维方式中，尝试将集合论形式化，这有点令人着迷。

46:23 It’s fascinating that humans can do that.

人类能做到这一点真是令人着迷。

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

我读了历史学家关于那个时期的一些历史记载，特别是关于策梅洛的公理和他对良序定理的证明。

46:37 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 Zermelo’s.

历史学家们说，在数学史上，从来没有一个数学定理像策梅洛定理那样受到如此公开和激烈的争论。

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 people were very suspicious of the well-order theorem because no one could imagine a well ordering, say, of the real numbers.

这也很有趣，因为选择公理一开始被广泛认为是一种基本原则，但后来人们对良序定理非常怀疑，因为没有人能够想象出实数的良好排序。

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

因此，在这种情况下，泽梅洛似乎从看似相当合理的原则出发，证明了这个明显的谎言。

And so people, mathematicians, were objecting.

因此，人们、数学家们都表示反对。

47:30 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.

但后来策梅洛和其他人实际上研究了一些一直强烈反对的人的数学论文等，发现在很多情况下，他们在自己的论证中隐含地使用了选择公理，尽管他们会公开反对它。

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

因为它的使用是如此自然，因为它在某种程度上是一个如此明显的原则。

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

我的意思是，如果你不够挑剔，甚至没有意识到你正在使用选择公理，那么很容易偶然使用它。

That’s true now, even.

即使现在也是如此。

People like to pay attention to when the axiom of choice is used or not used in mathematical arguments, up until this day.

直到今天，人们仍然喜欢关注数学论证中何时使用或不使用选择公理。

It used to be more important.

它曾经更重要。

48:13 In the early 20th century it was very important because people didn’t know if it was a consistent theory or not, and there were these antinomies arising, and so there was a worry about consistency of the axioms.

在20世纪初，它非常重要，因为人们不知道它是否是一个一致的理论，并且出现了这些矛盾，因此担心公理的一致性。

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

但是，当然，最终，随着哥德尔和科恩等人的结果，这个专门关于选择公理的一致性问题就消失了。

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

我们知道，选择公理本身永远不会成为集合论中不一致的根源。

If there’s inconsistency with the axiom of choice, then it’s already inconsistent without the axiom of choice.

如果与选择公理不一致，那么没有选择公理就已经不一致。

So it’s not the cause of inconsistency.

所以这不是不一致的原因。

And so in that…

所以在那...

48:49 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.

从这个角度来看，从一致性的角度来看，关注是否使用它的需要在某种程度上不太重要。

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

但是，基于我之前提到的这些建构主义思想，仍然有理由关注它。

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

我们应该说，在集合论中，一致性意味着不可能从理论的公理中推导出矛盾。

49:05 So it means that there are no contradictions.

所以这意味着不存在矛盾。

That’s a…

那是一个……

That’s right

没错

a consistent axiomatic system is that there are no contradictions.

一个一致的公理系统是不存在矛盾。

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

一致的理论是指你无法证明与该理论相矛盾的理论。

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

也许是短暂的暂停、短暂的休息、短暂的上厕所。

49:23 You mentioned to me offline we were talking about Russell’s paradox and that there’s another kind of anthropomorphizable proof of uncountability.

你在线下向我提到我们正在谈论罗素悖论，还有另一种拟人化的不可数性证明。

I was wondering if you can lay that out.

我想知道你是否可以把它列出来。

Oh yeah, sure.

哦，是的，当然。

49:41 Absolutely.

绝对地。

Both Russell’s paradox and the proof.

罗素悖论和证明。

Right.

正确的。

49:44 So we talked about Cantor’s proof that the real numbers, the set of real numbers is an uncountable infinity, it’s a strictly larger infinity than the natural numbers.

所以我们讨论了康托尔的证明，即实数、实数集合是不可数无穷大，它是比自然数严格更大的无穷大。

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.

但康托实际上证明了一个更普遍的事实，即对于任何集合，该集合的幂集是一个严格更大的集合。

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

所以幂集就是包含原集合所有子集的集合。

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

因此，如果你有一个集合，并且查看其所有子集的集合，那么康托尔证明这是一个更大的集合。

They’re not equinumerous.

他们不是马类。

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

当然，总有至少与元素一样多的子集，因为对于任何元素，您都可以创建只有那个人作为成员的单例子集，对吧？

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

因此，子集的数量至少与元素的数量一样多。

50:36 But the question is whether it’s strictly more or not.

但问题是严格来说是更多还是更多。

And so Cantor reasoned like this.

所以康托尔是这样推理的。

It’s very simple.

这很简单。

It’s a kind of distilling the abstract diagonalization idea without being encumbered by the complexity of the real numbers.

这是一种抽象对角化思想的提炼，而不受实数复杂性的阻碍。

So we have a set X and we’re looking at all of its subsets.

所以我们有一个集合 X，我们正在查看它的所有子集。

That’s the power set of X.

这就是 X 的幂集。

Suppose that X and the power set of X have the same size, suppose towards contradiction, they have the same size.

假设X和X的幂集具有相同的大小，矛盾地假设它们具有相同的大小。

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

这意味着我们可以将 X 的每个个体关联到一个子集。

And so now let me define a new set.

现在让我定义一个新集合。

I mean, another set, I’m going to define it.

我的意思是，另一组，我将定义它。

Let’s call it D.

我们姑且称之为D吧。

And D is the subset of X that contains all the individuals that are not in their set.

D 是 X 的子集，其中包含不在其集合中的所有个体。

51:28 Every individual was associated with a subset of X, and I’m looking at the individuals that are not in their set.

每个人都与 X 的一个子集相关联，我正在查看不在其集合中的个人。

Maybe nobody’s like that.

也许没有人是这样的。

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.

也许 X 中没有一个元素是这样的，或者也许它们都是那样，或者也许其中一些是，一些不是。

It doesn’t really matter for the argument.

这对于争论来说并不重要。

I defined a subset D consisting of the individuals that are not in the set that’s attached to them, but that’s a perfectly good subset.

我定义了一个子集 D，其中包含不属于所附集合中的个体，但这是一个非常好的子集。

And so because of the equinumerosity, it would have to be attached to a particular individual, you know?

因此，由于马数，它必须依附于特定的个人，你知道吗？

And- Let’s call that person, it should be a name starting with D, so Diana.

还有——我们称呼那个人吧，应该是D开头的名字，所以戴安娜。

52:10 And now we ask, is Diana an element of D or not?

现在我们问，戴安娜是否是 D 的元素？

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

但如果戴安娜是 D 的一个元素，那么她就在她的集合中。

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

所以她不应该这样，因为集合 D 是不属于其集合的个体的集合。

So if Diana is in D, then she shouldn’t be.

所以如果戴安娜在 D 中，那么她就不应该在 D 中。

But if she isn’t in D, then she wouldn’t be in her set.

但如果她不在 D 组，那么她就不会出现在她的组中。

And so she should be in D.

所以她应该在D。

That’s a contradiction.

这是一个矛盾。

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

因此，子集的数量总是大于任何集合的元素数量。

And the anthropomorphizing idea is the following.

拟人化的想法如下。

I’d like to talk about it this way.

我想这样谈谈。

For any collection of people, you can form more committees from them than there are people, even if you have infinitely many people.

对于任何人的集合，即使你有无限多的人，你也可以由他们组成比实际人数还要多的委员会。

53:03 Suppose you have an infinite set of people, and what’s a committee?

假设你有无限多的人，那么什么是委员会？

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

嗯，委员会只是委员会成员的名单，即委员会的成员。

So 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.

因此，有所有的两人委员会，有所有的单人委员会，还有一个普遍的、最糟糕的委员会，每个人都参加的委员会。

Okay.

好的。

The best committee is the empty committee.

最好的委员会是空委员会。

With no members and never meets and so on.

没有成员，从不见面等等。

Or is the empty committee meeting all the time?

还是委员会会议一直空着？

I’m not sure.

我不知道。

Yeah.

是的。

53:29 That’s…

那是……

wow, that’s a profound question.

哇，这是一个深刻的问题。

And does a committee with just one member meet also?

只有一名成员的委员会也会开会吗？

Yeah.

是的。

53:35 Maybe it’s always in session.

也许它总是在会话中。

I don’t know.

我不知道。

So the claim is that there are more committees than people.

因此，可以说委员会的数量比人员的数量还多。

Okay.

好的。

Suppose not.

假设没有。

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

好吧，那么我们可以在人民和委员会之间建立联系。

So we would have a kind of…

所以我们会有一种……

every committee could be named after a person in a one-to-one way.

每个委员会都可以以一对一的方式以一个人的名字命名。

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

我并不是说这个人是否在以他们名字命名的委员会中，或者不在其中，无论如何。

Maybe sometimes that happens, sometimes it doesn’t.

也许有时会发生，有时则不会。

I don’t know.

我不知道。

It doesn’t matter.

没关系。

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.

但让我们组建一个我所说的 D 委员会，该委员会由所有不在以他们名字命名的委员会中的人组成。

54:16 Okay.

好的。

Maybe that’s everyone, maybe it’s no one, maybe it’s half the people.

也许是所有人，也许不是任何人，也许是一半人。

It doesn’t matter.

没关系。

That’s a committee, it’s a set of people.

这是一个委员会，是一群人。

And so it has to be named after someone.

所以它必须以某人的名字命名。

Let’s call that person Daniella.

我们称那个人为丹妮拉吧。

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

现在我们要问，丹妮拉是以她名字命名的委员会的成员吗？

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

好吧，如果她是，那么她不应该是，因为这是一个由不属于自己委员会的人组成的委员会。

And if she isn’t, then she should be.

如果她不是，那么她应该是。

So again, it’s a contradiction.

如此看来，这又是一个矛盾。

So when I was teaching at Oxford, one of my students came up with the following different anthropomorphization of Cantor’s argument.

因此，当我在牛津教书时，我的一位学生对康托的论点提出了以下不同的拟人化。

Let’s consider all possible fruit salads.

让我们考虑所有可能的水果沙拉。

We have a given collection of fruits.

我们有一定的水果集合。

55:07 You know, apples and oranges and grapes, whatever.

你知道，苹果、橙子和葡萄，等等。

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

水果沙拉由这些水果的一些集合组成。

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

所以就有了香蕉、梨、葡萄沙拉等等。

There’s a lot of different kinds of salad.

沙拉有很多种。

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

每一套水果都可以做成沙拉，水果沙拉。

Okay…

好吧……

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.

我们想要证明，对于任何水果集合，即使有无限多种不同种类的水果，对于任何水果集合，可能的水果沙拉比水果的数量还要多。

So if not, then you can put a one-to-one correspondence between the fruits and the fruit salads, so you could name every fruit salad after a fruit.

因此，如果没有，那么您可以在水果和水果沙拉之间建立一一对应，这样您就可以用水果来命名每个水果沙拉。

That fruit might not be in that salad, it doesn’t matter.

那种水果可能不在沙拉里，没关系。

We’re just…

我们只是……

it’s a naming, a one-to-one correspondence.

这是一种命名，一种一一对应的关系。

55:53 And then, of course, we form the diagonal salad, which consists…

然后，当然，我们制作对角沙拉，其中包括……

Of all the fruits that are not in the salad that’s named after them.

所有不在沙拉中的水果都以它们的名字命名。

And that’s a perfectly good salad.

这是一道非常好的沙拉。

It might be a kind of diet salad, if it was the empty salad, or it might be the universal salad…

它可能是一种减肥沙拉，如果它是空沙拉，或者它可能是通用沙拉......

56:12 which had all fruits in it, if all the fruits were in it.

里面有所有的水果，如果所有的水果都在里面的话。

Or it might have just some and not all.

或者它可能只是一些而不是全部。

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

因此，对角线沙拉必须以某种水果命名。

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

那么我们假设它以榴莲命名，意味着它与榴莲一一对应。

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

然后我们问，好吧，榴莲在它命名的沙拉中吗？

And if it is, then it shouldn’t be.

如果是的话，那么就不应该是。

And if it isn’t, then it should be.

如果不是，那么也应该是。

And so it’s, again, the same contradiction.

这又是同样的矛盾。

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.

因此，所有这些论点都与康托证明任何集合的幂集都大于该集合相同。

56:48 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 because if it were, then we could form the set of all sets that are not elements of themselves.

这与罗素悖论中出现的逻辑完全相同，因为罗素认为所有集合的类都不能是集合，因为如果是集合，那么我们就可以形成所有不是其自身元素的集合的集合。

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

所以基本上，罗素证明的是集合的集合多于元素的集合。

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

因为我们可以形成对角线类，你知道，所有不属于其自身元素的集合的类。

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

如果这是一个集合，那么当且仅当它不是自身的元素时，它才是自身的元素。

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

这四个论点的逻辑完全相同。

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.

所以不可能存在所有集合的类，因为如果存在，那么就必须存在所有不是其自身元素的集合的类。

57:40 But that set would be an element of itself if and only if it’s not an element of itself, which is a contradiction.

但当且仅当该集合不是自身的元素时，它才是自身的元素，这是一个矛盾。

So this is the essence of the Russell paradox.

所以这就是罗素悖论的本质。

I don’t call it the Russell paradox.

我不称其为罗素悖论。

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

事实上，当我教它时，我称之为罗素定理。

There’s no universal set.

没有通用的集合。

And it’s not really confusing anymore.

而且它不再令人困惑了。

At the time, it was very confusing, but now we’ve absorbed this nature of set theory into our fundamental understanding of how sets are, and it’s not confusing anymore.

当时，这非常令人困惑，但现在我们已经将集合论的这种性质吸收到我们对集合如何的基本理解中，并且不再令人困惑了。

I mean, the history is fascinating though, 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.

我的意思是，关于罗素悖论的历史很有趣，因为在那之前，弗雷格正在从事他的不朽工作，实施逻辑主义哲学，试图将所有数学简化为逻辑。

58:30 So Frege wanted to give an account of all of mathematics in terms of logical notions, and he was writing this monumental work and had formulated his basic principles.

因此，弗雷格想用逻辑概念来解释所有的数学，他正在写这部不朽的著作并阐述了他的基本原理。

And those principles happened to imply that for any property whatsoever, you could form the set of objects with that property.

这些原则恰好意味着，对于任何属性，您都可以使用该属性形成一组对象。

This is known as the general comprehension principle.

这就是所谓的一般理解原则。

And he was appealing to the principles that support that axiom throughout his work.

他在整个工作中都诉诸支持这一公理的原则。

I mean, it was really…

我的意思是，那真的是……

It wasn’t just an incidental thing, he was really using this principle.

这不只是偶然的事情，他确实在运用这个道理。

59:11 And Russell wrote him a letter when he observed the work in progress, that there was this problem, because if you accept the principle that for any property whatsoever you can make a set of objects with that property, then you could form the set of all sets that are not members of themselves.

罗素在观察正在进行的工作时给他写了一封信，说存在这个问题，因为如果你接受这样的原则，即对于任何属性，无论你可以用该属性创建一组对象，那么你就可以形成所有不属于其自身成员的集合的集合。

That’s just an instance of the general comprehension principle.

这只是一般理解原则的一个例子。

And…

还有……

But the set of all sets that aren’t elements of themselves can’t be a set, because if it were, then it would be an element of itself if and only if it’s not a member of itself, and that’s a contradiction.

但是，所有不是自身元素的集合组成的集合不能是集合，因为如果是，那么当且仅当它不是自身的成员时，它才是自身的元素，这是一个矛盾。

And so Russell wrote this letter to Frege, and it was just at the moment when Frege was finishing his work.

于是罗素给弗雷格写了这封信，而此时正是弗雷格完成他的工作的时候。

It was already at the publishers and, you know, in press basically.

它已经在出版商那里了，你知道，基本上已经在印刷中了。

But it’s completely devastating.

但这完全是毁灭性的。

59:58 I mean, it must have been such a horrible situation for Frege to be placed in, because he’s finished this monumental work, you know, years of his life dedicated to this, and Russell finds this basically one-line proof of a contradiction in the fundamental principles of the thesis that completely destroys the whole system.

我的意思是，对于弗雷格来说，这一定是一个非常可怕的处境，因为他完成了这项具有里程碑意义的工作，你知道，他一生致力于此，而罗素发现这个基本上单行的证明完全摧毁了整个系统的论文基本原理中的矛盾。

And Frege had put in the appendix of his work a response to Russell’s letter in which he explained what happened, and he wrote very gracefully, “Hardly anything more unwelcome can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished.

弗雷格在他的著作的附录中对罗素的信作出了回应，他在信中解释了所发生的事情，他非常优雅地写道：“对于一名科学作家来说，没有什么比在著作完成后他的大厦的根基发生动摇更不受欢迎的事情了。

This is the position into which I was put by a letter from Mr.

这就是先生的一封信给我安排的位置。

1:00:46 Bertrand Russell as the printing of this volume was nearing completion.” And then he goes on to explain the matter, it concerns his basic law five and so on.

伯特兰·罗素 (Bertrand Russell) 表示，该书的印刷已接近完成。然后他接着解释这件事，涉及到他的基本法五等等。

It’s heartbreaking.

真是令人心碎。

1:00:54 I mean, there’s nothing more traumatic to a person who dreams of constructing mathematics all from logic, to get a very clean, simple contradiction.

我的意思是，对于一个梦想完全从逻辑构建数学的人来说，没有什么比得到一个非常干净、简单的矛盾更令人痛苦的了。

I mean, that’s just…

我的意思是，那只是……

You devote your life to…

你将一生奉献给……

1:01:08 This work, and then it’s shown to be contradictory, and that must have been heartbreaking.

这项工作，然后被证明是矛盾的，这一定令人心碎。

What do you think about the Frege project, the philosophy of logic, the dream of the power of logic…

您如何看待弗雷格计划、逻辑哲学、逻辑力量的梦想……

1:01:16 To construct a mathematical universe?

构建一个数学宇宙？

So, of course, the project of logicism did not die with Frege, and it was continued, and, you know, there’s a whole movement, the neologicists and so on, in contemporary times even.

所以，当然，逻辑主义的计划并没有随着弗雷格而消亡，它仍在继续，而且，你知道，甚至在当代，还有一个完整的运动，新逻辑主义者等等。

1:01:24 But my view of the matter is that really, we should view the main goals of logicism are basically completely fulfilled in the rise of set-theoretic foundationalism.

但我对此事的看法是，实际上，我们应该认为逻辑主义的主要目标在集合论基础主义的兴起中基本上完全实现了。

I mean, when you view ZFC as the foundation of mathematics, and in my view, the principles of ZFC are fundamentally logical in character, including the axiom of choice, as I mentioned, as a principle of logic.

我的意思是，当你将 ZFC 视为数学的基础时，在我看来，ZFC 的原理本质上是逻辑性质的，包括选择公理，正如我提到的，作为逻辑原理。

This is a highly disputed point of view, though, because a lot of people take even the axiom of infinity as mathematical, inherently mathematical and not logical and so on.

不过，这是一个备受争议的观点，因为很多人甚至将无穷大公理视为数学，本质上是数学的而不是逻辑的等等。

1:02:14 But I think if you adopt the view that the principles of ZFC have to do with the principles of abstract, you know, set formation, which is fundamentally logical in character, then it’s complete success for logicism.

但我认为，如果你认为 ZFC 的原则与抽象的原则有关，你知道，集合形成的原则在本质上是逻辑的，那么逻辑主义就完全成功了。

So the fact that set theory is able to serve as a foundation means that mathematics can be founded on logic.

所以集合论能够作为基础意味着数学可以建立在逻辑之上。

I think this is a good moment to talk about Gödel’s incompleteness theorems.

我认为现在是谈论哥德尔不完备定理的好时机。

1:02:35 So, can you explain them and what do they teach us about the nature of mathematical truth?

那么，你能解释一下它们吗？它们教给我们什么关于数学真理的本质？

Absolutely.

绝对地。

1:02:47 It’s one of the most profound developments in mathematical logic.

这是数理逻辑最深刻的发展之一。

I mean, the incompleteness theorems are when mathematical logic, in my view, first became sophisticated.

我的意思是，在我看来，不完备性定理是数理逻辑首次变得复杂的时候。

It’s a kind of birth of the subject of mathematical logic.

这是数理逻辑学科的诞生。

But to understand the theorems, you really have to start a little bit earlier with Hilbert’s program because at that time, you know, with the Russell Paradox and so on, there were these various contradictions popping up in various parts of set theory and the Burali-Forti paradox and so on.

但要理解这些定理，你确实必须早一点开始了解希尔伯特纲领，因为当时，你知道，随着罗素悖论等的出现，集合论和布拉里-福蒂悖论等的各个部分都出现了各种矛盾。

And Hilbert was famously supportive of set theory.

众所周知，希尔伯特对集合论的支持。

1:03:25 I mean, there’s this quote of him saying, “No one shall cast us from the paradise that Cantor has created for us.” And what I take him to mean by that is he was so captured by the idea of using set theory as a foundation of mathematics, and it was so powerful and convenient and unifying in a way that was extremely important.

我的意思是，他引用了这句话：“没有人会把我们从康托尔为我们创造的天堂中赶走。”我认为他的意思是，他对使用集合论作为数学基础的想法如此着迷，它如此强大、方便，并且以一种极其重要的方式统一。

And he didn’t want to give that up, despite the danger of these paradoxes, these contradictions, basically, is how some people viewed them.

他不想放弃这一点，尽管存在这些悖论的危险，这些矛盾基本上就是一些人的看法。

And so, it’s…

所以，这是……

This minefield of paradoxes.

这是一个充满悖论的雷区。

1:03:58 Yeah.

是的。

Right.

正确的。

1:04:00 A minefield.

一个雷区。

That’s a really good way of describing the situation.

这是描述情况的一个很好的方式。

And so Hilbert said, “Well, look, we have to fix this problem, you know.

所以希尔伯特说：“好吧，你看，我们必须解决这个问题，你知道。

We want to use the set theory foundations, but we want to do it in a way that is trustworthy and reliable.

我们想要使用集合论基础，但我们希望以一种值得信赖和可靠的方式来做到这一点。

We can’t allow that the foundations of mathematics are in question, you know.” This is a kind of attitude, I think, that underlies Hilbert and the Hilbert program.

你知道，我们不能允许数学基础受到质疑。”我认为，这是希尔伯特和希尔伯特纲领背后的一种态度。

And so he proposed, “Look, we’re going to have this strong theory, this set theory that we want to be proving our theorems in.

所以他提出，“看，我们将拥有这个强大的理论，这个集合论，我们想用它来证明我们的定理。

But on the one hand, we want it to be as strong as possible.

但一方面，我们希望它尽可能强大。

1:04:40 We would like it to answer all the questions.” There’s another famous quote of Hilbert in his retirement address where he proclaims, “Wir müssen wissen, wir werden wissen,” so, “We must know, we will know,” in which he’s very optimistic about the ability of mathematics to answer all of the questions of mathematics that we have posed.

我们希望它能够回答所有问题。”希尔伯特在他的退休演说中还有另一句名言，他宣称：“Wir müssen wissen, wir werden wissen”，即“我们必须知道，我们将会知道”，在这句话中，他对数学回答我们提出的所有数学问题的能力非常乐观。

We have all these problems we want to solve, and he is saying, “We’re going to do it.

我们有所有这些问题想要解决，他说：“我们会解决的。

We’re going to solve all these problems.” So we want to propose this strong theory, and one has the sense that he had in mind set theory in which all the questions are going to be answered.

我们将解决所有这些问题。”因此，我们想提出这一强有力的理论，人们会感觉他心中已经有了一套理论，所有问题都将在其中得到解答。

Okay?

好的？

1:05:21 But secondly, we want to combine that with a very weak arithmetic, purely finitistic theory; we want to prove that the reasoning process of the strong theory is safe.

但其次，我们想将其与非常弱的算术、纯粹有限论理论结合起来；我们要证明强理论的推理过程是安全的。

Okay?

好的？

So in order to make sense of that point of view, you basically have to invent the philosophy of formalism where we can look at what a proof is, what is the nature of mathematical reasoning.

因此，为了理解这种观点，你基本上必须发明形式主义哲学，我们可以在其中看看证明是什么，数学推理的本质是什么。

And on Hilbert’s way of thinking about this, a proof is basically itself a finitistic kind of object.

根据希尔伯特的思考方式，证明本身基本上就是一种有限性的对象。

It’s a sequence of…

这是一系列...

If you think about the nature of what a proof is, it’s a sequence of assertions which can be viewed as sort of sequences of symbols that conform with certain rules of logical reasoning.

如果你考虑一下证明的本质，它就是一系列断言，可以被视为符合某些逻辑推理规则的符号序列。

And this is a formalist way of understanding the nature of proof.

这是理解证明本质的形式主义方式。

1:06:16 So we think about a proof in a kind of syntactic, formal way.

所以我们以一种句法、形式化的方式来思考证明。

Even though the contents of those statements might be referring to infinite uncountable objects, the statements themselves are not infinite uncountable objects.

尽管这些语句的内容可能引用无限不可数对象，但语句本身并不是无限不可数对象。

The statements themselves are just finite sequences of symbols.

语句本身只是有限的符号序列。

So when you kind of think of proof as…

所以当你认为证据是......

1:06:33 Maybe it’s fair to say almost, like, outside of math?

也许可以公平地说，几乎在数学之外？

It’s, like, tools operating on math.

就像数学运算的工具一样。

And then for Hilbert, he thought proof is inside the axiomatic system.

然后对于希尔伯特来说，他认为证明存在于公理系统内部。

Something like this.

像这样的东西。

Yeah, that’s helpful.

是的，这很有帮助。

That’s wild.

那太疯狂了。

The main thing about formalism is that you think of the process of doing mathematics.

形式主义最重要的是你思考做数学的过程。

1:06:48 You divorce it from the meaning of the mathematical assertions, right?

你把它与数学断言的意义分开了，对吗？

So the meaning of the mathematical assertions that you make in this infinitary theory has to do with these huge uncountable infinities and so on, possibly.

因此，您在这个无限理论中所做的数学断言的含义可能与这些巨大的不可数无穷等有关。

And that’s a very sort of uncertain realm, maybe, and the source of the paradoxes and so on in some people’s minds.

也许，这是一个非常不确定的领域，也是某些人头脑中悖论等的根源。

But the reasoning process itself consists of writing down sequences of symbols on your page and, you know, undertaking an argument with them which is following these finitary rules.

但是推理过程本身包括在页面上写下符号序列，并且您知道，与遵循这些有限规则的符号进行论证。

And so, if we divorce the meaning of the symbols from just the process of manipulating the symbols, it’s a way of looking at the nature of mathematics as a kind of formal game in which…

因此，如果我们将符号的含义与操作符号的过程分开，那就可以将数学的本质视为一种形式游戏，其中……

1:07:43 the meaning may be totally absent.

意义可能完全不存在。

I don’t think it’s necessarily part of the formalist view that there is no meaning behind, but rather it’s emphasizing that we can divorce the meaning of the sentences from the process of manipulating those sentences.

我认为这不一定是形式主义观点的一部分，即背后没有意义，而是强调我们可以将句子的含义与操纵这些句子的过程分开。

And then Hilbert wanted to prove in this purely finitary theory that if we follow the rules of that game, we’re never going to get a contradiction.

然后希尔伯特想在这个纯粹有限的理论中证明，如果我们遵循游戏规则，我们永远不会遇到矛盾。

So those were the two aims of the Hilbert program: to found the strong infinitary theory, probably set theory, which is going to answer all the questions.

这就是希尔伯特纲领的两个目标：找到强无穷理论，可能是集合论，它将回答所有问题。

And then secondly, prove in the finitary theory that the strong theory is safe.

其次，在有限理论中证明强理论是安全的。

In other words, consistent, yeah?

换句话说，一致，是吗？

What does the word “finitary” in finitary theory mean?

有限理论中的“有限”一词是什么意思？

Yeah.

是的。

1:08:34 Well, this is, of course, philosophically contentious, and people have different ideas about what exactly it should mean.

当然，这在哲学上是有争议的，人们对于它到底意味着什么有不同的想法。

And so there’s hundreds of papers on exactly that question.

因此，有数百篇论文专门讨论了这个问题。

But I like to take it just kind of informally.

但我喜欢以非正式的方式看待它。

I mean, it means that we’re talking about finite sequences of symbols, and we’re going to have a theory, you know, finite strings of symbols.

我的意思是，这意味着我们正在谈论有限的符号序列，并且我们将有一个理论，你知道，有限的符号串。

And a finitary theory would be one whose subject matter is about those kinds of things so that we can conceivably argue about the nature of these finite strings.

有限理论的主题就是关于这些事情，这样我们就可以对这些有限弦的性质进行争论。

1:09:05 So a- a proof is just a finite sequence of statements, so that every statement is either one of the axioms or follows by the laws of logic from the earlier statements in some specified manner, like using modus ponens or some other law of logic like that.

因此，证明只是陈述的有限序列，因此每个陈述要么是公理之一，要么以某种特定的方式遵循早期陈述的逻辑定律，例如使用肯定前件或类似的其他逻辑定律。

And such that the last line on the list is, you know, the theorem that you’re proving.

列表中的最后一行就是您要证明的定理。

So that’s what a proof is in this kind of way of thinking.

这就是这种思维方式下的证明。

1:09:29 To take a specific example, I mean, I always conceive of the, perhaps the most natural finitary theory that one would be called upon to exhibit would be Peano arithmetic, the theory of Peano arithmetic, which is a first order theory of the nature of arithmetic.

举一个具体的例子，我的意思是，我总是设想，也许人们被要求展示的最自然的有限理论是皮亚诺算术，皮亚诺算术的理论，它是算术本质的一阶理论。

But okay, so some people say, “Well, Peano arithmetic has these strong first order induction axioms, and there’s much, much weaker versions of arithmetic, like I-sigma-naught or I-sigma-1 and so on, which are even more finitary than Peano arithmetic.” So different philosophical positions take different attitudes about what does it take to be finitary?

但是好吧，所以有人说，“嗯，皮亚诺算术有这些强大的一阶归纳公理，而且还有很多很弱的算术版本，比如 I-sigma-naught 或 I-sigma-1 等等，它们甚至比皮亚诺算术更有限。”那么不同的哲学立场对有限性采取不同的态度呢？

How finitary do you have to be to be truly finitary?

你必须有多有限才能成为真正的有限？

So according to Perplexity, Peano arithmetic is a foundational system for formalizing the properties and operations of natural numbers using a set of axioms called the Peano axioms.

因此，根据 Perplexity 的说法，皮亚诺算术是一个基础系统，它使用一组称为皮亚诺公理的公理来形式化自然数的属性和运算。

1:10:10 Peano arithmetic provides a formal language and axioms for arithmetic operations, such as addition and multiplication over the natural numbers.

皮亚诺算术为算术运算（例如自然数的加法和乘法）提供了形式语言和公理。

The axioms define the existence of a first natural number, usually zero or one; the concept of successor function, which generates the next natural number; rules for addition and multiplication built from these concepts; the principle of induction allowing proofs around all natural numbers.

公理定义了第一个自然数的存在，通常是零或一；后继函数的概念，生成下一个自然数；从这些概念构建的加法和乘法规则；归纳法原理允许围绕所有自然数进行证明。

And it goes on.

它还在继续。

So it’s a very particular kind of arithmetic that is a finitary.

所以这是一种非常特殊的算术，即有限算术。

You know, I view it as finitary, but this is a contentious view.

你知道，我认为这是有限的，但这是一个有争议的观点。

1:10:51 Not everyone agrees with that.

并非所有人都同意这一点。

That’s what I was trying to hint at.

这就是我试图暗示的。

Okay.

好的。

1:10:58 I got it.

我得到了它。

All right.

好的。

Peano arithmetic is one of the hugely successful theories of the natural numbers and elementary number theory.

皮亚诺算术是自然数和初等数论中非常成功的理论之一。

Essentially, all of classical number theory, whatever kind of theorems you want to be proving about the prime numbers or factorization or any kind of finitary reasoning about finite combinatorial objects, all of it can be formalized in Peano arithmetic.

本质上，所有经典数论，无论你想证明什么类型的关于素数或分解的定理，或者任何类型的关于有限组合对象的有限推理，所有这些都可以用皮亚诺算术形式化。

That’s the basic situation.

这就是基本情况。

Of course, one has to qualify those statements in light of Gödel’s incompleteness theorem, but for the most part, the classical number theoretic analysis of the finite numbers is almost entirely developable inside Peano arithmetic.

当然，人们必须根据哥德尔不完备性定理来限定这些陈述，但在大多数情况下，有限数的经典数论分析几乎完全可以在皮亚诺算术中展开。

1:11:45 So if we go back to the Hilbert program, Hilbert has these two goals: produce a strong theory which is going to answer all the questions, and then prove by purely finitary means that that theory will never lead into contradiction.

因此，如果我们回到希尔伯特纲领，希尔伯特有两个目标：产生一个能够回答所有问题的强有力的理论，然后通过纯粹有限的方法证明该理论永远不会导致矛盾。

And one can think about, well, the incompleteness theorem should be viewed as a decisive refutation of the Hilbert program.

人们可以想一想，不完备性定理应该被视为对希尔伯特纲领的决定性反驳。

It defeats both of those goals decisively, completely.

它彻底地、彻底地挫败了这两个目标。

But before explaining that, maybe one should think about, you know, what if Hilbert had been right?

但在解释这一点之前，也许人们应该想一想，如果希尔伯特是对的怎么办？

What would be the nature of mathematics in the world that Hilbert is telling us to search for?

希尔伯特告诉我们要寻找的世界中数学的本质是什么？

And if I may, going to Perplexity’s definition of Hilbert’s program, it was David Hilbert’s early 20th-century project to give all of classical mathematics a completely secure finitary foundation.

如果可以的话，我想谈谈 Perplexity 对希尔伯特纲领的定义，这是大卫希尔伯特在 20 世纪初的一个项目，旨在为所有经典数学提供一个完全安全的有限基础。

1:12:24 In essence, the goal was to formalize all of mathematics in precise axiomatic systems and then prove using only very elementary finitary reasoning about symbols that these systems are free of contradiction.

从本质上讲，其目标是将所有数学形式化为精确的公理系统，然后仅使用关于符号的非常基本的有限推理来证明这些系统不存在矛盾。

Right, exactly right.

对，完全正确。

1:12:51 Let’s imagine what it would be like if he had been right.

让我们想象一下，如果他是对的，那会是什么样子。

So we would have this finitary theory, and it would prove that the strong theory was free of contradiction.

这样我们就有了这个有限理论，它将证明强理论是不矛盾的。

So we could start enumerating proofs from the strong theory.

所以我们可以开始从强理论中列举证明。

I mean, right now, we can write a computer program that would systematically generate all possible proofs from a given theory.

我的意思是，现在我们可以编写一个计算机程序，根据给定的理论系统地生成所有可能的证明。

And so we could have this theorem enumeration machine that just spit out theorems all day long in such a manner that every single theorem would eventually be produced by this device.

因此，我们可以拥有这个定理枚举机，它整天都以这样的方式吐出定理，以至于每个定理最终都会由该设备产生。

And so if you had a mathematical question of any kind, you could answer it by just waiting for either the answer to come out yes or the answer to come out no.

因此，如果您有任何类型的数学问题，您只需等待答案为“是”或答案为“否”即可回答。

1:13:51 So the nature of mathematical investigation in Hilbert’s world is one of just turning the crank of the theorem enumeration machine.

因此，希尔伯特世界中数学研究的本质就是转动定理枚举机的曲柄。

Devoid of creative thinking or imagination, it’s just getting the answer by rote procedure.

缺乏创造性思维或想象力，只是通过死记硬背的方式得到答案。

So Hilbert, in effect, is telling us, with his program, that the fundamental nature of mathematics is rote computation.

因此，希尔伯特实际上是通过他的程序告诉我们，数学的基本本质是死记硬背。

I mean, the way I think about the Hilbert program seems extremely attractive in the historical context of being worried about the antinomies, the inconsistencies, and so how can we kind of block them.

我的意思是，在担心自相矛盾、不一致的历史背景下，我对希尔伯特纲领的思考方式似乎非常有吸引力，所以我们如何才能阻止它们。

And so-

所以——

1:14:31 It seems natural, first of all, to have a strong theory that’s going to answer all the questions, because the idea of logical independence and pervasiveness that we now know exists just wasn’t, you know, there was no known…

首先，拥有一个能够回答所有问题的强有力的理论似乎是很自然的，因为我们现在知道存在的逻辑独立性和普遍性的想法并不存在，你知道，没有已知的……

They didn’t know anything like that happening ever.

他们不知道曾经发生过这样的事情。

And so it’s natural to think that it wouldn’t happen, and also that they would be able to guard against this inconsistency.

因此，很自然地认为这种情况不会发生，而且他们能够防范这种不一致的情况。

So it seems like the goals of the Hilbert program are quite natural in that historical context.

因此，希尔伯特纲领的目标在当时的历史背景下似乎是很自然的。

But, you know, when you think a little more about what the nature of it would be like, it shows you this kind of rote procedure.

但是，你知道，当你更多地思考它的本质时，它会向你展示这种死记硬背的过程。

1:15:09 And now you’re saying, well, that doesn’t seem so unlikely maybe, in the light of increasing computer power and so on, it’s actually maybe turning into our everyday experience, where the machines are calculating more and more for us and in a way that could be alarming.

现在你会说，好吧，考虑到计算机能力的不断增强等，这似乎不太可能，它实际上可能会变成我们的日常体验，机器正在为我们计算越来越多的数据，并且以一种可能令人担忧的方式进行计算。

Okay, but to talk about the alternative to the Hilbert point of view, I mean, if he’s wrong, then what is the nature of mathematical reality?

好吧，但是要谈论希尔伯特观点的替代方案，我的意思是，如果他错了，那么数学现实的本质是什么？

Well, it would mean that we couldn’t ever maybe, for the first goal, we couldn’t ever write down a theory that answered all the questions.

好吧，这意味着我们永远不可能，对于第一个目标，我们永远无法写下一个可以回答所有问题的理论。

So we would always be in a situation where our best theory, even the infinitary theories, would have questions that they stumble with and are unable to answer.

因此，我们总是处于这样一种情况：我们最好的理论，甚至是无限理论，都会遇到一些问题，而且无法回答。

Independence would occur.

独立将会发生。

1:16:02 But then also, because of the failure of the second goal, we would also have to be constantly worrying about whether our theories were consistent or not, and we wouldn’t have any truly convincing means of saying that they were free from contradiction.

但同时，由于第二个目标的失败，我们也不得不不断担心我们的理论是否一致，并且我们没有任何真正令人信服的手段来证明它们不存在矛盾。

And the fact of Gödel’s Incompleteness Theorem shows that that is exactly the nature of mathematical reality, actually.

哥德尔不完备定理的事实表明，这实际上正是数学现实的本质。

Those are the two incompleteness theorems.

这就是两个不完备性定理。

So the first incompleteness theorem says you cannot write down a computably axiomatizable theory that answers all the questions.

因此，第一个不完备性定理表明，你无法写出一个可计算公理化的理论来回答所有问题。

Every such theory will be incomplete, assuming it includes a certain amount of arithmetic.

假设每一个这样的理论都包含一定量的算术，那么它都是不完整的。

And secondly, no such theory can ever prove its own consistency.

其次，任何这样的理论都无法证明其自身的一致性。

1:16:47 So not only is it the case that the finitary theory can’t prove the consistency of the strong infinitary theory, but even the infinitary theory can’t prove its own consistency, right?

那么不仅有限理论无法证明强无限理论的一致性，甚至无限理论也无法证明其自身的一致性，对吗？

That’s the second incompleteness theorem.

这就是第二个不完备性定理。

And so it’s, in that sense, a decisive takedown of the Hilbert program, which is really quite remarkable, the extent to which his theorem just really answered that whole puzzle.

因此，从这个意义上说，这是对希尔伯特纲领的决定性推翻，这确实非常了不起，他的定理在多大程度上真正回答了整个难题。

It’s quite amazing.

这真是太神奇了。

There’s another aspect, kind of easy to think about.

还有另一个方面，很容易思考。

I mean, if you’re wondering about theories that prove their own consistency, then would you trust a theory that proves of itself that it’s consistent?

我的意思是，如果你想知道能够证明其自身一致性的理论，那么你会相信一个能够证明其自身一致性的理论吗？

I mean, that’s like…

我的意思是，这就像……

1:17:35 It’s like the used car salesman telling you, “Oh, I’m trustworthy.” I mean, it’s not a reason to trust the used car salesman, is it?

这就像二手车销售员告诉你：“哦，我值得信赖。”我的意思是，这不是信任二手车销售员的理由，不是吗？

Just because he says that.

只因为他这么说。

So similarly, if you have a theory that proves its own consistency, well, I mean, even an inconsistent theory would prove its own consistency.

同样，如果你有一个理论可以证明其自身的一致性，那么，我的意思是，即使是一个不一致的理论也会证明其自身的一致性。

And so it doesn’t seem to be a logical reason to believe in the consistency, if you have a theory that proves itself consistent.

因此，如果你有一个理论证明自己是一致的，那么相信一致性似乎就不是一个合乎逻辑的理由。

Just for clarification, you used the word theory.

只是为了澄清起见，您使用了“理论”一词。

1:17:59 Is it, in this context, synonymous with axiomatic system?

在这种情况下，它是公理系统的同义词吗？

Right.

正确的。

1:18:07 So in mathematical logic, “theory” is a technical term.

所以在数理逻辑中，“理论”是一个技术术语。

And it means any set of sentences in a formal language.

它指的是正式语言中的任何一组句子。

And so if you say “axiomatic system,” it’s basically synonymous to my usage with theory.

因此，如果你说“公理系统”，它基本上与我对理论的使用同义。

So a theory means, you know, the consequences of a set of axioms or…

所以，你知道，理论意味着一组公理的后果或……

People are sometimes unclear on whether they just mean the axioms or the consequences of the axioms, but…

人们有时不清楚他们只是指公理还是公理的后果，但是……

So theory includes both the axioms and the consequences of the axioms, and you use it interchangeably and the context is supposed to help you figure out which of the two you’re talking about?

因此，理论包括公理和公理的后果，并且您可以互换使用它，并且上下文应该可以帮助您弄清楚您正在谈论的两者中的哪一个？

1:18:29 The axioms or the consequences?

公理还是后果？

Or maybe to you, they’re basically the same?

或者也许对你来说，它们基本上是相同的？

Yeah, well, they’re so closely connected, although, you know, all the features aren’t the same.

是的，嗯，它们是如此紧密地联系在一起，尽管，你知道，所有的功能并不相同。

1:18:44 So if you have a computable list of axioms for a theory, then you can start enumerating the consequences of the axioms, but you won’t be able to computably decide whether a given statement is a consequence or not.

因此，如果你有一个理论的可计算公理列表，那么你可以开始枚举公理的结果，但你将无法通过计算来确定给定的陈述是否是结果。

You can enumerate the consequences, so you can semi-decide the consequences, but you won’t be able to decide yes or no whether a given statement is a consequence or not.

你可以列举后果，这样你就可以半决定后果，但你无法决定给定的陈述是否是后果。

So it’s the distinction between a problem being computably decidable and a problem being computably enumerable, which was made clear following the work of Turing and others that came from that.

因此，这是可计算可判定的问题和可计算可枚举的问题之间的区别，这是在图灵和其他人的工作之后明确的。

I mean, it…

我的意思是，它……

1:19:30 So that’s one difference between the list of axioms of the theory and the theory itself.

这就是理论公理列表和理论本身之间的一个区别。

The axioms could be…

公理可能是……

You can decide, maybe computably, whether something is an axiom or not, but that doesn’t mean that you can decide computably whether or not something is a theorem or not.

你可以通过计算来决定某件事是否是公理，但这并不意味着你可以通过计算来决定某件事是否是定理。

Usually, you only get to decide the positive instances.

通常，您只能决定积极的情况。

If something is a theorem, you will eventually come to recognize that, but if something isn’t a theorem, maybe at no point will you be able to say, “No, that’s not a theorem.”

如果某件事是定理，你最终会认识到这一点，但如果某事不是定理，也许你永远无法说：“不，那不是定理。”

And that’s of course connected to the halting problem- …

这当然与停机问题有关——……

1:19:56 and all of these-…

而所有这些——……

all of these contradictions and paradoxes are all nicely beautifully interconnected.

所有这些矛盾和悖论都完美地相互关联。

That’s right.

这是正确的。

1:20:04 Absolutely.

绝对地。

So, can we just linger on Gödel’s incompleteness theorem?

那么，我们能不能只停留在哥德尔不完备定理上呢？

Sure.

当然。

You mentioned the two components there.

You mentioned the two components there.

1:20:09 You know, there are so many questions to ask.

You know, there are so many questions to ask.

Like, what is the difference between provability and truth?

Like, what is the difference between provability and truth?

What is true and what is provable?

What is true and what is provable?

Maybe that’s a good line to draw.

Maybe that’s a good line to draw.

Yeah, this is a really core distinction that it’s fascinating to me to go back and read even the early 20th-century people before Gödel and Tarski, and they were totally sloppy about this distinction between truth and proof.

是的，这是一个非常核心的区别，让我很着迷的是回顾甚至阅读哥德尔和塔斯基之前的 20 世纪早期的人们，他们对真理和证明之间的区别完全草率。

1:20:21 It wasn’t clear at all until Gödel, basically.

It wasn’t clear at all until Gödel, basically.

Although even as late as Bourbaki has a kind of confusion in this foundational work, so this standard graduate-level textbooks used in France in the presentation of logic, they are conflating truth and proof.

尽管即使晚到布尔巴基在这项基础性工作中也存在某种混乱，所以法国使用的这套标准研究生水平教科书在逻辑的表述中，他们将真理和证明混为一谈。

To be true for them means to be provable.

To be true for them means to be provable.

So in the early days, maybe it wasn’t clear enough that the concept of truth needed a mathematical investigation or analysis.

So in the early days, maybe it wasn’t clear enough that the concept of truth needed a mathematical investigation or analysis.

Maybe it was already taken to be fully clear.

Maybe it was already taken to be fully clear.

1:21:19 But because of the incompleteness theorem, we realized that actually there are quite subtle things happening, right?

But because of the incompleteness theorem, we realized that actually there are quite subtle things happening, right?

And so why don’t we talk about this distinction a bit?

And so why don’t we talk about this distinction a bit?

To me, it’s absolutely core and fundamental to our understanding of mathematical logic now.

对我来说，它绝对是我们现在理解数理逻辑的核心和基础。

1:21:34 …this distinction between truth and proof.

…this distinction between truth and proof.

So truth is on the semantic side of the syntax-semantics dichotomy.

因此，真理位于句法语义二分法的语义方面。

Truth has to do with the nature of reality.

真理与现实的本质有关。

I mean, okay, when I talk about reality, I’m not talking about physical reality.

I mean, okay, when I talk about reality, I’m not talking about physical reality.

I’m talking about mathematical reality.

我说的是数学现实。

So we have a concept of something being true in a structure, a statement being true in a mathematical structure.

So we have a concept of something being true in a structure, a statement being true in a mathematical structure.

Like maybe you have the real field or something, and you want to know, does it satisfy this statement or that statement?

Like maybe you have the real field or something, and you want to know, does it satisfy this statement or that statement?

Or you have a group of some kind, or maybe you have a graph.

Or you have a group of some kind, or maybe you have a graph.

This is a particular kind of mathematical structure that has a bunch of vertices and edges, and you want to know, does this graph satisfy that statement?

这是一种特殊的数学结构，有一堆顶点和边，你想知道，这个图是否满足该陈述？

1:22:19 And Tarski gave this absolutely wonderful account of the nature of truth in what’s now known as the disquotational theory of truth.

塔斯基在现在被称为真理的非引用理论中对真理的本质做出了绝对精彩的描述。

And what Tarski says is the sentence, quote, “Snow is white,” unquote, is true if and only if snow is white.

塔斯基所说的是这句话，引用，“雪是白的”，未引用的，当且仅当雪是白的时才是正确的。

And what he means by that is…

他的意思是……

Look, to say truth is a property of an assertion, so we can think of the assertion as it syntactically.

看，说真值是断言的一个属性，所以我们可以从语法上考虑断言。

So the sentence is true if and only if the content of the sentence is the case, you know?

So the sentence is true if and only if the content of the sentence is the case, you know?

So the sentence, “Snow is white,” you know, in quotations, is true.

所以，“雪是白的”这句话，在引文中是正确的。

That just means that snow is white, and that’s why it’s called the disquotational theory because we remove the quotation marks.

That just means that snow is white, and that’s why it’s called the disquotational theory because we remove the quotation marks.

1:23:16 …from the assertion, right?

…from the assertion, right?

And you can use this idea of disquotation to give a formal definition of truth in a mathematical structure of a statement in a formal language.

And you can use this idea of disquotation to give a formal definition of truth in a mathematical structure of a statement in a formal language.

So for example, if I have a formal language that allows me to make atomic statements about the objects and relations of the structure, and I can build up a formal language with, you know, with the logical connectives of and, and or, and implies, and not, and so on, and maybe I have quantifiers, right?

举例来说，如果我有一种形式语言，它允许我对结构的对象和关系做出原子陈述，并且我可以使用逻辑连接词“and”、“and or”、“andimplied”、“not”等等来建立一种形式语言，也许我有量词，对吗？

Then, for example, to say that the structure satisfies phi and psi, that that single statement, phi and psi, I’m thinking of that as one statement, just means that it satisfies phi and it satisfies psi.

然后，例如，如果说该结构满足 phi 和 psi，那么单个语句 phi 和 psi，我将其视为一个语句，仅意味着它满足 phi 并满足 psi。

And if you notice what happened there, I…

如果你注意到那里发生了什么，我……

1:24:06 At first, the “and” was part of the sentence inside the sentence, but then in the second part, I was using the word “and” to refer to the conjunction of the two conditions.

起初，“and”是句子中的一部分，但后来在第二部分中，我用“and”一词来指代两个条件的连接。

So…

所以……

Yeah, it has the disquotation.

是的，它有折扣。

Yeah, it has the disquotation.

Yeah, it has the disquotation.

1:24:18 And so this idea can be done for all the logical connectors and quantifiers and everything.

所以这个想法可以适用于所有逻辑连接器和量词以及一切。

You’re applying Tarski’s idea of disquotation, and it allows you to define by induction the truth of any assertion in a formal language inside any mathematical structure.

你正在应用塔斯基的反引用思想，它允许你通过归纳来定义任何数学结构中形式语言中任何断言的真实性。

And so to say that a sentence is true, first of all, it’s ambiguous unless you tell me which structure you’re talking about it being true in.

因此，要说一个句子是正确的，首先，除非你告诉我你所说的它在哪种结构中是正确的，否则它是模棱两可的。

And so maybe we have in mind the standard model of arithmetic or something with the natural numbers and the arithmetic structure, and I want to know is a given statement true in that structure.

所以也许我们想到的是算术的标准模型或具有自然数和算术结构的东西，我想知道在该结构中给定的陈述是否正确。

Then we have a formal definition of what that means according to the Tarski recursive definition of truth.

然后，根据塔斯基递归真理定义，我们对这意味着什么有了正式的定义。

Okay, that’s truth.

好吧，这是事实。

1:25:04 Proof, on the other hand, is, you know, in this Hilbert way of thinking, we can develop proof theory.

另一方面，证明是，你知道，按照希尔伯特的思维方式，我们可以发展证明理论。

What is a proof for a mathematician, for a mathematical logician?

对于数学家、数理逻辑学家来说，证明是什么？

A proof is a certain sequence or arrangement of sentences in the formal language that accord with the logical rules of a proof system.

证明是形式语言中符合证明系统逻辑规则的一定序列或句子排列。

So there are certain modes of reasoning that are allowed.

因此，某些推理模式是允许的。

So if you know A and you know A implies B in the proof, then at a later step you’re allowed to write B as a consequence.

因此，如果你知道 A 并且你知道 A 在证明中暗示 B，那么在后面的步骤中你可以写出 B 作为结果。

So if you know A and you know A implies B, those are both two statements that are known, then you can deduce B as a consequence according to the rule of modus ponens.

因此，如果你知道 A 并且你知道 A 蕴含 B，这两个陈述都是已知的，那么你可以根据肯定前件规则推导出 B 作为结果。

This is the rule of modus ponens.

这就是肯定前件规则。

And, you know, there are a lot of other rules.

而且，你知道，还有很多其他规则。

Some people would call this implication elimination.

Some people would call this implication elimination.

1:25:56 There are different kinds of proof systems.

有不同类型的证明系统。

There are a lot of different formal proof systems that exist that are studied by the proof theorists, and all of them have the property that they’re sound, which means that if the premises of the argument are all true in a structure and you have a proof to get a conclusion, then the conclusion is also true in that structure.

证明理论家研究了许多不同的形式证明系统，它们都具有健全的属性，这意味着如果论证的前提在一个结构中全部为真，并且你有证据得出结论，那么结论在该结构中也为真。

So that’s what it means to be sound.

这就是健全的意义。

That proofs preserve truth.

证明保留了真理。

They’re truth-preserving arguments.

它们是维护真理的论点。

Okay?

好的？

But also the proof systems are also generally complete.

But also the proof systems are also generally complete.

1:26:35 They’re both sound and complete, and complete means that whenever a statement is a consequence, a logical consequence of some other statements, which means that whenever the assumptions are true, then the consequence is also true in the structure.

它们既健全又完整，完整意味着只要一个陈述是一个结果，是其他一些陈述的逻辑结果，这意味着只要假设为真，那么结果在结构中也为真。

So whenever you have a logical consequence, then there is a proof of it.

因此，只要你有一个逻辑结果，就会有一个证明。

Okay?

好的？

And the proof systems generally have both of those properties; they’re sound and complete.

证明系统通常具有这两个属性；它们是健全且完整的。

There’s a third property, a lot of logicians talk about sound and complete this, sound and complete that.

There’s a third property, a lot of logicians talk about sound and complete this, sound and complete that.

But actually, there’s a hidden third adjective that they should always be talking about in any such case, which is that you should be able to recognize whether or not something is a proof or not.

但实际上，在任何这种情况下他们都应该总是谈论一个隐藏的第三个形容词，那就是你应该能够识别出某件事是否是证明。

1:27:19 So there’s a computable aspect to the proof systems.

所以证明系统有一个可计算的方面。

We want to be able to recognize whether something is a proof.

We want to be able to recognize whether something is a proof.

It should be computably decidable whether a given sequence of statements is a proof or not.

It should be computably decidable whether a given sequence of statements is a proof or not.

So we don’t want a proof system in which someone claims to have a proof, but we can’t check that fact whether it’s a proof or not.

所以我们不想要一个证明系统，其中有人声称拥有证明，但我们无法检查该事实是否是证明。

We want to be able to, you know, to correctly adjudicate all claims to having a proof.

你知道，我们希望能够正确裁决所有有证据的主张。

Yeah, a mathematician comes to mind that said he has a proof, but the margins are too small…

是的，我想到了一位数学家，他说他有一个证明，但余量太小了……

That’s right

没错

…

……

1:27:51 to continue.

继续。

Exactly.

确切地。

1:27:52 So…

所以……

So that doesn’t count as a proof.

所以这不能算作证据。

Yeah.

是的。

1:27:55 So generally, all the classical proof systems that are used are sound and complete, and also computably decidable in the sense that we can decide whether something is a proof or not.

因此，一般来说，使用的所有经典证明系统都是健全和完整的，并且在我们可以决定某物是否是证明的意义上也是可计算判定的。

So what is, again, the tension between truth and proof?

So what is, again, the tension between truth and proof?

1:28:04 Which is more powerful, and how do the two interplay with the contradictions that we’ve been discussing?

哪个更强大，两者如何与我们一直在讨论的矛盾相互作用？

Right.

正确的。

1:28:15 So the incompleteness theorem is the question whether we could, say, write down a theory for arithmetic.

So the incompleteness theorem is the question whether we could, say, write down a theory for arithmetic.

Say, for the standard model of arithmetic where we have the natural numbers and plus and times and zero, one, and less than, and so on.

Say, for the standard model of arithmetic where we have the natural numbers and plus and times and zero, one, and less than, and so on.

In that formal language, we can express an enormous number of statements about the nature not only of arithmetic, but actually by various coding methods, we can express essentially all of finite mathematics in that structure.

在这种形式语言中，我们不仅可以表达关于算术本质的大量陈述，而且实际上通过各种编码方法，我们基本上可以表达该结构中的所有有限数学。

So the question would be, can we write down a computable list of axioms that will answer all those questions by proof?

所以问题是，我们能否写出一个可计算的公理列表，通过证明来回答所有这些问题？

In other words, we want to have a complete theory, a theory of arithmetic that proves all and only the true statements.

In other words, we want to have a complete theory, a theory of arithmetic that proves all and only the true statements.

That would be the goal.

这就是目标。

Hilbert would love that.

希尔伯特会喜欢的。

1:29:03 I mean, that would be supportive of Hilbert’s program to have such a complete theory of arithmetic, and Godel proved that this is impossible.

I mean, that would be supportive of Hilbert’s program to have such a complete theory of arithmetic, and Godel proved that this is impossible.

You cannot write down a computable list of axioms that is complete in that sense.

你无法写出在这个意义上完整的可计算公理列表。

There will always be statements…

总会有一些说法……

if the theory is consistent, there will always be statements that you cannot prove and you cannot refute.

if the theory is consistent, there will always be statements that you cannot prove and you cannot refute.

So they are independent of that theory.

所以它们独立于该理论。

How traumatic is that, that there’s statements that are independent from the theory?

一些独立于理论的陈述是多么令人痛苦的事情？

I mean, my view is that, yeah, this isn’t traumatic at all.

I mean, my view is that, yeah, this isn’t traumatic at all.

1:29:30 This is rather completely eye-opening in terms of our understanding of the nature of mathematical reality.

就我们对数学现实本质的理解而言，这真是令人大开眼界。

I mean, we’re not…

我的意思是，我们不是……

we understand this profound fact about our situation with regard to mathematical truth.

we understand this profound fact about our situation with regard to mathematical truth.

The incompleteness theorem tells us, look, we just can’t write down a list of axioms that is going to be consistent and it’s going to answer all the questions.

不完备性定理告诉我们，看，我们只是无法写出一系列一致的公理，并且能够回答所有问题。

It’s impossible.

这是不可能的。

And so I don’t think of it as trauma.

所以我不认为这是创伤。

I just think, look, this is the nature of mathematical reality, and it’s good that we know it, and so now we need to move on from that and, you know, do what we can in light of that.

我只是想，看，这就是数学现实的本质，我们知道它是件好事，所以现在我们需要继续前进，并且，你知道，根据这一点做我们能做的。

Is it fair to say that in general it means if I give you a statement, you can’t know if your axiomatic system would be able to prove it?

Is it fair to say that in general it means if I give you a statement, you can’t know if your axiomatic system would be able to prove it?

That’s right.

这是正确的。

1:30:31 In general, you cannot…

一般来说，你不能...

the provability problem, we can formulate it as a decision problem.

the provability problem, we can formulate it as a decision problem.

Given a theory and given a statement, is that statement a consequence of that theory?

Given a theory and given a statement, is that statement a consequence of that theory?

Yeah.

是的。

This is one of the most famous decision problems.

这是最著名的决策问题之一。

In fact, the very first one, because it’s equivalent to the Hilbert-Ackermann Entscheidungsproblem, which is also appearing in the title of Turing’s 1936 paper that was so important for computability theory.

事实上，这是第一个，因为它相当于 Hilbert-Ackermann Entscheidungsproblem，该问题也出现在图灵 1936 年论文的标题中，该论文对可计算性理论非常重要。

So it’s a formulation of the Entscheidungsproblem.

所以这是Entscheidungsproblem 的表述。

Does a given theory have a given statement as a logical consequence?

给定的理论是否有给定的陈述作为逻辑结果？

1:31:08 Which, because of Godel’s completeness theorem, not his incompleteness theorem, but his earlier completeness theorem, Godel had proved that the proof systems that they studied did have this completeness property that I mentioned.

其中，由于哥德尔的完备性定理，不是他的不完备性定理，而是他早期的完备性定理，哥德尔证明了他们研究的证明系统确实具有我提到的这种完备性。

So provability is the same as logical consequence, so…

所以可证明性与逻辑结果相同，所以……

and this is an undecidable decision problem.

这是一个不可判定的决策问题。

Turing proved, and we now know it’s equivalent to the halting problem.

图灵证明了，我们现在知道它相当于停机问题。

Can you describe the halting problem?

您能描述一下停机问题吗？

1:31:30 Because it’s a thing that shows up in a very useful and, again, traumatic way through a lot of computer science, through a lot of mathematics.

因为它通过大量的计算机科学和大量的数学以一种非常有用但又令人痛苦的方式出现。

Yeah.

是的。

1:31:39 The halting problem is expressing a fundamental property of computational processes.

停止问题表达了计算过程的基本属性。

So given a program, or maybe we think of it as a program together with its input, but let me just call it a program.

因此，给定一个程序，或者我们可能将其视为一个程序及其输入，但让我将其称为程序。

So given a program, we could run that program, but I want to pose it as a decision problem.

因此，给定一个程序，我们可以运行该程序，但我想将其视为一个决策问题。

Will this program ever complete its task?

这个程序会完成它的任务吗？

Will it ever halt?

它会停止吗？

And the halting problem is the question, given a program, will it halt?

停止问题是，给定一个程序，它会停止吗？

Yes or no?

是还是不是？

And of course, for any one instance, the answer’s either yes or no.

当然，对于任何一种情况，答案要么是，要么不是。

That’s not what we’re talking about.

这不是我们要讨论的。

We’re talking about whether there’s a computable procedure to answer all instances of this question.

我们正在讨论是否有一个可计算的过程来回答这个问题的所有实例。

1:32:23 So it’s a decision problem given as a scheme of instances for all possible programs that you could ask about.

因此，这是一个决策问题，作为您可以询问的所有可能程序的实例方案给出。

What I want to know is, is there a computable procedure that will answer those questions?

我想知道的是，是否有一个可计算的程序可以回答这些问题？

And it turns out the answer’s no.

事实证明答案是否定的。

The halting problem is computably undecidable.

停止问题是不可计算的。

There is no computable procedure that will correctly answer all instances of whether a given program will halt.

没有可计算的过程可以正确回答给定程序是否会停止的所有情况。

And of course, we can get half the answers in the sense that you give me a program and you say, “Will this halt?” And I could take that program and I could run it.

当然，我们可以得到一半的答案，因为你给我一个程序，然后你说：“这会停止吗？”我可以接受该程序并且可以运行它。

1:32:59 And I could keep running it, and maybe in a week, it would halt.

我可以继续运行它，也许一周后，它就会停止。

And at that time, I could say, “Yes, it halted.” So I can get the yes answers correctly for halting, all the yes answers.

那时，我可以说：“是的，它停止了。”所以我可以正确地得到停止时的“是”答案，所有“是”答案。

But the problem is if it didn’t halt yet, like maybe I waited, you know, a thousand years and it still hasn’t halted, I don’t seem entitled to say, “No, it’s not going to halt” yet, because maybe in a thousand and one years, it’ll halt.

但问题是，如果它还没有停止，就像我等了，你知道，一千年，它仍然没有停止，我似乎没有资格说，“不，它不会停止”，因为也许一千年后，它就会停止。

And so at no point can I seem to say no.

所以我似乎永远都不能说不。

In order to say, “No, it won’t ever halt,” it seems like I would have to really understand how the program worked and what it was doing.

为了说“不，它永远不会停止”，我似乎必须真正理解该程序是如何工作的以及它在做什么。

So giving the yes answers was sort of trivial.

所以给出肯定的答案有点微不足道。

You didn’t have to understand it; you just needed to run it, which is a kind of rote task.

你不必理解它；你可以理解它。你只需要运行它，这是一种死记硬背的任务。

1:33:48 But to give the no answers, you need to have a kind of deep insight into the nature of the program and what it’s doing in such a way that you would understand it and be able to see, “Oh, no, I can see this program is never going to halt.” Because, you know, it’s a much more difficult task to say, “No, it won’t halt,” than it is to say, “Yes, it halted because I ran it and it halted.” And it turns out to be impossible to have a computable procedure that gives the no answers, you know?

但要给出“否”的答案，你需要对程序的本质及其正在做什么有深入的了解，这样你才能理解它并能够看到，“哦，不，我可以看到这个程序永远不会停止。”因为，你知道，说“不，它不会停止”比说“是的，它停止了，因为我运行了它，它停止了”要困难得多。事实证明，不可能有一个可计算的程序给出“否”的答案，你知道吗？

And the argument is not very difficult.

而且论证并不是很难。

Should we do it?

我们应该这样做吗？

Yes, let’s do it.

是的，让我们开始吧。

Okay.

好的。

1:34:19 Suppose toward contradiction, I mean, all these proofs are by contradiction, and this argument is going to be a diagonal argument in the same style as the Russell argument and the Cantor argument and Godel’s argument that we haven’t talked about yet.

假设对于矛盾，我的意思是，所有这些证明都是通过矛盾来证明的，并且这个论证将是一个对角论证，其风格与我们还没有讨论过的罗素论证、康托论证和哥德尔论证相同。

So many diagonal arguments come in.

出现了很多对角线参数。

So suppose towards contradiction that we had a procedure for determining whether a given program halted on a given input.

因此，假设我们有一个程序来确定给定的程序是否在给定的输入上停止。

Now, let me describe.

现在，让我描述一下。

I’m going to use that procedure as a subroutine in the following process.

我将在接下来的过程中使用该过程作为子例程。

And my process, let’s call it Q, process Q, and it takes as input a program P, okay?

我的进程，我们称之为 Q，进程 Q，它接受程序 P 作为输入，好吗？

And the first thing it does is it asks that subroutine, “Hey, would P halt if I ran it on P itself?” Okay.

它做的第一件事是询问该子例程，“嘿，如果我在 P 本身上运行它，P 会停止吗？”好的。

That’s the diagonal part, because we’re applying P to P, right?

那是对角线部分，因为我们将 P 应用于 P，对吗？

1:35:14 Okay, so I’m describing program Q, and program Q takes as input P, which is itself a program.

好的，我正在描述程序 Q，程序 Q 作为输入 P，它本身就是一个程序。

And the first thing it does is it asks the halting subroutine program, “Would P halt on P?” And if the answer comes back from the subroutine, “Yeah, that would halt,” then what I do in program Q is I immediately jump into an infinite loop.

它做的第一件事是询问停止子例程：“P 会在 P 上停止吗？”如果从子程序返回答案“是的，那会停止”，那么我在程序 Q 中所做的就是立即跳入无限循环。

So I don’t halt.

所以我没有停下来。

If P halts on P, I don’t halt.

如果 P 在 P 上停止，我就不会停止。

But if the answer came back, “No, P is never going to halt on P,” then I halt immediately.

但如果答案是：“不，P 永远不会在 P 上停止”，那么我会立即停止。

Okay, and that’s it.

好的，就是这样。

I’ve described what Q does.

我已经描述了 Q 的作用。

1:35:53 And the thing about Q is that Q’s behavior on P was the opposite of P’s behavior on P.

关于 Q 的事情是，Q 对 P 的行为与 P 对 P 的行为相反。

I mean, that’s how we designed Q specifically so that Q on P had the opposite behavior as P on P.

我的意思是，这就是我们专门设计 Q 的方式，以便 Q 在 P 上的行为与 P 在 P 上的行为相反。

Okay, so now, of course, what do we do?

好吧，现在，当然，我们该怎么办？

Well, the same thing that Russell did, and so forth, and Cantor, we ask, “Well, what would Q do on Q?” And because of this opposite behavior, Q would halt on Q if and only if Q does not halt on Q, which is a contradiction, because Q has to have the opposite behavior on Q than Q does, but that’s just contradictory.

好吧，与拉塞尔和康托所做的一样，我们会问，“好吧，Q 对 Q 会做什么？”由于这种相反的行为，当且仅当 Q 不在 Q 上停止时，Q 才会在 Q 上停止，这是一个矛盾，因为 Q 必须在 Q 上具有与 Q 上相反的行为，但这只是矛盾。

What a beautiful proof.

多么美丽的证明啊。

1:36:38 Simple.

简单的。

It’s absolutely beautiful.

这绝对是美丽的。

1:36:39 I agree.

我同意。

It’s following the same logic of Russell and Cantor.

它遵循拉塞尔和康托的相同逻辑。

I mean, going back to Cantor basically, because Russell is also quoting Cantor in his letter to Frege.

我的意思是，基本上回到康托尔，因为罗素在给弗雷格的信中也引用了康托尔。

Therefore, the conclusion is that the halting problem is not computably decidable.

因此，结论是停机问题不可计算确定。

Now we can immediately prove Godel’s theorem using this, actually.

现在我们实际上可以立即用它来证明哥德尔定理。

It’s an immediate consequence.

这是一个直接的后果。

So why don’t we just do that?

那么我们为什么不这样做呢？

I view this as the simplest proof of Godel’s theorem.

我认为这是哥德尔定理最简单的证明。

You don’t need the Godel sentence to prove Godel’s theorem.

你不需要哥德尔句子来证明哥德尔定理。

You can do it with the halting problem.

您可以通过停止问题来做到这一点。

So suppose that we could write down a computable axiomatization of all of the true facts of elementary mathematics, meaning arithmetic and finite combinatorial things such as Turing machine computations and so on.

因此，假设我们可以写出初等数学所有真实事实的可计算公理化，即算术和有限组合事物，例如图灵机计算等。

1:37:35 So in fact, all those finite combinatorial processes are formalizable inside arithmetic with the standard arithmetization coding process.

所以事实上，所有这些有限组合过程都可以通过标准算术编码过程在算术内部形式化。

But let me just be a little bit informal and say suppose we could write down a complete theory of elementary finite mathematics.

但让我稍微非正式地说，假设我们可以写出一个完整的初等有限数学理论。

So we have an axiomatization of that theory.

所以我们对该理论有了公理化。

Then we could produce all possible theorems from those axioms in the way that I was describing earlier with Hilbert’s program.

然后我们可以按照我之前用希尔伯特程序描述的方式从这些公理中产生所有可能的定理。

If we had a complete theory of elementary mathematics, we could construct a theorem enumeration machine that produced all the theorems and only the theorems from that theory.

如果我们有一个完整的初等数学理论，我们就可以构造一个定理枚举机，它产生所有定理，并且只产生该理论的定理。

So now, I have this theorem enumeration device on my desk, and I announce that I’m open for business to solve the halting problem.

所以现在，我的桌子上有这个定理枚举设备，我宣布我愿意营业以解决停机问题。

1:38:25 So you give me a program and input that you want to run that program on, and I’m going to answer the halting problem.

所以你给我一个程序和你想要运行该程序的输入，我将回答停止问题。

The way I’m going to do it is I’m just going to wait for the statement coming out of the theorem enumeration device that asserts either that P does halt on that input, or I wait for the statement that P does not halt on that input.

我要做的方法是等待定理枚举设备发出的语句，该语句断言 P 确实在该输入上停止，或者等待 P 不会在该输入上停止的语句。

But one of them’s going to happen because it was a complete theory that was enumerating all the true statements of elementary mathematics.

但其中一个将会发生，因为它是一个完整的理论，列举了初等数学的所有真实陈述。

So therefore, if I had such a system, I could solve the halting problem, but we already proved that you cannot solve the halting problem, so therefore you cannot have such a complete theory of arithmetic.

因此，如果我有这样一个系统，我可以解决停机问题，但是我们已经证明你无法解决停机问题，因此你不可能有如此完整的算术理论。

So that proves Godel’s theorem.

这样就证明了哥德尔定理。

Maybe to take a little bit of a tangent, can you speak…

也许稍微扯一下话题，你能说一下吗……

1:39:05 You’ve written a wonderful book about proofs and the art of mathematics.

你写了一本关于证明和数学艺术的精彩书。

So what can you say about proving stuff in mathematics?

那么关于证明数学中的东西你有什么想说的呢？

What is the process of proof?

举证过程是怎样的？

What are the tools?

有哪些工具？

What is the art?

什么是艺术？

What is the science of proving things in mathematics?

用数学证明事物的科学是什么？

This is something that I find so wonderful to teach young mathematicians who are learning how to become mathematicians and learning about proof, and I wrote that book when I was teaching such a proof-writing class in New York.

我发现教授那些正在学习如何成为数学家和学习证明的年轻数学家是一件非常美妙的事情，我在纽约教授这样一个证明写作课程时写了这本书。

1:39:37 Many universities have such a course, the proof-writing course, which is usually taken by students who have learned some mathematics.

许多大学都有这样一门课程，即校对写作课程，通常是学过一些数学的学生选修的。

Usually, they’ve completed maybe the calculus sequence and are making the kind of transition to higher mathematics, which tends to involve much more proof, and it’s a kind of challenging step for them.

通常，他们可能已经完成了微积分序列，并且正在向高等数学过渡，这往往涉及更多的证明，这对他们来说是一种具有挑战性的步骤。

So many math departments have this kind of course on proof-writing where the students would get exposed to how to write proofs.

很多数学系都有这种关于证明写作的课程，学生可以在那里学习如何写证明。

I wasn’t happy with most of the other books that exist for those kind of courses, and the reason was that they were so often so dull because they would concentrate on these totally uninteresting parts of what it’s like to write a proof, these kind of mechanistic procedures about how to write a proof.

我对大多数此类课程的其他书籍都不满意，原因是它们常常很乏味，因为它们会专注于编写证明的这些完全无趣的部分，这些关于如何编写证明的机械程序。

1:40:28 You know, if you’re going to prove an implication, then you assume the hypothesis and argue for the conclusion, and so on.

你知道，如果你要证明一个蕴涵，那么你就假设假设并论证结论，等等。

All of that is true and fine, and that’s good to know, except if that’s all that you’re saying about the nature of proof, then I don’t think you’re really learning very much.

所有这些都是真实而美好的，很高兴知道这一点，除非这就是你所说的关于证明本质的全部，那么我认为你并没有真正学到很多东西。

So I felt that it was possible to have a much better kind of book, one that was much more interesting and that had interesting theorems in it that still admitted elementary proof.

所以我觉得有可能有​​一本更好的书，一本更有趣的书，里面有有趣的定理，但仍然承认基本的证明。

So I wrote this book and tried to fill it with all of the compelling mathematical statements with very elementary proofs that exhibited lots of different proof styles in it.

因此，我写了这本书，并试图用所有令人信服的数学陈述和非常基本的证明来填充它，这些证明展示了许多不同的证明风格。

So, I found that the students appreciated it a lot.

所以，我发现学生们都很欣赏它。

We should say, “We dedicate the book to my students, may all their theorems be true, proved by elegant arguments that flow effortlessly from hypothesis to conclusion while revealing fantastical mathematical beauty.” Is there some interesting proofs that maybe illustrate, for people outside of mathematics or for people who just take math classes…

我们应该说，“我们将这本书献给我的学生，愿他们所有的定理都是正确的，通过优雅的论证来证明，这些论证毫不费力地从假设到结论，同时揭示了奇妙的数学之美。”对于数学以外的人或只上数学课的人来说，是否有一些有趣的证明可以说明......

Right

右

…in high school and so on?

……在高中等等？

Yeah, let’s do a proof.

是的，我们来做个证明。

1:41:39 There’s one in the book.

书上有一个。

We can talk about it.

我们可以谈谈。

I think it’s a nice problem.

我认为这是一个很好的问题。

It’s in discrete math, yeah, the 5.1, that one, more pointed at than pointing.

这是离散数学中的，是的，5.1，那个，更多的是指向而不是指向。

Okay.

好的。

So this is the following problem.

所以这就是下面的问题。

Suppose you’re gathered with some friends, you know, in a circle, and you can point at each other however you want, or yourself, whatever, it doesn’t matter, and you can point at more than one person, you know, use all your fingers or your feet or whatever you want.

假设你和一些朋友聚集在一起，你知道，围成一圈，你可以随意指着对方，也可以指着你自己，无论如何，没关系，你可以指着多个人，你知道，用你所有的手指或脚或任何你想要的东西。

So maybe you point at three of your friends or something and they point at two or three of their friends or whatever, and one person is pointing at 10 people and somebody isn’t pointing at anybody maybe, and various people are pointed at also, right?

所以也许你指着你的三个朋友或其他东西，他们指着两三个朋友或其他什么，一个人指着 10 个人，而另一个人可能没有指着任何人，而且也指着不同的人，对吗？

1:42:20 So the question is, could we arrange a pattern of pointing so that everyone was more pointed at than they are pointing at others?

所以问题是，我们是否可以安排一种指向模式，使每个人都比他们指向其他人更多地被指向？

So in other words, maybe there’s seven people pointing at me, but I’m only pointing at five people and maybe there’s, you know, 20 people pointing at you, but you’re only pointing at 15 people or something like that, right?

换句话说，也许有 7 个人指着我，但我只指着 5 个人，也许有 20 个人指着你，但你只指着 15 个人或类似的人，对吧？

So I want to know.

所以我想知道。

There’s a similar question on Twitter.

Twitter 上也有类似的问题。

For a group of people on Twitter, could you arrange that everyone has more followers than following?

对于 Twitter 上的一群人，你能否安排每个人的关注者数量多于关注者数量？

Yeah, it’s the same question.

是的，这是同一个问题。

Mathematically, it’s identical.

从数学上来说，它是相同的。

Although, I don’t know, it’s not identical, because I said you could point at yourself, and I think that’s not…

虽然，我不知道，但并不相同，因为我说你可以指着你自己，我认为那不是……

Can you follow yourself?

你能跟随自己吗？

No, I don’t think so, no.

不，我不这么认为，不。

I don’t think you can.

我认为你不能。

1:43:10 Okay.

好的。

So can you arrange it so that everyone is more pointed at than pointing?

那么能不能安排得大家都被指着而不是指着呢？

In my book, I give a couple of different proofs of this.

在我的书中，我给出了几个不同的证明。

I think I give an induction proof and then there’s another proof.

我想我给出了一个归纳证明，然后还有另一个证明。

I think there’s three different proofs in there.

我认为里面有三个不同的证据。

But why don’t we just talk about my favorite proof?

但我们为什么不直接谈谈我最喜欢的证明呢？

Suppose it were possible to arrange that we’re all more pointed at than pointing, okay?

假设可以安排我们都更多地被指向而不是指向，好吗？

Now what we’re going to do, we’re going to agree, we’re going to give a dollar to everyone that we’re pointing at.

现在我们要做的，我们要达成一致，我们要给我们所指的每个人一美元。

1:43:40 Okay?

好的？

So what happens?

那么会发生什么呢？

Everybody made money, because I was pointed at by more people than I’m pointing, so I got $10 but I only paid out $7.

每个人都赚了钱，因为指向我的人比我指向的人还多，所以我得到了 10 美元，但我只支付了 7 美元。

And similarly, you got paid $20 but you only paid out $15.

同样，你得到了 20 美元，但你只支付了 15 美元。

So if everyone is more pointed at than pointing, then everyone makes money.

所以如果每个人都被指而不是指，那么每个人都赚钱。

But it’s obviously impossible for us to make money as a group by just trading money with ourselves.

但我们作为一个群体，仅仅通过与自己交易金钱显然是不可能赚钱的。

And therefore, it can’t be possible that we’re all more pointed at than pointing.

因此，我们不可能都被指向而不是指向。

And this proof illustrates something.

这个证明说明了一些事情。

It’s one of my habits that I suggest in the book: to anthropomorphize your mathematical ideas.

我在书中建议的习惯之一是：将你的数学思想拟人化。

So,

所以，

1:44:22 you should imagine that the mathematical objects that are playing a role in your question are people, or active, somehow, animals or something that maybe have a will and a goal and so on.

你应该想象在你的问题中发挥作用的数学对象是人，或者是活跃的，不知何故，动物或可能有意志和目标的东西等等。

This is this process of anthropomorphizing.

这就是这个拟人化的过程。

And it often makes the problems easier to understand, because we all are familiar with the fact that it’s difficult to make money, and the proof is totally convincing because of our knowledge that we can’t make money as a group by trading dollars between us, without any new money coming into the group.

这通常会使问题更容易理解，因为我们都熟悉赚钱很难的事实，而且证据是完全令人信服的，因为我们知道，如果没有任何新资金进入团队，我们就无法通过在我们之间交易美元作为一个团队来赚钱。

1:45:01 But that by itself is actually a difficult mathematical claim.

但这本身实际上是一个困难的数学主张。

I mean, if someone had to prove that you can’t make money by trading within a group, you know, it can’t be that everyone in the group makes money just by shifting money around in the group.

我的意思是，如果有人必须证明你不能通过在团体内交易来赚钱，你知道，团体中的每个人都不可能仅仅通过在团体内转移资金来赚钱。

Maybe you think that’s obvious, and it is obvious if you think about money.

也许你认为这是显而易见的，如果你考虑到金钱，这也是显而易见的。

But if you had asked the question about mathematical functions of a certain kind and so on, then maybe it wouldn’t be as clear as it is when you’re talking about this money thing, because we can build on our human experience about the difficulty of getting money and other resources.

但如果你问的是某种数学函数等问题，那么也许它不会像你谈论金钱问题时那么清楚，因为我们可以根据人类经验来了解获得金钱和其他资源的难度。

It doesn’t have to be money; it could be candy, whatever.

不一定是钱；它可能是糖果，无论如何。

You know, we just know that you can’t easily get more things in that kind just by trading within a group.

你知道，我们只是知道，仅仅通过在团体内进行交易，你无法轻易获得更多此类东西。

And we should say that sometimes the power of proof is such that the non-obvious can be shown, and then over time that becomes obvious.

我们应该说，有时证明的力量是这样的：非显而易见的事情可以被证明，然后随着时间的推移，它变得显而易见。

1:45:48 So, in the context of- money or social systems, there’s a bunch of things that are non-obvious.

因此，在金钱或社会体系的背景下，有很多事情是不明显的。

And the whole point is that proof can guide us to the truth, to the accurate description of reality.

重点是，证据可以引导我们了解真相，引导我们准确描述现实。

We just proved a property of money.

我们刚刚证明了金钱的财产。

It’s interesting to think about, well, what if there were infinitely many people in your group?

想想很有趣，如果你的团队中有无限多个人怎么办？

1:46:14 Then it’s not true anymore.

那么它就不再是真的了。

The theorem fails.

该定理失败。

In fact, you can arrange that everyone is strictly more pointed than pointing.

其实你可以安排大家严格比指指点点。

And also, if everyone has even just one dollar bill-

而且，如果每个人都只有一美元钞票的话——

1:46:35 then you can arrange that afterwards everyone has infinitely many dollar bills.

那么你可以安排以后每个人都有无限多的美元钞票。

Because in terms of cardinality, that’s the same.

因为就基数而言，这是一样的。

It’s just, say, countable infinity in each case.

比如说，每种情况下都是可数无穷大。

If you had countably many friends and everyone has one dollar bill, then you can arrange a pattern of passing those dollar bills amongst each other so that afterwards everyone has infinitely many dollar bills.

如果你有可数的朋友，并且每个人都有一张一美元的钞票，那么你可以安排一种将这些钞票互相传递的模式，这样之后每个人都有无限多的钞票。

What you need is for each person to be attached to, you know, one of the train cars or something.

What you need is for each person to be attached to, you know, one of the train cars or something.

So, think of everyone as coming from Hilbert’s train, but also think of them as fitting into Hilbert’s Hotel.

So, think of everyone as coming from Hilbert’s train, but also think of them as fitting into Hilbert’s Hotel.

So, just have everyone on the Nth car give all their money to the person who ends up in the Nth room.

So, just have everyone on the Nth car give all their money to the person who ends up in the Nth room.

So, they each give one dollar to that person.

因此，他们每人给那个人一美元。

1:47:16 So afterwards, that person has infinitely many dollars, but everyone only paid out one dollar.

So afterwards, that person has infinitely many dollars, but everyone only paid out one dollar.

So it’s a way of making it happen.

所以这是实现这一目标的一种方式。

To what degree, sticking on the topic of infinity, should we think of infinity as something real?

围绕无穷大这个话题，我们应该在多大程度上将无穷大视为真实的东西？

That’s an excellent question.

这是一个很好的问题。

1:47:33 I mean, a huge part of the philosophy of mathematics is about this kind of question, that what is the nature of the existence of mathematical objects, including infinity?

我的意思是，数学哲学的很大一部分是关于这类问题的，即数学对象（包括无穷大）存在的本质是什么？

But I think asking about infinity specifically isn’t that different than asking about the number five.

But I think asking about infinity specifically isn’t that different than asking about the number five.

What is- what does it mean for the number five to exist?

What is- what does it mean for the number five to exist?

What are the numbers really, right?

到底数字是多少，对吧？

1:47:58 This, this is maybe one of the fundamental questions of mathematical ontology.

这个,这可能是数学本体论的基本问题之一。

I mean, there’s many different positions to take on the question of the nature of the existence of mathematical objects or abstract objects in general.

我的意思是，对于数学对象或一般抽象对象的存在本质问题，有许多不同的立场。

And there’s a certain kind of conversation that sometimes happens when you do that.

And there’s a certain kind of conversation that sometimes happens when you do that.

And it goes something like this: sometimes people find it problematic to talk about the existence of abstract objects such as numbers, and there seems to be a kind of wish that we could give an account of the existence of numbers or other mathematical objects or abstract objects that was more like, you know, the existence of tables and chairs and rocks and so on.

事情是这样的：有时人们发现谈论数字等抽象对象的存在是有问题的，而且似乎有一种希望我们能够解释数字或其他数学对象或抽象对象的存在，这更像是桌子、椅子和岩石等的存在。

1:48:42 And so there seems to be this desire to reduce mathematical existence to something, you know, that we can experience physically in the real world.

因此，人们似乎希望将数学存在简化为我们可以在现实世界中实际体验到的东西。

But my attitude about this attempt is that it’s very backward, I think, because I don’t think we have such a clear existence of the nature of physical objects, actually.

但我对这种尝试的态度是，我认为它非常落后，因为我认为我们实际上对物理对象的本质没有如此清晰的存在。

I mean, we all have experience about existing in the physical world, as we must, because we do exist in the physical world, but I don’t know of any satisfactory account of what it means to exist physically.

我的意思是，我们都有关于存在于物质世界中的经验，这是我们必须的，因为我们确实存在于物质世界中，但我不知道有任何关于物质存在意味着什么的令人满意的解释。

1:49:28 I mean, if I ask you, say, “Imagine, you know, a certain kind of steam locomotive,” you know, and I describe the engineering of it and the weight of it and the nature of the gear linkages, and, you know, and I show you schematic drawings of the whole design and so on, and, you know, we talk in detail about every single detailed aspect of this steam locomotive.

我的意思是，如果我问你，“想象一下，你知道，某种蒸汽机车”，你知道，我会描述它的工程设计、重量和齿轮连杆的性质，而且，你知道，我会向你展示整个设计的示意图等等，而且，你知道，我们会详细讨论这台蒸汽机车的每一个细节。

But then suppose after all that conversation, I say, “Okay, now I would like you to tell me what would it mean for it to exist physically, I mean, as opposed to just being an imaginary steam locomotive?” Then what, what could you possibly say about it?

但假设在所有谈话之后，我说：“好吧，现在我想让你告诉我，它的物理存在意味着什么，我的意思是，而不是仅仅作为一个想象中的蒸汽机车？”那么，你能对此说些什么呢？

I mean, except by saying, “Oh, I just mean that it exists in the physical world.” But what does that mean?

我的意思是，除了说“哦，我只是说它存在于物质世界中”。但这意味着什么？

That’s the question, right?

这就是问题所在，对吧？

It’s not an answer to the question.

这不是问题的答案。

1:50:18 That is the question.

这就是问题所在。

So I don’t think that there’s anything sensible that we can say about the nature of physical existence.

So I don’t think that there’s anything sensible that we can say about the nature of physical existence.

It is a profound mystery.

这是一个深奥的谜。

In fact, it becomes more and more mysterious the more physics we know.

In fact, it becomes more and more mysterious the more physics we know.

I mean, back in, say, Newtonian physics, one had a picture of the nature of physical objects as, you know, little billiard balls or something, or maybe they’re infinitely divisible or something like that.

我的意思是，在牛顿物理学中，人们对物理物体的本质有一幅图画，比如，你知道，小台球之类的东西，或者它们可能是无限可分的或者类似的东西。

Okay, but then this picture is upset with the atomic theory of matter.

Okay, but then this picture is upset with the atomic theory of matter.

But then that picture’s upset when we realize that the atoms actually can be split and consist of electrons and protons and neutrons and so on.

但当我们意识到原子实际上可以分裂并且由电子、质子和中子等组成时，这种情况就变得混乱了。

1:50:56 But then that picture’s upset when we realize that those things themselves are built out of quarks and leptons and so on, and who knows what’s coming.

但当我们意识到这些东西本身是由夸克和轻子等构成的，谁知道会发生什么时，情况就变得混乱了。

And furthermore, all of those things, the nature of their existence is actually as wave functions in some cloud of probability and so on.

And furthermore, all of those things, the nature of their existence is actually as wave functions in some cloud of probability and so on.

And so it just becomes more and more mysterious the more we learn, and not at all clarifying.

And so it just becomes more and more mysterious the more we learn, and not at all clarifying.

And so the nature of what it means to say that there’s an apple on my desk, and to give an account of what that physical existence really is at bottom, I think, is totally absent.

因此，我认为，说我桌上有一个苹果，以及解释那个物理存在的本质是什么，其本质是完全不存在的。

Whereas we do seem to have a much more satisfactory account of the nature of abstract existence.

然而，我们似乎确实对抽象存在的本质有了更令人满意的解释。

I mean, I can talk about the nature of the empty set.

I mean, I can talk about the nature of the empty set.

1:51:43 You know, this is the predicate which is never true or something like that.

You know, this is the predicate which is never true or something like that.

I can talk about those kind of logical properties or the singleton of the empty set and so on.

I can talk about those kind of logical properties or the singleton of the empty set and so on.

I mean, of course, it’s very difficult if you go very far with it, but the point is that it doesn’t get more and more mysterious.

我的意思是，当然，如果你走得很远，那是非常困难的，但关键是它不会变得越来越神秘。

The more that you say, it becomes only more and more clear.

你说得越多，它就变得越来越清楚。

And so it seems to me that we don’t really have any understanding of what the physical world is as opposed to the abstract world, and it’s the abstract world where existence is much more clear.

因此，在我看来，我们对与抽象世界相反的物理世界并没有任何了解，而抽象世界的存在更加清晰。

It is very true that we don’t know anything about the soda bottle or the steam locomotive just because we can poke at it.

It is very true that we don’t know anything about the soda bottle or the steam locomotive just because we can poke at it.

1:52:19 Again, we anthropomorphize, and that actually gets us into trouble sometimes because I’m not feeling the quantum mechanics when I’m touching it.

再说一次，我们拟人化，这实际上有时会给我们带来麻烦，因为当我触摸它时，我感觉不到量子力学。

That’s right.

这是正确的。

And therefore, it’s easy to forget and feel like this is real and mathematical objects are not, but you’re making the opposite argument.

因此，很容易忘记并感觉这是真实的，而数学对象不是，但你却提出了相反的论点。

1:52:34 When you draw a distinction between numerals and numbers, which numerals are the representation of the number on the page and so on, could you say that a number is real?

当你区分数字和数字时，哪些数字是页面上数字的表示等等，你能说数字是真实的吗？

Do numbers exist?

数字存在吗？

I happen to think so.

我碰巧这么认为。

1:52:57 I mean, I’m on the side of realism in mathematics, and I think that these abstract objects do have a real existence in a way that we can give an account of, in a way I just tried to describe.

我的意思是，我站在数学实在论的一边，我认为这些抽象对象确实以一种我们可以解释的方式真实存在，以我刚刚试图描述的方式。

So, like, you would describe it as the size of a set with four elements in it?

那么，您会将其描述为包含四个元素的集合的大小吗？

Well, there are different ways to understand the nature of four.

嗯，有不同的方法来理解四的本质。

1:53:14 I mean, actually, this gets into the question of structuralism, which is maybe a good place to talk about it.

我的意思是，实际上，这涉及到结构主义问题，这也许是谈论它的好地方。

What is structuralism?

什么是结构主义？

Structuralism is a philosophical position in mathematics, or the philosophy of mathematics, by which one emphasizes that what’s important about mathematical objects is not what they’re made out of or what their substance or essence is, but rather how they function in a mathematical structure.

结构主义是数学或数学哲学中的一种哲学立场，它强调数学对象的重要之处不是它们是由什么构成的，也不是它们的实质或本质是什么，而是它们如何在数学结构中发挥作用。

1:53:25 And so, what I call the structuralist attitude in mathematics is that we should only care about our mathematical structures up to isomorphism.

因此，我所说的数学结构主义态度是，我们应该只关心我们的数学结构直至同构。

1:53:53 If I have a mathematical structure of a certain kind and I make an exact copy of it using different individuals to form the elements of that structure, then the isomorphic copy is just as good mathematically, and there’s no important mathematical difference that would ever arise from working with this isomorphic copy instead of the original structure.

如果我有某种数学结构，并且我使用不同的个体来形成该结构的元素来制作它的精确副本，那么同构副本在数学上同样好，并且使用此同构副本而不是原始结构不会产生重要的数学差异。

And so, therefore, that’s another way of saying that the substance of individuals in a mathematical structure is irrelevant with regard to any mathematical property of that structure.

因此，这是数学结构中个体的实质与该结构的任何数学属性无关的另一种说法。

And so…

所以……

1:54:33 So to ask a question like, “What is the number four really?” is an anti-structuralist thing, because if you have a structure, say, the natural numbers, with all the numbers in it, 0, 1, 2, 3, 4, and so on, then I could replace the number four with something else, like, you know, this bottle of water could play the role of the number four in that structure, and it would be isomorphic.

So to ask a question like, “What is the number four really?”是一个反结构主义的东西，因为如果你有一个结构，比如说自然数，里面有所有的数字，0、1、2、3、4等等，那么我可以用其他东西代替数字四，就像，你知道，这瓶水可以在该结构中扮演数字四的角色，而且它是同构的。

And it wouldn’t matter at all for any mathematical purpose to use this alternative mathematical system, you know?

对于任何数学目的来说，使用这种替代数学系统根本不重要，你知道吗？

That’s to say that we don’t care what the number four is really.

这就是说，我们并不关心数字四到底是什么。

That is irrelevant.

那是无关紧要的。

What…

什么……

1:55:12 The only thing that matters is what are the properties of the number four in a given mathematical system, and recognizing that there are other isomorphic copies of that system, and the properties of that other system’s number four are going to be identical to the properties of this system’s number four with regard to any question that’s important about the number four.

唯一重要的是给定数学系统中数字 4 的属性是什么，并认识到该系统存在其他同构副本，并且对于任何与数字 4 相关的重要问题，其他系统的数字 4 的属性将与该系统的数字 4 的属性相同。

But those questions won’t be about essence.

但这些问题不会涉及本质。

So in a sense, structuralism is anti-essential in mathematics.

所以从某种意义上说，结构主义在数学中是反本质的。

So is it fair to think of numbers as a kind of pointer to a deep underlying structure?

那么，将数字视为一种指向深层底层结构的指针是否公平？

Yeah, I think so, because I guess part of the point of structuralism is that it doesn’t make sense to consider mathematical objects or individuals in isolation.

是的，我这么认为，因为我猜结构主义的部分观点是孤立地考虑数学对象或个体是没有意义的。

1:55:48 What’s interesting and important about mathematical objects is how they interact with each other and how they behave in a system, and so maybe one wants to think about the structural role that the objects play in a larger system, a larger structure.

数学对象的有趣和重要之处在于它们如何相互作用以及它们在系统中的行为方式，因此也许人们想要思考这些对象在更大的系统、更大的结构中所扮演的结构角色。

There’s a famous question that Frege had asked actually when he was looking into the nature of numbers, because in his logicist program, he was trying to reduce all of mathematics to logic.

弗雷格在研究数字的本质时实际上提出了一个著名的问题，因为在他的逻辑主义纲领中，他试图将所有数学简化为逻辑。

1:56:26 And in that process, he was referring to the Cantor-Hume principle that, whenever two sets are equinumerous, then they have the same number of elements, if and only if.

在此过程中，他引用了康托-休谟原理，即只要两个集合是等数的，那么它们就具有相同数量的元素，当且仅当。

And he founded his theory of number on this principle, but he recognized that there was something that dissatisfied him about that situation, which is that the Cantor-Hume principle does not seem to give you criteria for which things are numbers.

他根据这个原理建立了他的数论，但他认识到，这种情况有一些令他不满意的地方，那就是康托-休谟原理似乎没有给出事物是数的标准。

It only tells you a kind of identity criteria for when are two numbers equal to each other.

它仅告诉您两个数字何时彼此相等的一种恒等标准。

Well, two numbers are equal just in case the sets of those sizes are equinumerous, so that’s the criteria for number identity.

嗯，两个数字相等，以防这些大小的集合相等，所以这就是数字同一性的标准。

But it is not a criteria for what is a number.

但这并不是数字的标准。

1:57:12 And so this problem has become known as the Julius Caesar problem because Frege said we don’t seem to have any way of telling from the Hume principle whether Julius Caesar is a number or not.

所以这个问题被称为凯撒大帝问题，因为弗雷格说我们似乎没有任何方法可以根据休谟原理判断凯撒大帝是否是一个数字。

So he’s asking about the essence of number and whether…

所以他在问数字的本质以及是否……

Of course, one has a sense that he picked maybe what he was trying to present as a ridiculous example…

当然，人们有一种感觉，他也许选择了他试图呈现的一个荒谬的例子……

…Because maybe you have the idea that, well, obviously Julius Caesar is not a number, and there’s a lot of philosophical writing that seems to take that line also, that obviously the answer is that Julius Caesar is not a number.

……因为也许你有这样的想法，嗯，显然朱利叶斯·凯撒不是一个数字，而且有很多哲学著作似乎也采取了这一点，显然答案是朱利叶斯·凯撒不是一个数字。

But the structuralists disagree with that position.

但结构主义者不同意这一立场。

The structuralist attitude is, “Look, you give me a number system.

结构主义的态度是，“看，你给了我一个数字系统。

If Julius Caesar isn’t a number, then I can just…

如果朱利叶斯·凯撒不是一个数字，那么我可以……

1:58:01 Let’s take the number 17 out of that system and plug in Julius Caesar for that role, and now I’ve got a new number system, and now Julius Caesar happens to be the number 17.” And that’s totally fine, you know.

让我们从那个系统中取出数字 17，并插入朱利叶斯·凯撒担任该角色，现在我有了一个新的数字系统，现在朱利叶斯·凯撒恰好是数字 17。”你知道，那完全没问题。

So the point of structuralism is that the question of whether Julius Caesar is a number or not is irrelevant to mathematics.

所以结构主义的要点是，凯撒大帝是否是一个数字的问题与数学无关。

It is irrelevant because it is not about structure, it’s about this essence of the mathematical objects.

这是无关紧要的，因为它与结构无关，而是与数学对象的本质有关。

So that’s the structuralist criticism of Frege’s point.

这就是结构主义对弗雷格观点的批评。

You’ve kind of made the case that you can say more concrete things about the existence of objects in mathematics than you can in our physical reality, about which to us human brains, things are obvious or not.

你已经证明，在数学中你可以比在我们的物理现实中更具体地描述物体的存在，而对于我们人类的大脑来说，事情是显而易见的或不明显的。

1:58:35 So what’s more real, the reality we see with our eyes or the reality we can express in mathematical theorems?

那么，我们用眼睛看到的现实和我们可以用数学定理表达的现实哪个更真实呢？

I’m not quite sure.

我不太确定。

1:58:59 I mean…

我的意思是……

I live entirely in the platonic realm, and I don’t really understand the physical universe at all.

我完全生活在柏拉图式的境界中，我根本不了解物理宇宙。

So I don’t have strong views.

所以我没有什么强烈的观点。

Let’s talk about the platonic realm.

我们来谈谈柏拉图式的境界。

1:59:13 Is it…

难道是……

Like, because you live there, is it real?

就像，因为你住在那里，这是真的吗？

Or-

或者-

Oh yeah, totally, yeah.

哦，是的，完全是的。

1:59:19 This is the realist position in mathematics is that abstract objects have a real existence.

这就是数学中实在论的立场，即抽象的对象具有真实的存在。

And okay, what’s meant by that is that there’s some sense of existence in which those objects can be regarded as real.

好吧，这意味着存在某种存在感，这些物体可以被视为真实的。

How should we think about that?

我们应该如何思考这个问题？

1:59:32 How should we try to visualize that?

How should we try to visualize that?

What does it mean to live amongst abstract objects?

生活在抽象物体之中意味着什么？

Because life is finite.

因为生命是有限的。

We’re all afraid of death.

我们都害怕死亡。

We fall in love with other physical manifestations of objects.

我们爱上物体的其他物理表现。

And you’re telling me that maybe reality actually exists elsewhere, and this is all just a projection…

你告诉我，也许现实确实存在于其他地方，而这一切都只是一个投影……

Well, I mean-

嗯，我的意思是——

…from the abstract realm?

…from the abstract realm?

Do abstract objects exist in a place and at a time?

抽象物体是否存在于某个地点和某个时间？

2:00:02 That’s very debatable, I think.

That’s very debatable, I think.

Right.

正确的。

2:00:07 And what does place and time mean?

And what does place and time mean?

All time, yeah, so…

一直以来，是的，所以……

So what’s more real: physics or the mathematical Platonic space?

那么，哪个更真实：物理学还是数学柏拉图空间？

Well, the mathematical Platonic realm is…

Well, the mathematical Platonic realm is…

2:00:16 I’m not sure I would say it’s more real, but I’m saying we understand the reality of it in a much deeper and more— …a more convincing way.

我不确定我会说它更真实，但我是说我们以更深入、更……更令人信服的方式理解它的现实。

I don’t think we understand the nature of physical reality very well at all, and I think most people aren’t even scratching the surface of the question as I intend to be asking it.

我认为我们根本不太了解物理现实的本质，而且我认为大多数人甚至没有触及我想要问的问题的表面。

So, you know, obviously we understand physical reality.

所以，你知道，显然我们了解物理现实。

I mean, I knock on the table— …and so on, and we know all about what it’s like to, you know, have a birthday party or to drink a martini or whatever.

我的意思是，我敲桌子……等等，我们都知道，你知道，举办生日聚会或喝马提尼或其他什么的感觉。

And so we have a deep understanding of existing in the physical world.

因此我们对物质世界的存在有了深刻的理解。

But maybe understanding is the wrong word.

But maybe understanding is the wrong word.

We have an experience of living in the world—

We have an experience of living in the world—

Yeah, experience.

是的，经验。

…and riding bicycles and all those things, but I don’t think we actually have an understanding at all, I mean, very, very little of the nature of physical existence.

……以及骑自行车和所有这些事情，但我认为我们实际上根本没有了解，我的意思是，对物质存在的本质知之甚少。

2:01:01 I think it’s a profound mystery.

I think it’s a profound mystery.

Whereas I think that we do have something a little better of an understanding of the nature of mathematical existence and abstract existence.

然而我认为我们确实对数学存在和抽象存在的本质有了更好的理解。

So that’s how I would describe the point.

So that’s how I would describe the point.

Somehow it feels like we’re approaching some deep truth from different directions, and we just haven’t traveled as far in the physics world as we have in the mathematical world.

不知怎的，感觉就像我们正在从不同的方向接近一些深刻的真理，我们只是在物理世界中没有像在数学世界中走得那么远。

Maybe I could hope that someone will give, you know, the convincing account, but it seems to be a profound mystery to me.

也许我希望有人能给出令人信服的说法，但这对我来说似乎是一个深刻的谜。

2:01:41 I, I can’t even imagine what it would be like to give an account of physical existence.

我，我什至无法想象描述物理存在会是什么样子。

Yeah, I wonder, like a thousand years from now as physics progresses- …

是的，我想知道，从现在起一千年后，随着物理学的进步——……

2:01:52 what this same conversation would look like.

what this same conversation would look like.

Right.

正确的。

2:01:59 That would be quite interesting.

That would be quite interesting.

Do you think there’s breakthroughs a thousand years from now on the mathematics side?

您认为一千年后数学方面会取得突破吗？

2:02:00 ‘Cause we’ve just discussed, and we’ll return to a lot of turmoil a century ago.

因为我们刚刚讨论过，我们将回到一个世纪前的混乱状态。

Do you think there’s more turmoil to be had?

Do you think there’s more turmoil to be had?

It’s interesting to me because I have my feet in two worlds: mathematics and philosophy, and to compare the differences between these subjects.

这对我来说很有趣，因为我涉足两个世界：数学和哲学，并比较这些学科之间的差异。

2:02:14 One of the big cultural differences is towards the idea of progress in the subject, because mathematics has huge progress.

最大的文化差异之一是对学科进步的看法，因为数学取得了巨大的进步。

We simply understand the mathematical ideas much, much better, continually improving our understanding, and there’s growth in knowledge.

我们只是更好地理解数学思想，不断提高我们的理解力，知识就会增长。

We understand the nature of infinity now better than they did 100 years ago.

我们现在比 100 年前更了解无穷的本质。

I mean, definitely better.

I mean, definitely better.

And they understood it better 100 years ago than they did, you know, for the previous thousands of years, and so on.

他们在 100 年前比他们之前的几千年更了解这一点，等等。

2:02:57 So, in almost every part of mathematics, there’s improved understanding of the core issues, so much so that the questions at hand become totally different, and the field sort of moves on to more difficult, interesting questions.

因此，几乎在数学的每个部分，对核心问题的理解都得到了提高，以至于手头的问题变得完全不同，并且该领域转向更困难、更有趣的问题。

Whereas in philosophy, that’s a little bit true that there’s progress.

而在哲学中，确实有进步。

But meanwhile, it’s also true that there are these eternal questions that have been with us for thousands of years, and in fact, so much so that you can find a lot of philosophers arguing the important contribution of philosophy is in asking the questions rather than answering them, because it’s hopeless to answer them.

但与此同时，这些永恒的问题确实已经伴随我们数千年了，事实上，很多哲学家认为哲学的重要贡献在于提出问题而不是回答问题，因为回答这些问题是无望的。

I mean, the nature of these deep philosophical questions is so difficult.

我的意思是，这些深刻的哲学问题的本质是如此困难。

Less of a sense of progress is what I’m trying to say.

我想说的是，进步感越来越少。

2:03:48 I don’t see any reason to think that the progress in mathematics, in the growth in our mathematical understanding and knowledge, won’t simply continue.

我认为没有任何理由认为数学的进步、我们数学理解和知识的增长不会简单地持续下去。

And so, a thousand years from now, maybe the mathematics that they will be doing at that time would probably be completely unrecognizable to me.

所以，一千年后，也许他们当时所做的数学对我来说可能完全无法识别。

I maybe wouldn’t even begin to understand what they’re talking about, even without sort of witnessing the intervening developments.

即使没有亲眼目睹事态的发展，我可能也不会开始理解他们在说什么。

So if you bring someone from ancient times to today, they maybe wouldn’t even understand what we’re talking about with some of the questions.

因此，如果你把某人从古代带到今天，他们可能甚至不明白我们在谈论的一些问题。

But I feel that, you know, if Archimedes came and we were able to communicate, I think I would be able to tell him about some of the things that are going on in mathematics now, and maybe, you know…

但我觉得，你知道，如果阿基米德来了，我们能够交流，我想我就能告诉他数学界现在正在发生的一些事情，也许，你知道……

2:04:43 Or anyone from that time, I mean.

Or anyone from that time, I mean.

So I think it is possible to have this kind of progress, even when the subject kind of shifts away from the earlier concerns as a result of the progress, basically.

所以我认为基本上有可能取得这种进展，即使主题由于进展而偏离了早期的关注点。

To take a tangent on a tangent, since you mentioned philosophy, maybe potentially more about the questions, and maybe mathematics is about the answers, I have to say you are a legend on MathOverflow, which is like Stack Overflow but for math.

切线切线，既然你提到了哲学，也许更多的是关于问题，也许数学是关于答案，我不得不说你是 MathOverflow 上的传奇人物，MathOverflow 就像 Stack Overflow，但针对数学。

2:04:57 You’re ranked number one all time on there with currently over 246,000 reputation points.

您一直排名第一，目前拥有超过 246,000 点声誉点。

How do you approach answering difficult questions on there?

您如何在那里回答困难的问题？

Well, MathOverflow has really been one of the great pleasures of my life.

嗯，MathOverflow 确实是我一生中最大的乐趣之一。

2:05:24 I’ve really enjoyed it.

我真的很喜欢它。

And I’ve learned so much from interacting on MathOverflow.

我从 MathOverflow 的互动中学到了很多东西。

I’ve been on there since 2009, which was shortly after it started.

我从 2009 年开始就在那里，那是在它开始后不久。

I mean, it wasn’t exactly at the start, but a little bit later.

我的意思是，这并不完全是在开始时，而是晚了一点。

And I think it gives you the stats for how many characters I typed, and I don’t know how many million it is, but this enormous amount of time that I’ve spent thinking about those questions, and it has really just been amazing to me.

我认为它为您提供了我输入了多少个字符的统计数据，我不知道有多少个字符，但是我花了大量时间思考这些问题，这对我来说真的很神奇。

How do you find the questions that grab you and how do you go about- …

你如何找到吸引你的问题以及你如何着手-......

2:06:06 answering them?

回答他们？

So, I’m interested in any question that I find interesting.

所以，我对任何我觉得有趣的问题都感兴趣。

2:06:11 So…

所以……

And it’s not all questions.

这并不是所有问题。

Sometimes certain kinds of questions just don’t appeal to me that much.

有时某些类型的问题对我来说没有太大吸引力。

So you go outside of set theory as well?

So you go outside of set theory as well?

When I first joined MathOverflow, I was basically one of the few people in logic who was answering.

当我第一次加入 MathOverflow 时，我基本上是逻辑领域为数不多的回答问题的人之一。

2:06:23 I mean, there were other people who know some logic, particularly from category theory and other parts of mathematics that aren’t in the most traditional parts of logic, but they were answering some of the logic questions.

我的意思是，还有其他人了解一些逻辑，特别是范畴论和数学的其他部分，这些部分不属于最传统的逻辑部分，但他们正在回答一些逻辑问题。

So I really found myself able to make a contribution in those very early days by engaging with the logic-related questions.

因此，我确实发现自己能够在很早的时候通过解决与逻辑相关的问题做出贡献。

But there weren’t many logic people asking questions either.

但提出问题的逻辑人士也不多。

But what I found was that there was an enormous amount of interest in topics that were logic-adjacent.

但我发现人们对逻辑相关的话题非常感兴趣。

2:07:04 So a question would arise, you know, in group theory, but it had a logic aspect or an analysis or whatever, and there would be some logic angle on it.

因此，在群论中会出现一个问题，但它有一个逻辑方面或分析或其他什么，并且会有一些逻辑角度。

And what I found was that I was often able to figure out an answer by learning enough about that other subject matter.

我发现，通过充分了解其他主题，我通常能够找到答案。

This is what was so rewarding for me, is because basically I had to learn enough.

这对我来说是非常有意义的，因为基本上我必须学习足够的东西。

My main expertise was logic, but someone would ask a question, you know, that was about, say, the axiom of choice in this other subject matter or the continuum hypothesis or something like that in the other subject matter.

我的主要专长是逻辑，但有人会问一个问题，你知道，这是关于其他主题中的选择公理或连续统假设或其他主题中的类似问题。

And I would have to learn enough about that other subject and the context of the question in order to answer, and I was often able to do that.

我必须对其他主题和问题的背景有足够的了解才能回答，而且我经常能够做到这一点。

And so I was quite happy to do that.

And so I was quite happy to do that.

2:07:50 And also I learned a lot by doing that, because I had to learn about these other problem areas.

通过这样做我也学到了很多东西，因为我必须了解这些其他问题领域。

And so it really allowed me to grow enormously as a mathematician.

所以它确实让我作为一名数学家得到了巨大的成长。

To give some examples of questions you’ve answered: What are some reasonable sounding statements that are independent of ZFC?

举一些您回答过的问题的例子：有哪些独立于 ZFC 的听起来合理的陈述？

2:08:01 What are the most misleading alternate definitions in taught mathematics?

数学教学中最具误导性的替代定义是什么？

Is the analysis as taught in universities in fact the analysis of definable numbers?

大学里教授的分析实际上是对可定义数字的分析吗？

Solutions to the continuum hypothesis?

连续统假设的解？

Most unintuitive application of the axiom of choice?

选择公理最不直观的应用？

Non-trivial theorems with trivial proofs?

具有平凡证明的非平凡定理？

Reductio ad absurdum or the contrapositive?

反证法还是反证法？

What is a chess piece mathematically?

数学上的棋子是什么？

We should say you worked quite a bit on infinite chess, which we should definitely talk about.

我们应该说你在无限国际象棋方面做了很多工作，我们绝对应该谈论这一点。

It’s awesome.

太棒了。

You’ve worked on so many fascinating things.

你从事过很多令人着迷的事情。

Has philosophy ever clarified mathematics?

哲学曾经澄清过数学吗？

Why do we have two theorems when one implies the other?

当一个定理暗示另一个定理时，为什么我们有两个定理？

2:08:48 And, of course, just as an example, you’ve given a really great, almost historical answer on the topic of the continuum hypothesis.

当然，作为一个例子，你对连续统假设的主题给出了一个非常好的、几乎历史性的答案。

Maybe that’s a good place to go.

也许那是一个值得去的地方。

We’ve touched on it a little bit, but it would be nice to lay out what is the continuum hypothesis that Cantor struggled with.

我们已经稍微触及了它，但最好能阐明康托尔所苦苦挣扎的连续统假设是什么。

And I would love to also speak to the psychology of his own life story, his own struggle with it.

我也很想谈谈他自己的人生故事的心理学，以及他自己与之斗争的经历。

The human side of mathematics is also fascinating.

数学的人性化一面也令人着迷。

So what is the continuum hypothesis?

那么什么是连续统假设呢？

The continuum hypothesis is the question that arises so naturally whenever you prove that there’s more than one size of infinity.

每当你证明无穷大的大小不止一种时，连续统假设就会很自然地出现这个问题。

2:09:16 So Cantor proved that the infinity of the real numbers is strictly larger than the infinity of the natural numbers.

于是康托证明了实数的无穷大严格大于自然数的无穷大。

But immediately when you prove that, one wants to know, “Well, is there anything in between?” I mean, what could be a more natural question to ask immediately after that?

但当你证明这一点时，人们立刻想知道，“那么，有什么介于两者之间吗？”我的意思是，在那之后立即问一个更自然的问题是什么？

And so Cantor did ask it, and he spent his whole life thinking about this question.

所以康托确实问了这个问题，他一生都在思考这个问题。

The continuum hypothesis is the assertion that there is no infinity in between the natural numbers and the real numbers.

连续统假设是指自然数和实数之间不存在无穷大。

And, of course, Cantor knew many sets of real numbers.

当然，康托尔知道许多实数组。

Everything in between…

之间的一切……

2:10:01 I mean, everything that’s in that interval would be equinumerous with some set of real numbers.

我的意思是，该区间内的所有内容都与一组实数等价。

But we know lots of sets of real numbers.

但我们知道很多实数集。

I mean, there are all these various closed sets, Cantor sets, and so on.

我的意思是，有各种不同的闭集、康托集等等。

There’s Vitali set.

有维塔利套装。

We have all kinds of sets of real numbers.

我们有各种实数集。

And so you might think, “Well, if the continuum hypothesis is false, then we’ve probably seen the set already.

所以你可能会想，“好吧，如果连续统假设是错误的，那么我们可能已经看到了这个集合。

We just have to prove that it’s strictly in between.” But it turned out that for all the sets that anyone ever could define or pick out or observe, for all the sets of real numbers, it was always the case either that they were countable, in which case they’re equinumerous with the natural numbers or else finite, or they were fully equinumerous with the whole real line.

我们只需证明它严格介于两者之间。”但事实证明，对于任何人都可以定义、挑选或观察的所有集合，对于所有实数集合，它们总是要么是可数的，在这种情况下它们与自然数等价，要么是有限的，或者它们与整个实数线完全等价。

2:10:44 And so they were never strictly in between.

所以他们从来都不是严格介于两者之间的。

I mean, you’re in this situation and you have hundreds, thousands of sets that are candidates to be in between, but in every single case, you can prove it’s on one side or the other and not strictly in between.

我的意思是，在这种情况下，你有成百上千的集合可供选择，但在每种情况下，你都可以证明它是在一侧或另一侧，而不是严格介于两者之间。

And so in every situation where you’re able to figure out whether it’s in between or not, it’s always never strictly in between.

因此，在每种情况下，只要你能够弄清楚它是否介于两者之间，它总是永远不会严格介于两者之间。

Now, Cantor was obsessed with this.

现在，康托对此很着迷。

I think he was.

我想他是。

2:11:11 Yeah, I’m not a historian, so I don’t know the exact history.

是的，我不是历史学家，所以我不知道确切的历史。

Well, everything I’ve seen, it seems to be the question that broke him, huh?

好吧，我所看到的一切似乎都是这个问题让他崩溃了，是吧？

2:11:15 I mean, just struggling with different opinions on the hypothesis within himself and- …desperately chasing, trying to prove it.

我的意思是，他只是在自己内心对这个假设的不同意见中挣扎，然后……拼命地追逐，试图证明它。

So he had a program for proving it, which has been affirmed in a certain respect.

所以他有一个程序来证明这一点，在某个方面已经得到了肯定。

2:11:29 Of course, the continuum hypothesis holds for open sets.

当然，连续统假设对于开集成立。

That’s easy to see.

这很容易看出。

If you have an open interval, then this is fully equinumerous with the whole real line.

如果有一个开区间，那么这与整条实线完全等价。

Any interval is equinumerous with the whole line because all you would need is a function, you know, like the arctangent function or something that maps the whole real line into an interval.

任何区间都与整条线等价，因为您所需要的只是一个函数，例如反正切函数或将整条实线映射到区间的函数。

And that’s a one-to-one function.

这是一个一对一的函数。

So we know the open sets have the property that they’re non-trivial open sets are all fully equinumerous with the whole real line.

所以我们知道开集具有这样的性质：它们是非平凡的开集，并且与整个实数线完全等价。

So never strictly in between.

所以永远不要严格介于两者之间。

But remarkably, Cantor proved it also for the closed sets, and that is using what’s called the Cantor-Bendixson theorem.

但值得注意的是，康托尔也在闭集上证明了这一点，那就是使用所谓的康托尔-本迪克森定理。

2:12:15 So it’s quite a remarkable result.

所以这是一个非常了不起的结果。

It’s definitely not obvious.

这绝对不明显。

And this theorem actually was the origin of the ordinals.

而这个定理实际上就是序数的起源。

Cantor had to invent the ordinals in order to make sense of his Cantor-Bendixson process.

康托必须发明序数词才能理解他的康托-本迪克森过程。

Can you define the open and the closed set in this context?

你能在这种情况下定义开集和闭集吗？

Oh, yeah.

哦，是的。

2:12:34 Sure.

当然。

So a set of reals is open if every point that it contains is surrounded by a little interval of points, the whole tiny little interval.

因此，如果一组实数包含的每个点都被一小段点（整个微小的区间）包围，那么它就是开集。

But that tiny little interval is already just by itself equinumerous with the whole line.

但这个微小的间隔本身就已经与整条线相等了。

So that’s why that question is sort of easy for open sets.

这就是为什么这个问题对于开集来说有点简单。

A closed set is a complement of an open set, and there are a lot of closed sets that are really complicated, of varying sizes.

闭集是开集的补集，并且有很多非常复杂、大小各异的闭集。

So of course, any closed interval is a closed set, but it’s not only those.

所以当然，任何闭区间都是闭集，但不仅仅是这些。

There are also things like the Cantor set, which you get by omitting middle thirds.

还有像康托集这样的东西，你可以通过省略中间三分之一来得到。

Maybe some people have seen this construction.

也许有些人见过这样的建筑。

Or you can imagine sort of randomly taking a lot of little tiny open intervals, you know, all over the line and so on.

或者你可以想象在整个线上随机地取很多微小的开区间，等等。

2:13:18 So that altogether would be an open set, and the complement of it would be a closed set.

因此，这将是一个开集，而它的补集将是一个闭集。

So you can imagine just kind of tossing down these open intervals, and what’s left over is the closed set.

所以你可以想象只是扔掉这些开区间，剩下的就是闭集。

Those sets can be quite complicated, and they can have isolated points, for example, if the two open intervals were just kissing and leaving only the one point between them.

这些集合可能非常复杂，并且它们可以有孤立的点，例如，如果两个空区间只是接吻并且在它们之间只留下一个点。

But also you could have sequences that are converging to a point that would also be a closed set, or convergent sequences of convergent sequences and so on.

但也可以有收敛到一个闭集的点的序列，或者收敛序列的收敛序列等等。

That would be a closed set also.

那也将是一个闭集。

The Cantor set is constructed by iteratively removing open intervals, middle thirds, like you mentioned, from the interval, and trying to see can we do a thing that- that goes in between?

康托集是通过迭代地从间隔中删除开放间隔、中间三分之二来构造的，就像你提到的那样，并试图看看我们可以做一件介于两者之间的事情吗？

Right.

正确的。

2:14:00 So the question would be, “Can you produce a set that has an intermediate size-” …an intermediate cardinality, right?” And Cantor proved with the closed set, “No, it’s impossible.

所以问题是，“你能产生一个具有中间大小的集合——”……中间基数，对吗？康托尔用闭集证明了，“不，这是不可能的。

Every closed set is either countable or equinumerous with the whole real line.”

每个闭集对于整条实线来说要么是可数的，要么是等数的。”

2:14:18 And what the Cantor program for solving the continuum hypothesis was, was a sort of working up.

解决连续统假设的康托程序是一种工作。

So you did it for open sets and for closed sets, and you sort of work up.

所以你做了开放组和封闭组，然后你就进行了锻炼。

Maybe he wants to go into what are called the Borel sets, which are sort of combinations of open and closed sets.

也许他想研究所谓的 Borel 集，这是开集和闭集的组合。

And there’s a vast hierarchy of Borel complexity.

Borel 复杂度有一个巨大的层次结构。

And it turns out that the continuum hypothesis has been proved also for the Borel sets in this hierarchy.

事实证明，对于该层次结构中的 Borel 集，连续统假设也得到了证明。

And, but then one wants to go beyond.

And, but then one wants to go beyond.

What about more complicated sets?

What about more complicated sets?

So there’s this hierarchy of complexity for sets of real numbers.

因此，实数集存在复杂的层次结构。

2:14:53 And Cantor’s idea was to sort of work your way up the hierarchy by proving that the continuum hypothesis was more and more true for those more and more complicated sets, based on our understanding of the earlier cases.

康托尔的想法是，根据我们对早期案例的理解，通过证明连续统假设对于那些越来越复杂的集合越来越正确，从而沿着层次结构向上推进。

And that has been carried out to a remarkable degree.

这已经达到了令人瞩目的程度。

It turns out that one needs, one begins to need large cardinal assumptions though in order to get to the higher realms, even at the level of projective hierarchy, which are sets that you can define by using quantifiers over the real numbers themselves.

事实证明，人们需要，人们开始需要大的基本假设，以便达到更高的领域，即使是在射影层次结构的水平上，这些假设是您可以通过在实数本身上使用量词来定义的集合。

So you get this hierarchy on top of the Borel hierarchy, the hierarchy of projectively definable sets.

因此，您可以将此层次结构置于 Borel 层次结构（可投影定义的集合的层次结构）之上。

2:15:35 And it turns out that if you have enough large cardinals, then the projective sets also are always either countable or equinumerous with the whole real line.

事实证明，如果你有足够大的基数，那么射影集也总是与整个实数线可数或等数的。

And then one can try to go beyond this and so on.

然后人们可以尝试超越这一点等等。

So I view all of those results, which came, you know, in the past 50 years, the later ones, as fulfilling this Cantor idea that goes back, you know, 120 years to his idea that we would prove the continuum hypothesis by establishing more and more instances for greater and greater complexity of sets.

所以我认为所有这些结果，你知道，在过去 50 年里，后来的结果，都实现了康托的想法，这个想法可以追溯到 120 年前他的想法，即我们将通过为越来越复杂的集合建立越来越多的实例来证明连续统假设。

But of course, even with what we know now, it hasn’t fully succeeded and it can’t because the hierarchy of complexity doesn’t include all sets of real numbers.

但当然，即使就我们现在所知，它还没有完全成功，而且也不可能成功，因为复杂性的层次结构并不包括所有实数集。

Some of them are sort of transcending this hierarchy completely in a way.

其中一些在某种程度上完全超越了这个层次结构。

2:16:27 And so the program can’t ever fully be successful, especially in light of the independence result.

因此，该计划永远不可能完全成功，尤其是考虑到独立结果。

Yeah.

是的。

2:16:34 Well, spoiler alert, can you go to the independence result?

好吧，剧透警报，你能看一下独立结果吗？

Sure.

当然。

So what does that mean?

那么这意味着什么呢？

2:16:39 So the continuum hypothesis was shown to be independent from the ZFC axioms of mathematics?

那么连续统假设被证明独立于 ZFC 数学公理吗？

Right.

对。

2:16:46 So the ZFC axioms were the axioms that were put forth first by Zermelo in 1908 in regard to his proof of the well-order theorem using the axiom of choice.

所以ZFC公理是Zermelo在1908年首先提出的公理，他用选择公理证明了良序定理。

That wasn’t fully ZFC.

那不完全是 ZFC。

At that time, it was just Zermelo theory because he sort of…

当时，这只是策梅洛的理论，因为他有点……

There was a kind of missing axiom, the replacement axiom and the foundation axiom were added later, and that’s what makes the Zermelo-Fraenkel axiomatization, which became sort of standard.

有一种缺失的公理，后来添加了替代公理和基础公理，这就是策梅洛-弗兰克尔公理化的原因，它成为了某种标准。

Actually, there’s another aspect, which is Zermelo’s original theory allowed for the existence of ur-elements, or these atoms, mathematical objects that are not sets but out of which we build the set theoretic universe, whereas set theorists today generally don’t use ur-elements at all.

实际上，还有另一个方面，即策梅洛的原始理论允许存在 ur 元素，或者这些原子，这些数学对象不是集合，但我们从中构建了集合论宇宙，而今天的集合论学家通常根本不使用 ur 元素。

2:17:34 I argue that it’s really the philosophy of structuralism that leads them to omit the ur-elements because it turns out that if you adopt ZFC axioms with ur-elements, ZFCU it’s called, or ZFA, then any structure that exists, any mathematical structure that exists in that set theoretic universe with the atoms is isomorphic to a structure that doesn’t use the atoms at all.

我认为，这实际上是结构主义哲学导致他们省略了 ur 元素，因为事实证明，如果你采用带有 ur 元素的 ZFC 公理，称为 ZFCU，或 ZFA，那么任何存在的结构，存在于带有原子的集合论宇宙中的任何数学结构都与根本不使用原子的结构同构。

And you don’t need the atoms if you’re a structuralist because you only care about the structures up to isomorphism anyway, and the theory is simply more elegant and clear without the atoms.

如果你是结构主义者，你就不需要原子，因为无论如何你只关心同构的结构，而且没有原子，理论会更加优雅和清晰。

They’re just not needed.

They’re just not needed.

And so that’s why today when we talk about set theory, generally we talk about the atom-free version, and ZFC has no ur-elements.

这就是为什么今天当我们谈论集合论时，通常我们谈论无原子版本，而 ZFC 没有 ur 元素。

Okay.

好的。

So we formulate the ZFC axioms of set theory.

So we formulate the ZFC axioms of set theory.

2:18:26 These are expressing the main principle ideas that we have about the nature of sets and set existence.

这些表达了我们关于集合本质和集合存在的主要原则思想。

And Cantor had asked about the continuum hypothesis in the late 19th century, and it remained open, totally open until 1938.

康托尔在 19 世纪末询问过连续统假说，直到 1938 年，它仍然是开放的，完全开放的。

We should mention, I apologize, that it was the number one problem in the Hilbert’s 23 set of problems formulated at the beginning of the century.

我们应该指出，我很抱歉，这是本世纪初提出的希尔伯特 23 组问题中的第一个问题。

That’s right.

没错。

Maybe you can comment on why he put that as number one.

也许你可以评论一下为什么他把它放在第一位。

So, right.

所以，对。

2:19:02 So Hilbert had introduced at his famous address at the turn of the century this list of problems that he thought could guide or were important to consider in the coming century of mathematics.

因此，希尔伯特在世纪之交的著名演讲中介绍了他认为可以指导或在下一个世纪的数学中需要考虑的重要问题清单。

I mean, that’s how people talk about it now, although I’m not sure at all…

我的意思是，人们现在就是这么谈论它的，尽管我一点也不确定……

Of course, I can’t really speak for Hilbert at all, but if you were a very prominent mathematician, I find it a little hard to believe that Hilbert would have conceived of his list in the same way that we now take his lists.

当然，我根本不能代表希尔伯特说话，但如果你是一位非常杰出的数学家，我发现有点难以相信希尔伯特会以我们现在采用他的列表的方式构思他的列表。

I mean, having observed the century unfold, we know that that list of 23 problems did in fact guide whole research programs, and it was extremely important and influential.

我的意思是，通过观察本世纪的发展，我们知道这 23 个问题清单实际上指导了整个研究项目，而且极其重要和有影响力。

2:19:46 But at the time, Hilbert would have no reason to think that that would be true, and he was just giving a lecture and had a list of problems that he thought were very important.

但当时希尔伯特没有理由认为这是真的，他只是在做演讲，并列出了他认为非常重要的问题清单。

And so I would find it more reasonable to think that he was just making a list of problems that he thought were extremely interesting and important and fundamental in a way without the heavy burden of guiding this 20th century research.

因此，我认为更合理的是，他只是列出了他认为极其有趣、重要和基础的问题清单，而没有承担指导这项 20 世纪研究的沉重负担。

Although it turns out that, in fact, that’s exactly what they did.

尽管事实证明，事实上，他们就是这么做的。

2:20:18 And we already discussed Hilbert’s views on the nature of set theory and the fundamental character, that quote where he said, “No one will cast us from the paradise that Cantor has created for us.” So I think Hilbert was convinced by Cantor on the importance and the fundamental nature of the continuum hypothesis for the foundations of mathematics, which was a critically important development for the unity of mathematics.

我们已经讨论了希尔伯特对集合论本质和基本特征的看法，他说：“没有人会把我们从康托尔为我们创造的天堂中赶走。”因此，我认为康托尔相信希尔伯特相信连续统假设对于数学基础的重要性和基本性质，这是数学统一性的一个至关重要的发展。

I mean, before set theory emerged as a foundation of mathematics, there were…

我的意思是，在集合论成为数学基础之前，有……

you know, there are different subjects in mathematics.

你知道，数学有不同的科目。

There’s algebra and there’s analysis, real analysis, and topology and geometry, and so on.

有代数，也有分析、实分析、拓扑和几何等等。

There’s all these disparate subjects with their own separate axioms, right?

所有这些不同的主题都有自己独立的公理，对吗？

2:21:03 And, but sometimes it happens, like when you’re proving, say, the fundamental theorem of algebra, you know, that the complex numbers are an algebraically closed field that you can solve any polynomial equation in.

而且，但有时会发生这种情况，比如当你证明代数基本定理时，你知道，复数是一个代数闭域，你可以在其中求解任何多项式方程。

But the proof methods for that theorem come from other parts of mathematics, you know, those topological proofs and so on.

但该定理的证明方法来自数学的其他部分，你知道，那些拓扑证明等等。

And so how does that work?

And so how does that work?

I mean, if you have totally different axiom systems, but you’re using results from one subject in another subject, it’s somehow incoherent unless there’s one underlying subject.

我的意思是，如果你有完全不同的公理系统，但你在另一个主题中使用一个主题的结果，那么除非有一个基础主题，否则它在某种程度上是不连贯的。

So the unity of mathematics was provided by the existence of a mathematical foundation like set theory.

因此，数学的统一性是由集合论等数学基础的存在提供的。

And at the time, it was set theory.

And at the time, it was set theory.

2:21:47 And so it’s critically important to be able to have a single theory in which one views all of mathematics as taking place to resolve that kind of transfer and borrowing phenomenon that was definitely happening.

因此，至关重要的是能够有一个单一的理论，在这个理论中，人们将所有数学视为正在发生的，以解决那种肯定正在发生的转移和借用现象。

So that must have been part of Hilbert’s thinking about why it’s so important to have a uniform foundation, and set theory was playing that role at the time.

因此，这一定是希尔伯特思考为什么拥有统一基础如此重要的一部分，而集合论当时就扮演了这个角色。

Now, of course, we have other possible foundations coming from category theory or type theory, and there’s univalent foundations now.

当然，现在我们有来自范畴论或类型论的其他可能的基础，并且现在有单价的基础。

So there are competing foundations now.

So there are competing foundations now.

There’s no need to just use one set theoretic foundation.

没有必要只使用一套理论基础。

2:22:25 Although set theory continues to, in my view, have an extremely successful metamathematical analysis as a foundation, I think is much more successful than set theory for any of those other foundations, but it’s much less amenable to things like computer proof and so on, which is part of the motivation to find these alternative foundations.

在我看来，虽然集合论仍然拥有极其成功的元数学分析作为基础，但我认为它比任何其他基础的集合论要成功得多，但它不太适合计算机证明等事物，这是寻找这些替代基础的部分动机。

So, yeah, just to talk about Hilbert, I think he was motivated by the need for unifying foundation of mathematics and set theory was playing that role, and the continuum hypothesis is such a core fundamental question to ask, so it seems quite natural that he would put it on the list.

所以，是的，只是谈论希尔伯特，我认为他的动机是需要统一数学基础，而集合论正在发挥这一作用，而连续统假设是一个需要提出的核心基本问题，所以他将其列入清单似乎是很自然的。

There were other logic-related questions, though, like Hilbert’s tenth problem is also related to logic.

不过，还有其他与逻辑相关的问题，例如希尔伯特的第十个问题也与逻辑相关。

2:23:08 This is the question about Diophantine equations, and he asked to provide an algorithm to decide whether a given Diophantine equation has a solution in the integers.

这是关于丢番图方程的问题，他要求提供一种算法来判断给定的丢番图方程是否有整数解。

So a Diophantine equation is just…

So a Diophantine equation is just…

I mean, it’s a, maybe a fancy way of talking about something that’s easy to understand, a polynomial equation, except it’s not just one variable, many variables.

我的意思是，这也许是一种谈论易于理解的事物的奇特方式，即多项式方程，只不过它不仅仅是一个变量，而是许多变量。

So you have polynomials in several variables over the integers, and you want to know, can you solve it?

因此，您在整数上有多个变量的多项式，并且您想知道，您能解决它吗？

So the problem is, as stated by Hilbert, provide an algorithm for answering the question whether a given polynomial equation has a solution in the integers.

因此，正如希尔伯特所说，问题是提供一种算法来回答给定多项式方程是否有整数解的问题。

So he’s sort of presuming that there is an algorithm, but he wants to know what it is.

所以他有点假设存在一种算法，但他想知道它是什么。

What is the algorithm?

算法是什么？

2:23:55 But the problem was solved by proving that there is no algorithm.

但通过证明没有算法，问题就解决了。

It’s an undecidable problem, like the halting problem.

这是一个不可判定的问题，就像停机问题一样。

There is no computable procedure that will correctly decide whether a given polynomial equation has a solution in the integers.

没有可计算的过程可以正确确定给定的多项式方程是否有整数解。

So that’s quite a remarkable development, I think.

所以我认为这是一个相当了不起的发展。

So there were also a few other logic-related questions on the list.

所以清单上还有一些其他与逻辑相关的问题。

And so eventually, the continuum hypothesis was shown to be independent from ZFC axioms, as we’ve mentioned.

因此，正如我们所提到的，最终，连续统假设被证明独立于 ZFC 公理。

2:24:20 So what…

那又怎样……

How does that make you feel?

这让你感觉如何？

What is independence and what does that mean?

What is independence and what does that mean?

But once you tell the story, the historical story-

但一旦你讲故事，历史故事——

Yes

是的

…

……

2:24:32 is really quite dramatic-

is really quite dramatic-

Yeah, that’s great

是的，那太好了

I think.

我想。

2:24:34 Because Cantor poses the question in the late 19th century.

因为康托尔在 19 世纪末提出了这个问题。

And then it’s totally open.

And then it’s totally open.

Hilbert asks about it, you know, at the turn of the 20th century.

你知道，希尔伯特在 20 世纪之交问过这个问题。

Nobody has any clue.

没有人有任何线索。

There’s no answer coming.

没有答案。

Until 1938, this is four decades later, right?

Until 1938, this is four decades later, right?

So a long time, and Godel, Kurt Godel proved half of it.

这么长时间了，哥德尔，库尔特·哥德尔证明了一半。

What he proved is that if the axioms of set theory are consistent, then there is a set theoretic world where both the axiom of choice and the continuum hypothesis are true.

他证明的是，如果集合论的公理是一致的，那么就存在一个集合论世界，其中选择公理和连续统假设都为真。

So what he’s doing is showing, this is called the constructible universe, Godel’s L.

所以他正在做的是展示，这就是所谓的可构造宇宙，哥德尔的L。

So he solved this…

于是他解决了这个问题……

this is the same result where he answers the safety question of the axiom of choice, but also for the continuum hypothesis.

这与他回答选择公理的安全问题的结果相同，而且对于连续统假设也是如此。

They’re true in the same set theoretic universe we get.

它们在我们所得到的同一个集合理论宇宙中都是正确的。

2:25:38 So if ZF, without the axiom of choice, is consistent, then so is ZFC plus the continuum hypothesis is the result, 1938.

因此，如果没有选择公理的 ZF 是一致的，那么 ZFC 加上连续统假设就是结果，1938。

It’s such a beautiful argument.

It’s such a beautiful argument.

It’s just incredible, I think, because he’s building an alternative mathematical reality.

我认为这简直令人难以置信，因为他正在构建另一种数学现实。

That’s the structure of the proof is that, okay, if there’s any mathematical reality, if there’s any set theoretic world, then we’re going to build another one, a separate one, a different one, maybe different.

证明的结构是，好吧，如果存在任何数学现实，如果存在任何集合理论世界，那么我们将构建另一个世界，一个单独的世界，一个不同的世界，也许是不同的世界。

Maybe it’s the same as the original one.

Maybe it’s the same as the original one.

It could be.

可能是。

If we started already in the one that he built, then it would be the same.

如果我们已经从他建造的那个开始，那么它会是一样的。

But there’s no reason to assume it was the same.

但没有理由认为它们是相同的。

2:26:15 So he has this kind of model construction method to build this alternative set theoretic reality, the constructible universe, and then he proves that the axiom of choice is true there and also the continuum hypothesis is true there, and it’s just amazing.

所以他有这种模型构建方法来构建这个替代的集合论现实，可构建的宇宙，然后他证明了选择公理在那里是正确的，连续统假设在那里也是正确的，这真是太神奇了。

Really beautiful argument.

真是美丽的论点。

Okay, so then for the other part of the independence, that’s only half of it, because Godel shows basically that you can’t refute the continuum hypothesis, but that’s not the same thing as proving that it’s true.

好吧，那么对于独立性的另一部分，这只是一半，因为哥德尔基本上表明你不能反驳连续统假设，但这与证明它是正确的不是一回事。

He showed that if set theory is consistent without the continuing hypothesis, then it’s consistent with the continuing hypothesis.

他证明，如果集合论在没有连续假设的情况下是一致的，那么它就与连续假设一致。

So that’s not the same thing as proving that it’s true.

所以这与证明它是真的不是一回事。

Yeah.

是的。

2:26:59 And then it didn’t come until 1963 when Paul Cohen invented the method of forcing and proved that if there’s a model of set theory, then there’s a model of set theory in which the continuum hypothesis is false.

直到1963年，保罗·科恩发明了强迫方法，并证明如果存在一个集合论模型，那么就存在一个连续统假设为假的集合论模型。

So Cohen also is giving us this extremely powerful tool for building alternative mathematical realities, is how I think about it.

因此，科恩还为我们提供了这个极其强大的工具来构建替代的数学现实，这就是我的想法。

He’s explained to us how to take any set theoretic world and build another different one in which the continuum hypothesis is false.

他向我们解释了如何采用任何集合理论世界并建立另一个不同的世界，其中连续统假设是错误的。

The forcing extension.

强制延伸。

It’s such a fascinating technique, a tool of forcing.

这是一种非常令人着迷的技术，一种强迫的工具。

2:27:37 Maybe I’m anthropomorphizing it, but it seems like a way to escape one mathematical universe into another, or to expand it or to alter it.

也许我正在将它拟人化，但这似乎是一种逃离一个数学宇宙进入另一个数学宇宙，或者扩展它或改变它的方法。

So you travel between mathematical universes.

所以你在数学宇宙之间旅行。

Can you explain the technique of forcing?

您能解释一下强制技术吗？

Yeah, exactly.

是的，完全正确。

2:27:57 It’s all those things.

就是这些事情。

It’s so wonderful.

太棒了。

I mean, that’s exactly how I think about it.

我的意思是，这正是我的想法。

I mean…

我的意思是……

And we should mention, maybe this is a good place to even give a bigger picture.

我们应该提到，也许这是一个提供更大视野的好地方。

2:28:03 One of your more controversial ideas in mathematics, as laid out in the paper, “The Set-Theoretic Multiverse,” you describe that there may not be one true mathematics, but rather multiple mathematical universes, and forcing is one of the techniques that gets you from one to the other, so…

您在数学方面最具争议性的想法之一，正如论文“集合论多元宇宙”中所述，您描述可能不存在一种真正的数学，而是存在多个数学宇宙，而强迫是让您从一个数学宇宙到另一个数学宇宙的技术之一，所以……

Can you explain the whole shebang?

你能解释一下整个事情吗？

The whole…

整个…

Yeah, sure, let’s get into it.

是的，当然，让我们开始吧。

2:28:27 So the lesson of Cohen’s result and Gödel’s result and so on, these producing these alternative set theoretic universes.

所以科恩的结果和哥德尔的结果等等的教训，这些产生了这些替代的集合论宇宙。

We’ve observed that the continuum hypothesis is independent and the axiom of choice is independent of the other axioms, but it’s not just those two.

我们观察到连续统假设是独立的，选择公理也独立于其他公理，但不仅仅是这两个公理。

We have thousands of independence results.

我们有数千个独立结果。

Practically every non-trivial statement of infinite combinatorics is independent of ZFC.

实际上，无限组合的每个重要陈述都独立于 ZFC。

I mean, this is the fact.

我的意思是，这就是事实。

It’s not universally true.

这并不普遍正确。

There are some extremely difficult prominent results where people proved things in ZFC, but for the most part, if you ask a non-trivial question about infinite cardinalities, then it’s very likely to be independent of ZFC.

人们在 ZFC 中证明了一些极其困难的突出结果，但在大多数情况下，如果你问一个关于无限基数的重要问题，那么它很可能独立于 ZFC。

2:29:15 And we have these thousands of arguments, these forcing arguments that are used to establish that.

我们有成千上万的论据，这些令人信服的论据被用来证明这一点。

And so how should we take that?

那么我们应该如何看待这一点呢？

I mean, on the one hand, if you have a theory and it doesn’t answer any of the questions that you’re interested in, okay, so what does that mean?

我的意思是，一方面，如果你有一个理论，但它不能回答你感兴趣的任何问题，那么这意味着什么呢？

If you’re following what I call the universe view or the monist view, you might naturally say, “Well, look, ZFC is a weak theory, and there’s the true set theoretic reality out there, and we need a better theory because the current theory isn’t answering the questions.

如果你遵循我所说的宇宙观或一元论观点，你可能会自然地说：“嗯，看，ZFC 是一个弱理论，并且存在真集理论现实，我们需要一个更好的理论，因为当前的理论没有回答这些问题。

Everything’s independent.” And so that seems like a quite reasonable thing to take if you think that there is…

一切都是独立的。”因此，如果您认为存在……，那么这似乎是一个相当合理的事情。

2:29:59 that every set theoretic question has a definite answer, and there’s a unique set theoretic truth or a unique fact of the matter, right?

每个集合论问题都有一个明确的答案，并且存在唯一的集合论真理或唯一的事实，对吗？

This is the universe view.

这就是宇宙观。

And by the way, to reiterate, independent means it cannot be proved or disproved within this axiomatic system, within this theory.

顺便说一句，重申一下，独立意味着它不能在这个公理体系、这个理论中被证明或反驳。

Right, exactly.

对，完全正确。

2:30:16 So to be independent means you can’t prove it, and also you can’t prove that it’s false.

所以独立意味着你无法证明它，也无法证明它是假的。

You can’t refute it.

你无法反驳它。

And you’re saying that’s why the statement is so traumatic or sad, that most of the interesting stuff, as you said, has been shown to be independent.

你说这就是为什么这个声明如此痛苦或悲伤，正如你所说，大多数有趣的东西都被证明是独立的。

2:30:22 Of ZFC.

ZFC的。

But that’s an interesting way to put it, I think, because it reminds me of this…

但我认为这是一种有趣的表达方式，因为它让我想起了这一点……

2:30:31 When I was a graduate student in Berkeley, there was another graduate student who was working with a non-logic professor in C*-algebras or something like this.

当我在伯克利读研究生时，有另一位研究生正在与一位非逻辑教授一起研究 C* 代数或类似的学科。

So it’s a part of analysis or functional analysis, and they were looking at a question, and it turned out to be independent of ZFC, right?

所以它是分析或功能分析的一部分，他们正在研究一个问题，结果证明它与 ZFC 无关，对吧？

And the attitude of this other professor was that, “Oh, I guess I asked the wrong question.” But my attitude and the attitude of all the set theorists was when you ask a question that turns out to be independent, then you asked exactly the right question because this is the one…

另一位教授的态度是，“哦，我想我问错了问题。”但我的态度和所有集合论学家的态度是，当你问一个结果是独立的问题时，你就问了完全正确的问题，因为这就是那个……

You know, it’s carving nature at its joints.

你知道，它是在关节处雕刻自然。

2:31:18 You’re adjudicating the nature of set theoretic reality by finding these two realms.

你通过找到这两个领域来判断集合论现实的本质。

You find one of these dichotomies.

你会发现这些二分法之一。

You know, there are the worlds where it’s true and the worlds where it’s false.

你知道，有些世界是真的，有些世界是假的。

And so when you ask that question, that’s to be celebrated.

所以当你问这个问题时，这是值得庆祝的。

It means you asked exactly the right, interesting, fascinating question.

这意味着你问了一个正确的、有趣的、引人入胜的问题。

So it’s not a kind of bleak thing that you can’t prove it and you can’t refute it, and that’s such a disaster.

所以这不是一种你无法证明、你无法反驳的惨淡事情，那就是一场灾难。

Rather, it means that you found this cleavage in mathematical reality, and it’s good to know about those when they happen, you know?

相反，这意味着你在数学现实中发现了这种分歧，当它们发生时了解它们是件好事，你知道吗？

Carving nature at its joints.

在关节处雕刻自然。

2:31:52 So what can you do about the things that are shown to be independent from ZFC?

那么对于那些被证明独立于 ZFC 的东西你能做什么呢？

Right.

对。

2:32:00 So…

所以……

What are the techniques?

有哪些技巧呢？

So one thing is that because of the incompleteness theorem, we know that there’s going to be…

所以一件事是，由于不完备性定理，我们知道将会有……

2:32:01 For any theory that we can write down, there’s going to be true things we can’t prove in it.

对于我们可以写下来的任何理论，其中都会有我们无法证明的真实事物。

So those things are going to be independent.

所以这些事情将是独立的。

And so we’re already aware of the fact that there will always be these independent phenomena for any theory that we write.

因此，我们已经意识到这样一个事实：对于我们所写的任何理论，总会存在这些独立的现象。

And furthermore, some of those theories we won’t even be able to prove that they’re consistent, you know, like the consistency of their own theory.

此外，有些理论我们甚至无法证明它们是一致的，你知道，就像他们自己的理论的一致性一样。

So that’s called the consistency-strength hierarchy.

这就是所谓的一致性强度层次结构。

2:32:37 So it’s a direct consequence of Gödel’s second incompleteness theorem that for any theory we can write down, then towering over it is this incredibly tall tower of consistency strength, where the strength in theories aren’t just adding another axiom, but they’re adding another axiom even whose consistency was not provable in the previous layers of the hierarchy.

因此，这是哥德尔第二不完备性定理的直接结果，对于我们可以写下的任何理论，然后耸立在它之上的是这座令人难以置信的高高的一致性强度塔，其中理论的强度不仅仅是添加另一个公理，而是添加另一个公理，即使其一致性在层次结构的前几层中无法证明。

And so how lucky we are to find the large cardinal axioms that instantiate exactly this feature of increasing consistency strength, this unending and extremely tall hierarchy of consistency strength of axioms.

因此，我们多么幸运地找到了大的基本公理，它们准确地实例化了一致性强度不断增加的这一特征，以及公理一致性强度的这种无休止且极高的层次结构。

And it exactly fulfills the prediction that Gödel’s theorem makes about that kind of thing.

而它恰恰满足了哥德尔定理对此类事情的预言。

2:33:28 Except, the axioms in the large cardinal hierarchy aren’t, you know, metalogical self-referential statements of the form that sometimes arise in the Gödel analysis, but rather they’re professing existence of big infinities, these large cardinal axioms.

除此之外，大基数层次中的公理并不是哥德尔分析中有时出现的元逻辑自我指涉陈述，而是它们声称存在大无穷大，即这些大基数公理。

And so it’s such a welcome development, and yet it’s also known that the continuum hypothesis is independent of all of the known large cardinal axioms.

因此，这是一个非常受欢迎的发展，但众所周知，连续统假设独立于所有已知的大基本公理。

So none of the large cardinal axioms can settle the continuum hypothesis.

因此，没有一个大的基本公理可以解决连续统假设。

So the independence phenomenon is still there for things like the continuum hypothesis and the cardinal combinatorics that I mentioned.

因此，对于我提到的连续统假设和基数组合学等事物，独立现象仍然存在。

So you’re building this incredible hierarchy of axiomatic systems that are more powerful than ZFC.

所以你正在构建这个令人难以置信的公理系统层次结构，它比 ZFC 更强大。

More powerful than ZFC and then more powerful than that, more powerful than that, and so on.

比ZFC更强大，然后比ZFC更强大，比ZFC更强大，等等。

2:34:24 It keeps going forever, and it will never be finished.

它会永远持续下去，而且永远不会结束。

And still, to this day, the continuum hypothesis does not…

直到今天，连续统假说仍然没有……

It’s not settled by any of the large cardinal axioms.

任何大的基本公理都无法解决这个问题。

Wow.

哇。

2:34:39 Wow.

哇。

How does that make you feel?

这让你感觉如何？

Will it ever be settled?

它会得到解决吗？

Well, it’s part of my multiverse view, I guess.

嗯，我想这是我多元宇宙观的一部分。

2:34:47 We started by describing the universe view, which is the view that there are facts about all of these questions, and it will turn out—if you’re a universe view person, which I’m not, but if you are—then you will hold that there is a right answer to the continuum hypothesis question, and there’s a right answer to the large cardinal questions, and so on.

我们首先描述宇宙观，即所有这些问题都有事实存在的观点，结果会是——如果你是一个宇宙观的人，而我不是，但如果你是——那么你就会认为连续统假设问题有一个正确的答案，对于大的基本问题也有一个正确的答案，等等。

And that what we should be aiming to do is figure out this one true set theory.

我们的目标应该是找出这一真集理论。

In contrast, I take the developments of set theory over the past half-century or more as evidence that there isn’t such a unique set-theoretic reality.

相比之下，我将集合论在过去半个世纪或更长时间里的发展视为不存在如此独特的集合论现实的证据。

2:35:33 Rather, what we’ve been doing for decades now is producing more and more alternative set-theoretic universes in which the fundamental truths differ from one to the other.

相反，我们几十年来一直在做的事情是产生越来越多的替代性集合论宇宙，其中的基本真理彼此不同。

And that is the answer to the continuum hypothesis question: the fact that given any model of set theory, there’s a forcing extension where the continuum hypothesis is true, and another one where it’s false.

这就是连续统假设问题的答案：事实上，给定任何集合论模型，都有一个强制外延，其中连续统假设为真，而另一个强制外延则为假。

You can sort of turn it on and off like a light switch.

您可以像电灯开关一样打开和关闭它。

And that’s the fundamental nature of the continuum hypothesis, that you can have it or you can have the negation as you like within a very closely related set-theoretic world.

这就是连续统假设的基本性质，在一个非常密切相关的集合论世界中，你可以随心所欲地拥有它，也可以拥有它的否定。

Wherever you happen to be living, there’s a closely related one where CH is true, where the continuum hypothesis is true, and one where it’s false.

无论你生活在哪里，都会有一个密切相关的假设，CH 为真，连续统假设为真，而连续统假设为假。

2:36:23 And that itself is a kind of answer.

这本身就是一种答案。

It’s not a singularist answer, a universe view answer.

这不是单一主义的答案，而是宇宙观的答案。

It’s a pluralist answer.

这是一个多元化的答案。

And this led me to my views on the multiverse view of set theory and pluralist truth, namely the fundamental nature of set-theoretic truth has this plural character in that there isn’t a singular meaning to the fundamental terms, but rather there’s this choice of alternative set-theoretic universes that have different truths.

这使我对集合论和多元真理的多元宇宙观产生了看法，即集合论真理的基本性质具有复数特征，因为基本术语没有单一含义，而是存在具有不同真理的替代集合论宇宙的选择。

So what does the multiverse view of mathematics enable you to do?

那么数学的多元宇宙观能让你做什么呢？

2:36:55 What does it empower you to do, and what are the limitations?

它能让您做什么？有哪些限制？

What are the things it breaks about mathematics as a field, as a space of knowledge, and what does it enable?

它打破了数学作为一个领域、作为一个知识空间的哪些方面，以及它带来了什么？

First of all, I guess one should say that these different philosophical positions that you might take in the philosophy of set theory, like the multiverse view or the universe view, we don’t ever disagree about the mathematics.

首先，我想人们应该说，你在集合论哲学中可能采取的这些不同的哲学立场，比如多元宇宙观或宇宙观，我们在数学上从来没有分歧。

2:37:10 We’re all agreeing on what the theorems are.

我们都同意这些定理是什么。

It’s a question of philosophical perspective on the underlying meaning or the context, or really what is a philosophy of mathematics for?

这是一个对潜在意义或背景的哲学视角的问题，或者说数学哲学到底有什么用？

And I mean, if you look back in history, for example, to the time of calculus with Newton and Leibniz, right?

我的意思是，如果你回顾历史，例如，回顾一下牛顿和莱布尼茨的微积分时代，对吧？

2:37:44 They famously developed the ideas of calculus using their concepts of infinitesimals, and those foundations were roundly mocked by Bishop Berkeley and so on, who talked about, you know, “What are these same evanescent increments?

他们利用无穷小概念发展了微积分思想，而这些基础遭到了伯克利主教等人的严厉嘲笑，他们谈到，你知道，“这些转瞬即逝的增量是什么？

And shall we not call them the ghosts of departed quantities?” But the foundations really were kind of completely suspect, I think, at the time.

难道我们不应该称它们为已逝数量的幽灵吗？”但我认为，当时的基金会确实是完全值得怀疑的。

And the foundations of infinitesimal calculus really only became rigorous in the 1950s or so with the development of non-standard analysis and Robinson’s work.

直到 20 世纪 50 年代左右，随着非标准分析和罗宾逊工作的发展，无穷微积分的基础才真正变得严谨。

Okay, so the point I’m trying to make is that, do you need a robust, rigorous foundation of mathematics to make enduring insights in mathematics?

好吧，我想说的是，你是否需要坚实、严格的数学基础才能获得持久的数学见解？

2:38:34 And the answer, regrettably, is apparently not because in calculus, even with that lousy, creaky foundation of infinitesimals not even well understood that Newton and Leibniz had, they proved all the fundamental theorems of calculus and, you know, they had all the main insights in those early days with that extremely bad foundation.

遗憾的是，答案显然不是因为在微积分中，即使牛顿和莱布尼茨拥有那个糟糕的、摇摇欲坠的无穷小基础，他们甚至都没有很好地理解，但他们证明了微积分的所有基本定理，而且，你知道，他们在早期的基础非常糟糕的情况下拥有所有主要的见解。

And so that shows you something about the relevance of the kind of foundational views on mathematics and how important they are for mathematical developments and progress and insight.

因此，这向您展示了数学基本观点的相关性，以及它们对于数学发展、进步和洞察力的重要性。

I mean, because I view those early mathematical developments in calculus as genuinely mathematical and extremely important and insightful, even though the foundations weren’t any good by contemporary perspectives.

我的意思是，因为我认为微积分的早期数学发展是真正的数学，并且极其重要和富有洞察力，尽管从当代的角度来看，其基础并不好。

Okay.

好的。

So, rather…

所以，宁可……

2:39:32 So when it comes to the philosophy of set theory and the dispute between the universe view and pluralism, my view is that the choice of the philosophical perspective doesn’t actually have to do with the mathematical developments directly at all.

所以当谈到集合论哲学以及宇宙观和多元论之间的争论时，我的观点是，哲学视角的选择实际上与数学的发展没有直接关系。

Rather, it tells us, “Where should set theory go?

相反，它告诉我们：“集合论应该走向何方？

What kind of set theory should we be looking at?

我们应该关注什么样的集合论？

What kind of questions should we be asking?” So if you have a universe mentality, the universe view, then you’re going to be pushed to try to find and articulate the nature of the one true set-theoretic universe.

我们应该问什么样的问题？”因此，如果你有一种宇宙心态、宇宙观，那么你就会被迫去尝试寻找并阐明唯一真正的集合论宇宙的本质。

And I think that remark is really well borne out by the developments with Hugh Woodin, who’s one of the most prominent mathematicians and philosophers with the universe view and his theory of ultimate L and so on.

我认为休·伍丁（Hugh Woodin）的发展确实很好地证实了这一点，他是最著名的数学家和哲学家之一，拥有宇宙观和终极 L 理论等。

And he’s really striving.

他真的很努力。

Who was also your advisor.

谁也是你的顾问。

He was also my supervisor, yeah, my graduate supervisor.

他也是我的导师，是的，我的研究生导师。

Which is a personal story as well.

这也是一个个人故事。

This fundamental dispute, yeah, on this question.

是的，在这个问题上存在根本性的争议。

2:40:32 But he is a very strong and successful research program, sort of trying to give legs to finding the nature of the one true set-theoretic universe.

但他是一个非常强大和成功的研究项目，有点试图为寻找唯一真正的集合论宇宙的本质提供支持。

And it’s driving the questions that he’s asking and the mathematical programs that he’s pursuing.

这推动了他提出的问题和他正在追求的数学程序。

Whereas if you have a pluralist view, as I do, then you’re going to be led and attracted to questions that have to do with the interaction of different set-theoretic universes, or maybe you want to understand the nature of how are the models of set theory related to their forcing extensions and so on.

然而，如果你像我一样有多元主义观点，那么你就会被引导和吸引到与不同集合论宇宙的相互作用有关的问题，或者你可能想了解集合论模型如何与其强制扩展相关的本质等等。

And so this led to things that I call, say, set-theoretic potentialism, where you think about a set-theoretic universe in a potentialist way.

因此，这导致了我所说的集合论势能主义，你以势能主义的方式思考集合论宇宙。

2:41:21 Not in the sense of potential infinity directly, because all of these universes have infinite sets inside them already, but they’re potentialist in the sense that we could have more sets.

不是直接意义上的势无穷大，因为所有这些宇宙内部已经有无限的集合，但它们是势论的，因为我们可以拥有更多的集合。

The universe could be wider and taller and so on, by forcing or by extending upward.

通过强迫或向上延伸，宇宙可以变得更宽、更高等等。

And so we want to understand the nature of this realm of set-theoretic universes.

因此，我们想要了解集合论宇宙领域的本质。

And that’s quite some exciting work.

这是相当令人兴奋的工作。

And so with Benedikt Loewe and I, we proved some theorems on the modal logic of forcing and set-theoretic potentialism under end extension.

因此，我和 Benedikt Loewe 证明了关于末端扩展下的强迫模态逻辑和集合论势能论的一些定理。

I’ve done a bunch of work on this topic.

我在这个主题上做了很多工作。

And also I mounted, together with Gunter Fuchs and Jonas Riets, who was one of my own PhD students, the topic of set-theoretic geology, which is studying…

我还与冈特·福克斯（Gunter Fuchs）和乔纳斯·里茨（Jonas Riets）（我自己的博士生之一）一起提出了集合论地质学的主题，该主题正在研究……

2:42:11 It’s taking the metaphor of forcing.

它用强迫来比喻。

I mean, in forcing, you have the ground model and the forcing extension.

我的意思是，在强制中，你有地面模型和强制扩展。

And when I was first working with Jonas, he said, “I want to undo forcing.

当我第一次与乔纳斯合作时，他说：“我想取消强迫。

I want to go backward.” And I at first said, “But Jonas, it doesn’t work that way.

我想向后退。”我一开始说：“但是乔纳斯，事情不是这样的。

You start in the model, in the ground model, and you go out, you go to the bigger one.

你从模型开始，在地面模型中，然后你出去，你去更大的模型。

You know, that’s how forcing works.” And he said, “No, no, I want to go backward.” And so he was quite persistent, actually.

你知道，这就是强迫的运作方式。”他说：“不，不，我想倒退。”事实上，他非常坚持。

And so finally, I said, “Okay, let’s do it.” Let’s take it seriously.” And so we sat down and started thinking more precisely and carefully and deeply about the nature of taking a set-theoretic universe and seeing where did it come from by forcing, which was a new way of thinking about forcing at the time.

最后，我说：“好吧，我们开始吧。”让我们认真对待吧。”于是我们坐下来，开始更精确、更仔细、更深入地思考集合论宇宙的本质，看看它是从哪里来的力，这在当时是一种新的思考力的方式。

Like reverse-engineering the forcing?

就像对强迫进行逆向工程一样？

Yeah, something like that.

是的，类似的事情。

2:43:00 Forcing is a way of producing a new universe.

强迫是产生新宇宙的一种方式。

And so you could start somewhere and go to that new universe, or you could look where you are and say, “Well, look, I got here by doing that already in the past.”

所以你可以从某个地方开始，进入那个新宇宙，或者你可以看看你所在的地方，然后说，“好吧，看，我过去已经这样做了，所以来到了这里。”

2:43:12 So we defined models of the bedrock model and ground, sort of undoing the forcing.

因此，我们定义了基岩模型和地面模型，某种程度上消除了强迫。

And really, it was quite fruitful.

确实，这是非常富有成果的。

And I view this as part of the pluralist perspective, except the difference is that set-theoretic geology is amenable to the universe view.

我认为这是多元主义观点的一部分，不同之处在于集合论地质学适合宇宙观。

So even though the work was inspired by this philosophical view on the multiverse view, nevertheless, the central ideas of geology have now been picked up by the people with the research program in the universe view, because it turns out that set-theoretic geology is helping them or us to discover the nature of the one true universe relates to its mantle.

因此，尽管这项工作的灵感来自于这种多元宇宙观的哲学观点，然而，地质学的中心思想现在已经被宇宙观研究项目的人们所接受，因为事实证明，集合论地质学正在帮助他们或我们发现与其地幔相关的唯一真实宇宙的本质。

There’s this concept of the set-theoretic mantle that I had introduced in a way that is extremely interesting.

我以一种非常有趣的方式引入了集合论的概念。

2:44:01 And so it’s historically quite funny, I think, because this research program that grew entirely out of the pluralist point of view ended up being picked up by the universe point of view research program in a way that is quite important.

所以我认为这在历史上是相当有趣的，因为这个完全出于多元观点的研究项目最终以一种非常重要的方式被宇宙观点研究项目所采用。

Can you prove something in the world that you arrived at through forcing and then take some of that back to the ground model?

你能证明你通过强迫到达的世界中的某些东西，然后将其中的一些带回地面模型吗？

Yeah, absolutely.

是的，绝对是。

2:44:30 And that’s a really powerful argument method, actually.

实际上，这是一种非常强大的论证方法。

People often want to do that.

人们常常想这样做。

Suppose you’re in some set-theoretic context.

假设您处于某种集合论背景中。

You know, you could think about it as living in a set-theoretic universe, and you want to prove something in that universe only.

你知道，你可以把它想象成生活在一个集合论宇宙中，并且你只想证明那个宇宙中的某些东西。

But maybe one way to do it is to first construct this forcing extension and then use the features about this forcing extension to realize that certain things must have already been true in the ground model.

但也许一种方法是首先构建这个强迫扩展，然后使用有关这个强迫扩展的特征来认识到某些事情在地面模型中一定已经是正确的。

And then you throw the forcing extensions away and you-

然后你扔掉强制扩展然后你-

Oh, cool

哦，酷

Yeah.

是的。

2:45:01 So this can happen.

所以这可能会发生。

To pick a more elementary example, if you think about the early days of people reasoning with the complex numbers before they really understood them.

举一个更基本的例子，如果你想想早期人们在真正理解复数之前就用它们进行推理。

So they would have these algebraic equations that they’re trying to solve.

所以他们会尝试解决这些代数方程。

They would have the tools and methods of doing it, but then in the course of, you know, they would have to do things to the polynomial and change the factors and so on, and produce other polynomials and solve them and so on.

他们会有这样做的工具和方法，但在这个过程中，你知道，他们必须对多项式进行处理并改变因子等等，并产生其他多项式并求解它们等等。

Sometimes, they could produce solutions.

有时，他们可以提出解决方案。

In the middle of their construction, they were led to, like, the square root of minus five or something in the construction.

在他们的构造过程中，他们被引导到，例如负五的平方根或构造中的其他东西。

And they didn’t have any meaning for that, but they would just do it symbolically, you know.

他们对此没有任何意义，但他们只是象征性地这样做，你知道。

2:45:48 And eventually, it would turn out, because of the methods that they had, they would combine and they would cancel and so on, and all the complex parts would cancel out and they’d end up with this actual answer, you know, three plus square root of 17 or whatever.

最终，结果是，由于他们拥有的方法，他们会结合起来，他们会取消等等，所有复杂的部分都会取消，他们最终会得到这个实际的答案，你知道，三加17的平方根或其他什么。

And they could check it and it worked.

他们可以检查它并且它起作用了。

It was a solution of the original equation.

这是原方程的解。

And so it must have been bewildering to them because they would start with this question purely in the real numbers, an algebraic question, and they would march on their method and proceed through the land of nonsense, you know, with these square roots of negative numbers and then end up with an answer that was real again that they could verify was correct.

所以这对他们来说一定是令人困惑的，因为他们会从这个纯粹的实数问题开始，一个代数问题，然后他们会继续他们的方法，并通过这些负数的平方根继续进行无意义的土地，然后最终得到一个再次真实的答案，他们可以验证它是否正确。

2:46:29 And so I view this kind of forcing argument that I was just describing in a similar way.

所以我以类似的方式看待我刚刚描述的这种强制论证。

You start in set theory and you go to this land of nonsense in the forcing extension, this imaginary world.

你从集合论开始，然后你进入了强制扩展中的这片胡言乱语的土地，这个想象的世界。

And you argue and you come back.

然后你争论然后你回来。

I mean, you make a consequence in the ground model, and it’s such a beautiful way of arguing.

我的意思是，你在地面模型中得出了一个结果，这是一种非常美妙的争论方式。

So speaking of the land of nonsense, I have to ask you about surreal numbers, but first, I need another bathroom break.

所以说到废话之地，我不得不问你一些超现实的数字，但首先，我需要再去趟厕所。

2:46:47 All right, we’re back, and there’s this aforementioned wonderful blog post on the surreal numbers and that there’s quite a simple surreal number generation process that can basically construct all numbers.

好吧，我们回来了，上面有一篇关于超现实数字的精彩博客文章，并且有一个非常简单的超现实数字生成过程，基本上可以构造所有数字。

So maybe this is a good spot to ask what are surreal numbers and what is the way we can generate all numbers?

所以也许这是一个很好的地方来问什么是超现实数字以及我们如何生成所有数字？

So the surreal number system is an amazing, an amazingly beautiful mathematical system that was introduced by John Conway.

因此，超现实数字系统是约翰·康威 (John Conway) 提出的一个令人惊叹的、令人惊叹的美丽数学系统。

Rest in peace, one of the great mathematicians ever on this earth.

安息吧，地球上最伟大的数学家之一。

Yes, absolutely.

是的，绝对是。

2:47:33 And I really admire his style of mathematical thinking and working in mathematics and the surreal number system is a good instance of this.

我真的很钦佩他的数学思维和数学工作风格，超现实的数字系统就是一个很好的例子。

So the way I think about the surreal numbers system is what it’s doing is providing us a number system that unifies all the other number systems.

因此，我对超现实数字系统的看法是，它正在为我们提供一个统一所有其他数字系统的数字系统。

So it extends the real numbers.

因此它扩展了实数。

Well, not only does it extend the integers, the natural numbers, the rational numbers, and the real numbers, but also the ordinals and the infinitesimals.

嗯，它不仅扩展了整数、自然数、有理数和实数，还扩展了序数和无穷小。

So they’re all sitting there inside the surreal numbers, and it’s this colossal system of numbers.

所以它们都坐在超现实的数字中，这就是这个巨大的数字系统。

It’s not a set even.

这甚至不是一套。

It’s a proper class, it turns out, because it contains all the ordinal numbers.

事实证明，这是一个真类，因为它包含所有序数。

2:48:19 But it’s generated from nothing by a single rule, and the rule is, so we’re going to generate the numbers in stages, in a transfinite sequence of stages.

但它是通过单一规则从无到有生成的，而规则是，所以我们将在超限的阶段序列中分阶段生成数字。

And at every stage, we take the numbers that we have so far and in all possible ways, we divide them into two sets, a lower set and an upper set, or a left set and a right set.

在每个阶段，我们都会采用迄今为止所掌握的数字，并以所有可能的方式将它们分为两组，一个较低的集合和一个较高的集合，或者一个左集和一个右集。

So we divide them into these two sets so that everything in the left set is less than everything in the right set, and then at that moment, we create a new number that fits in the gap between L and R.

因此，我们将它们分为这两个集合，以便左侧集合中的所有内容都小于右侧集合中的所有内容，然后在那一刻，我们创建一个适合 L 和 R 之间间隙的新数字。

Okay?

好吧？

That’s it.

就是这样。

That’s all we do.

这就是我们所做的一切。

So let me say it again.

那么让我再说一遍。

2:49:05 The rule is we proceed in stages, and at any stage, in all possible ways, we divide the numbers we have into two collections, the left set and the right set, so that everything in the left set is less than everything in the right set.

规则是我们分阶段进行，在任何阶段，以所有可能的方式，我们将我们拥有的数字分为两个集合，左集和右集，以便左集中的所有内容都小于右集中的所有内容。

And we create a new number, a new surreal number that will fit in that gap.

我们创建了一个新的数字，一个新的超现实数字来填补这个空白。

Okay.

好的。

So for example, we could start…

例如，我们可以开始……

Well, at the beginning, we don’t have any numbers.

嗯，一开始，我们没有任何数字。

We haven’t created anything yet, and so, we could take nothing and we could divide it into two sets, the empty lower set and the empty upper set.

我们还没有创建任何东西，所以，我们什么都不能拿，我们可以把它分成两个集合，空的下集合和空的上集合。

I mean, the two empty sets.

我的意思是，两个空集。

And everything in the empty set is less than everything in the empty set because that’s a vacuous statement.

空集中的所有内容都小于空集中的所有内容，因为这是一个空洞的陈述。

2:49:48 So we’re, we satisfy the conditions and we apply the number generation rule, which says we should create a new number.

所以我们满足条件并应用数字生成规则，该规则表明我们应该创建一个新数字。

And this is what I call the big bang of numbers, the surreal genesis when the number zero is born.

这就是我所说的数字大爆炸，即数字零诞生时的超现实起源。

Zero is the firstborn number that is bigger than everything in the empty set and less than everything in the empty set.

零是第一个出生的数字，它比空集中的所有数字都大，并且比空集中的所有数字都小。

Okay, but now we have this number zero, and so therefore, we now can define new gaps.

好的，但现在我们的数字为零，因此我们现在可以定义新的间隙。

Because if we put zero into the left set and have an empty right set, then we should create a new number that’s bigger than zero and less than everything in the empty set, and that number is called the number one.

因为如果我们将零放入左集合并有一个空的右集合，那么我们应该创建一个大于零且小于空集合中的所有数字的新数字，该数字称为数字一。

2:50:30 And similarly, at that same stage, we could have put zero into the right set, and so that would be the firstborn number that’s less than zero, which is called minus one.

同样，在同一阶段，我们可以将零放入正确的集合中，这样这将是小于零的第一个数字，称为负一。

So now we have three numbers, minus one, zero, and one, and they have four gaps because there could be a number below minus one or between minus one and zero or between zero and one or above one, and so we create those four new numbers.

现在我们有三个数字，负一、零和一，它们有四个间隙，因为可能存在低于负一、负一和零之间、零和一之间或高于一的数字，所以我们创建了这四个新数字。

The first number above one is called two.

1 上面的第一个数字称为 2。

The first number between zero and one is called 1/2, and then on the negative side, we have minus 1/2 and minus two and so on.

0 和 1 之间的第一个数字称为 1/2，然后在负数方面，我们有负 1/2 和负 2，依此类推。

So now we have, what is that, seven numbers.

现在我们有七个数字。

So there’s eight gaps between them.

所以它们之间有八个间隙。

2:51:10 So at the next birthday, they call them, the next stage will be born all the numbers between those gaps, and then between those and between those and so on.

因此，在下一个生日时，他们称它们为下一个阶段将诞生这些间隙之间的所有数字，然后是这些间隙之间的所有数字，然后是这些间隙之间的数字，依此类推。

And as the days progress, we get more and more numbers.

随着时间的推移，我们得到的数字越来越多。

But those are just the finite birthdays, because as I said, it’s a transfinite process.

但这些只是有限的生日，因为正如我所说，这是一个超限的过程。

So at day omega, that’s the first infinite day, we’re going to create a lot of new surreal numbers.

因此，在欧米茄日，这是第一个无限日，我们将创建许多新的超现实数字。

So every real number will be born at that stage, because every real number fills a gap in the previously born rational numbers that we had just talked about.

所以每一个实数都会在那个阶段诞生，因为每一个实数都填补了我们刚才谈到的之前诞生的有理数的空白。

It’s not all the rationals, because actually the rational numbers that are born at the finite stages are just the rationals whose denominator is a power of two, it turns out.

并不是所有的有理数，因为实际上在有限阶段产生的有理数只是分母是 2 的幂的有理数。

Those are called the dyadic rationals.

这些被称为二元理性。

2:51:57 So the real numbers are all born on day omega, but also some other numbers are born on day omega.

所以实数都是在欧米茄日出生的，但也有一些其他数字是在欧米茄日出生的。

Namely, the ordinal omega itself is the firstborn number that’s bigger than all those finite numbers, and minus omega is the firstborn number that’s less than all those finite numbers.

也就是说，序数 omega 本身是比所有这些有限数大的第一个数，而负 omega 是小于所有这些有限数的第一个数。

But also, we have the number epsilon, which is the firstborn number that’s strictly bigger than zero and strictly less than all the positive rational numbers.

而且，我们还有数字 epsilon，它是第一个严格大于零且严格小于所有正有理数的数字。

So that’s going to be an infinitesimal number in that gap, and so on.

所以在这个间隙中这将是一个无穷小的数字，依此类推。

On day omega plus one, we get more numbers, and then omega plus two and so on.

在欧米茄加一这一天，我们得到更多的数字，然后是欧米茄加二，依此类推。

And the numbers just keep coming forever.

而且这个数字一直在不断增加。

So, this is how you build the surreal number system.

所以，这就是构建超现实数字系统的方式。

2:52:39 And then it turns out you can define the arithmetic operations of addition and multiplication in a natural way that is engaging with this recursive definition.

然后事实证明，您可以以与此递归定义相关的自然方式定义加法和乘法的算术运算。

So we have sort of recursive definitions of plus and times for the surreal numbers.

因此，我们对超现实数有加法和乘法的递归定义。

And it turns out you can prove that they make the surreal numbers into what’s called an ordered field.

事实证明，你可以证明他们将超现实的数字变成了所谓的有序域。

So they satisfy the field axioms, which means that you have distributivity and commutativity of addition and multiplication, and also you have reciprocals for every non-zero number.

因此它们满足场公理，这意味着加法和乘法具有分配性和交换性，并且每个非零数都有倒数。

You can divide by the number.

您可以除以数字。

So you can add and multiply and divide and subtract.

所以你可以加、乘、除、减。

And furthermore, you can take square roots.

此外，您还可以求平方根。

2:53:21 And furthermore, every odd degree polynomial has a root, which is true in the real numbers, because if you think about, say, a cubic or a fifth degree polynomial, then you know it’s going to cross the axis, because it has opposite behaviors on the two infinities, because it’s an odd degree polynomial.

此外，每个奇数次多项式都有一个根，这在实数中是正确的，因为如果你考虑三次或五次多项式，那么你知道它会穿过轴，因为它在两个无穷大上有相反的行为，因为它是一个奇数次多项式。

So on the positive side, it’s going to the positive infinity.

所以从积极的一面来看，它会趋于正无穷大。

On the negative side, it would be going to minus infinity.

从消极的一面来看，它将变为负无穷大。

So it has to cross.

所以必须要穿越。

So we know in the real numbers, every odd degree polynomial has a root.

所以我们知道在实数中，每个奇数次多项式都有一个根。

And that’s also true in the surreal numbers.

超现实的数字也是如此。

So that makes it what’s called a real closed field which is a very nice mathematical theory.

这就是所谓的实闭域，这是一个非常好的数学理论。

So it’s really quite interesting how we can find copies of all these other number systems inside the surreal numbers.

因此，我们如何在超现实数字中找到所有其他数字系统的副本，这真的很有趣。

But the surreal numbers are fundamentally discontinuous as you’re worried about.

但正如你担心的那样，超现实的数字从根本上来说是不连续的。

2:54:09 What are the consequences of this?

这会带来什么后果？

Right.

正确的。

2:54:14 So the surreal numbers have a property that they form a non-standard model of the real field, which means that they provide a notion of infinitesimality that one can use to develop calculus on the grounds of Robinson’s non-standard theory that I had mentioned earlier.

因此，超现实数有一个属性，即它们形成了实场的非标准模型，这意味着它们提供了一种无穷小的概念，人们可以用它来根据我之前提到的罗宾逊非标准理论来发展微积分。

But they don’t have the least upper bound property for subcollections.

但它们没有子集合的最小上限属性。

There’s no set of surreal numbers, no non-trivial set of surreal numbers has at least upper bound, and there are no convergent sequences in the surreal numbers.

不存在超现实数集，不存在非平凡的超现实数集至少有上限，并且超现实数中不存在收敛序列。

And so for the sort of ordinary use in calculus based on limits and convergence, that method does not work in the surreal numbers at all.

因此，对于基于极限和收敛性的微积分中的普通应用，该方法根本不适用于超现实数。

So that’s what I mean when I say the surreal numbers are fundamentally discontinuous.

这就是我所说的超现实数字从根本上来说是不连续的。

They have a fundamental discontinuity going on.

它们之间存在着根本性的不连续性。

2:55:07 But you can still do calculus with them, because you have infinitesimals if you use these non-standard methods, the infinitesimal based methods to calculus.

但是您仍然可以用它们进行微积分，因为如果您使用这些非标准方法（基于无穷小的微积分方法），您就会得到无穷小。

And people do that.

人们就是这样做的。

I once organized a conference in New York, and we had John Conway as a speaker at that conference.

我曾经在纽约组织过一次会议，约翰·康威（John Conway）在该会议上担任演讲者。

And there was a question session, and someone asked him, I mean, it’s a bit of a rude question, I think, but they asked it and the question was, “What is your greatest disappointment in life?” I mean, I would never ask a question like that at a conference in a very public setting.

有一个提问环节，有人问他，我的意思是，我认为这是一个有点粗鲁的问题，但他们问的问题是，“你一生中最大的失望是什么？”我的意思是，我永远不会在非常公开的会议上提出这样的问题。

2:55:41 But Conway was extremely graceful and he answered by saying that, “The surreal numbers…” Not the numbers themselves, but the reception of the surreal numbers, because he had ambition that the surreal numbers would become a fundamental number system used throughout mathematics and science, because it was able to do non-set analysis, it was able to do calculus, it unified the ordinals and so on.

但康威非常优雅地回答说，“超现实数……”不是数字本身，而是超现实数的接受，因为他有野心，超现实数将成为整个数学和科学中使用的基本数字系统，因为它能够进行非集合分析，能够进行微积分，它统一了序数等等。

And it’s such a unifying, amazing structure, beautiful structure with elegant proofs and sophisticated ideas all around it.

这是一个如此统一、令人惊叹的结构，美丽的结构，优雅的证明和复杂的想法。

And he was disappointed that it never really achieved that unifying status that he had the ambition for.

他感到失望的是，它从未真正实现他所渴望的统一地位。

And this, he mentioned as his greatest disappointment.

他说这是他最大的失望。

Yeah, Donald Knuth tried to celebrate it, but it never quite took hold.

是的，唐纳德·高德纳 (Donald Knuth) 试图庆祝这一点，但它从未完全流行。

So I don’t want to give the impression, though, that the surreal numbers are not widely studied, because there are thousands of people who are…

所以我不想给人留下超现实数字没有被广泛研究的印象，因为有成千上万的人……

Sure

当然可以

…studying it.

……正在研究。

2:56:42 In fact, Philip Ehrlich, who is one of the world experts on the surreal numbers, mentioned to me once that Conway was his own worst enemy with regard to that very issue because in the Conway style, everything is a game.

事实上，菲利普·埃利希（Philip Ehrlich）是世界超现实数字专家之一，他曾经向我提到，康威在这个问题上是他自己最大的敌人，因为在康威风格中，一切都是游戏。

And he treated the surreal numbers as a kind of plaything, a toy, and maybe that makes people not take it seriously.

他把这些超现实的数字当作一种玩物、一种玩具，也许这会让人们不认真对待它。

Although my view is that it is extremely serious and useful and profound, and I’ve been writing a whole series of essays on the surreal numbers for my Substack at Infinitely More.

尽管我的观点是它非常严肃、有用和深刻，而且我一直在为我的 Infinitely More 的 Substack 写一系列关于超现实数字的文章。

And I just find the whole subject so fascinating and beautiful.

我发现整个主题是如此迷人和美丽。

I mean, it’s true.

我的意思是，这是真的。

I’m not applying it in engineering, which maybe was part of this Conway ambition.

我没有将其应用到工程中，这可能是康威雄心勃勃的一部分。

And I just wanted to, before I forget, mention Conway turning everything into a game.

我只是想在我忘记之前提一下康威将一切变成了游戏。

2:57:30 It is a fascinating point that I didn’t quite think about, which I think the Game of Life is just an example of exploration of cellular automata.

这是一个令人着迷的观点，我没有仔细考虑过，我认为生命游戏只是探索元胞自动机的一个例子。

I think cellular automata is one of the most incredible, complicated, fascinating…

我认为元胞自动机是最令人难以置信、最复杂、最迷人的......

It feels like an open door into a world we have not quite yet explored.

这感觉就像一扇打开的门，进入我们尚未探索的世界。

And it’s such a beautiful illustration of that world, the Game of Life, but calling it a game…

这是那个世界的美丽例证，生命的游戏，但称其为游戏……

Maybe life balances it, because that’s your powerful word, but it’s not quite a game.

也许生活会平衡它，因为这是你强有力的词，但这不完全是一场游戏。

It’s a fascinating invitation to an incredibly complicated and fascinating mathematical world.

这是对一个极其复杂和迷人的数学世界的迷人邀请。

2:58:09 I think every time I see cellular automata and the fact that we don’t quite have mathematical tools to make sense of that world, it fills me with awe.

我想，每次我看到元胞自动机，以及我们没有足够的数学工具来理解这个世界的事实，我都会充满敬畏。

Speaking of a thousand years from now, it feels like that is a world we might make some progress on.

说到一千年后，感觉那是一个我们可能会取得一些进步的世界。

The Game of Life is a sort of playground for computably undecidable questions because, in fact, you can prove that the question of whether a given cell will ever become alive is computably undecidable.

生命游戏是可计算不可判定问题的游乐场，因为事实上，您可以证明给定细胞是否会存活的问题是可计算不可判定的。

2:58:23 In other words…

换句话说……

Yeah

是啊

…given a configuration, and you ask, “Will this particular cell ever, you know, be alive—” …in the evolution?” And you can prove that that question is equivalent to the halting problem.

......给定一个配置，你会问，“这个特定的细胞，你知道，会活着吗......”......在进化中？你可以证明这个问题等价于停机问题。

2:58:39 It’s computably undecidable.

它是不可计算的。

It’s semi-decidable in the sense that if it will become alive, then you will know it at a finite stage because you could just run the Game of Life algorithm and let it run.

从某种意义上说，它是半可判定的，如果它会变得活跃，那么你会在有限阶段知道它，因为你可以运行生命游戏算法并让它运行。

And if it ever did come alive, you could say, “Yeah, it was alive.” But if you’ve run it for a thousand years and it hasn’t come alive yet, then you don’t necessarily seem to have any basis for saying, “No, it won’t ever come alive,” if the behavior was very complicated.

如果它真的复活了，你可以说：“是的，它还活着。”但如果你运行它一千年而它还没有复活，那么如果行为非常复杂，你似乎不一定有任何理由说“不，它永远不会复活”。

2:59:18 Maybe if you have a complete understanding of the evolution of the behavior, then you can say no, but you can prove you won’t always have that understanding— …precisely because the problem is equivalent to the halting problem.

也许如果你对行为的演变有完整的理解，那么你可以说不，但你可以证明你不会总是有这种理解——……正是因为这个问题相当于停机问题。

And nevertheless, when you sit back and look and visualize the thing, some little mini cellular automata civilizations are born and die quickly, and some are very predictable and boring, but some have this rich, incredible complexity.

尽管如此，当你坐下来观察并想象这个事物时，一些小型的微型细胞自动机文明诞生和消亡得很快，有些是非常可预测和无聊的，但有些则具有如此丰富、令人难以置信的复杂性。

2:59:28 And maybe that speaks to a thing I wanted to ask on the halting problem and decidability.

也许这说明了我想问的关于停止问题和可判定性的问题。

You’ve mentioned this thing where if you understand the program deeply, you might be able to say something.

你提到了这件事，如果你对程序了解得很深的话，你也许能说出来。

So can we say something interesting about, maybe, how many programs, statistically, we know something about in terms of whether they halt or not?

那么我们可以说一些有趣的事情吗？从统计数据来看，我们对有多少程序是否停止有所了解？

Or what does it mean to understand a program deeply enough—

或者说足够深入地理解一个程序意味着什么——

Right

右

…to be able to make a prediction?

……能够做出预测吗？

The main lesson of computability theory, in my view, is that it’s never the case that you can have a thorough understanding of the behavior of a program by looking at the program, and that the content of what you learn from a program, I mean, in the most general case, is always obtained just by running it and looking at the behavior.

在我看来，可计算性理论的主要教训是，你永远不可能通过查看程序来彻底了解程序的行为，而且你从程序中学到的内容，我的意思是，在最一般的情况下，总是通过运行它并查看行为来获得。

3:00:11 And the proof of that is there’s a theorem called Rice’s Theorem, which makes that idea completely robust.

证明这一点的是有一个叫做赖斯定理的定理，它使这个想法完全站得住脚。

But I want to just take a little detour towards another question riffing on something that you just said.

但我想稍微绕一下另一个问题，即重复你刚才所说的内容。

Namely, one can ask the question, what is the behavior of a random program?

也就是说，人们可以问这样一个问题：随机程序的行为是什么？

So you have some formal computing language, you know, and you want to look at the collection of all programs of a certain size.

所以你有一些正式的计算语言，你知道，你想查看一定大小的所有程序的集合。

3:01:06 Maybe there are only finitely many.

也许数量有限。

And can you say something about the behavior of a randomly chosen one, like with a certain likelihood it will have a certain behavior?

你能谈谈随机选择的一个人的行为吗，比如它有一定的可能性会有某种行为？

And the answer turns out to be extremely interesting.

事实证明答案非常有趣。

Once, years ago, Alexey Myasnikov asked me a question.

几年前，阿列克谢·米亚斯尼科夫曾经问过我一个问题。

He had this concept of a decision problem with a black hole, and what that means is it’s a decision problem which is possibly difficult in the worst case, but the difficulty was concentrated in a very tiny region called the black hole.

他提出了黑洞决策问题的概念，这意味着这是一个在最坏情况下可能很困难的决策问题，但困难集中在一个称为黑洞的非常小的区域。

And outside of that black hole, it was very easy.

在那个黑洞之外，一切都很容易。

3:01:42 And so, for example, this kind of problem is a terrible problem to use if you’re basing your encryption scheme.

因此，举例来说，如果您的加密方案基于这种问题，那么这种问题就是一个可怕的问题。

You don’t want to use a black hole problem because if someone can rob the bank 95% of the time, then that’s not what you want, or even any nontrivial percent of the time is too dangerous.

你不想使用黑洞问题，因为如果有人可以在 95% 的情况下抢劫银行，那么这不是你想要的，甚至任何不平凡的时间都太危险了。

So you don’t want to use problems that are almost every case is easily solved as the basis of your encryption.

因此，您不想使用几乎所有情况都可以轻松解决的问题作为加密的基础。

And the question Alexey asked me was, “Does the halting problem have a black hole?”

阿列克谢问我的问题是：“停机问题是否存在黑洞？”

3:02:17 And so if we take, say, the standard model of Turing machines—it’s one-way infinite tape with zeros and ones on the tape and so on, the head moving back and forth, and it stops when it gets into the halt state—then it turns out we proved that there is a black hole.

因此，如果我们采用图灵机的标准模型——它是单向无限磁带，磁带上有零和一，等等，磁头来回移动，当它进入停止状态时就会停止——那么事实证明我们证明了黑洞的存在。

And what that means is there’s a computer procedure that decides correctly almost every instance of the halting problem.

这意味着有一个计算机程序可以正确决定几乎每个停机问题的实例。

Even though the halting problem is not decidable, we can decide almost every instance.

尽管停止问题是不可判定的，但我们几乎可以判定每个实例。

So more precisely, there’s a collection of Turing machine programs such that we can easily decide whether a program’s in that collection or not.

更准确地说，有一个图灵机程序的集合，这样我们就可以轻松地确定程序是否在该集合中。

And for the programs in the collection, we can decide the halting problem for those programs easily.

对于集合中的程序，我们可以轻松地确定这些程序的停机问题。

3:03:03 And furthermore, almost every program is in the collection in the sense that as the number of states becomes large, the proportion of programs in the collection goes to 100%.

此外，几乎每个程序都在集合中，因为随着状态数量变大，程序在集合中的比例达到 100%。

So the asymptotic density of the programs is one.

所以程序的渐近密度是一。

And the proof was quite fascinating because it’s one of these situations where the theorem sounds really surprising, I think, to many people when I first tell it, I mean, to computability experts.

这个证明非常令人着迷，因为我认为，当我第一次告诉可计算性专家时，这个定理对很多人来说听起来真的很令人惊讶。

Then it’s sort of intriguing to think that you can solve almost every instance of a halting problem.

然后，想到您可以解决几乎所有停止问题的实例，这有点有趣。

But then when they hear the proof, it’s completely a letdown.

但当他们听到证据时，却完全失望了。

Unfortunately, nobody likes the theorem after the proof.

不幸的是，证明后没有人喜欢这个定理。

3:03:47 And so the proof is so simple, though.

不过，证明非常简单。

If you know how a Turing machine operates, there’s this infinite paper tape on which the machine writes zeros and ones, and the head moves back and forth according to rigid instructions.

如果你知道图灵机是如何运作的，就会发现有一条无限的纸带，机器在上面写下零和一，并且头部根据严格的指令来回移动。

And the instructions are all of the form: if the machine is in such and such a state and it’s reading such and such a symbol on the tape, then it should write this symbol on the tape, it should change to this new state specified, and it should either move left or right as specified.

指令都是这样的形式：如果机器处于这样那样的状态，并且它正在读取磁带上的这样那样的符号，那么它应该在磁带上写下这个符号，它应该改变到指定的新状态，并且应该按照指定的向左或向右移动。

So a program consists of instructions like that.

所以程序是由这样的指令组成的。

If you look at a program, one of the states is the halt state, and that’s when the program halts.

如果您查看一个程序，其中一种状态是暂停状态，此时程序将暂停。

3:04:30 But you can calculate how many programs don’t have any instruction that transitions to the halt state.

但是您可以计算有多少程序没有任何转换到暂停状态的指令。

You can easily calculate the proportion.

您可以轻松计算出比例。

And in the limit, it goes to 1 over E squared, 13 and a half percent.

在极限情况下，它会达到 E 平方的 1，即 13%.5。

If you calculate the limit, the proportion of programs with end states that don’t ever halt because they don’t have any instruction saying halt— —those programs obviously never halt because they can’t halt.

如果计算限制，则具有最终状态的程序的比例永远不会停止，因为它们没有任何指令说停止——这些程序显然永远不会停止，因为它们无法停止。

They don’t have any instruction that says halt.

他们没有任何指示停止。

So 13% of programs, you could say—

所以 13% 的程序，你可以说——

13%, you can say they don’t halt because you just look at them and you can understand them.

13%，你可以说他们不会停下来，因为你只要看看他们，你就能理解他们。

There’s no halt state.

没有停止状态。

They never change to the halt state, so they can’t halt.

它们永远不会更改为停止状态，因此无法停止。

I mean, that nevertheless is beautiful to know.

我的意思是，知道这一点还是很高兴的。

3:05:13 To show.

展示。

So that’s a kind of trivial reason for non-halting.

所以这是一种不停止的微不足道的理由。

3:05:17 And when I first made that observation, I thought, “Okay, this is the proof strategy.” Because I wanted to say at first the goal was, “Look, that’s a stupid reason for a program not to halt.

当我第一次做出这个观察时，我想，“好吧，这就是证明策略。”因为我一开始想说的目标是，“看，这是一个让程序不停止的愚蠢理由。

And I just want to pile up as many stupid reasons as I can think of—” —until it gets more than 50%, and then I can say most.

我只想尽可能多地堆砌我能想到的愚蠢理由——”——直到超过50%，然后我就可以说出大部分了。

That was brilliant.

那太棒了。

Yeah, that was my goal.

是的，这就是我的目标。

I love this.

我喜欢这个。

Yeah, so we thought more about it, though, and we hit the jackpot because we found one gigantic stupid reason that converged to 100%, in the limit.

是的，所以我们想得更多了，不过，我们中了大奖，因为我们发现了一个巨大的愚蠢原因，在极限范围内收敛到 100%。

3:05:45 And so, the stupid reason for a program not to halt is that, well, if you think about the behavior: the head is sitting there.

因此，程序不停止的愚蠢原因是，好吧，如果你考虑一下这种行为：头坐在那里。

It’s on the leftmost cell of the tape at the very beginning.

它位于磁带一开始最左边的单元格上。

It’s in the start state, and the head is following an instruction.

它处于启动状态，头部正在遵循指令。

And the instruction says, “When you’re in the start state,” which it is, “and you’re reading something on the tape, then you should write something and you should change to a new state, and you should either move left or right.” But half of them move left.

指令说，“当你处于开始状态时”，事实上，“你正在读磁带上的一些东西，那么你应该写一些东西，你应该改变到一个新的状态，你应该向左或向右移动。”但其中一半向左移动。

But if you move left and you are already at the end, then the head falls off.

但如果你向左移动并且已经到了尽头，那么头就会掉下来。

3:06:32 And so the computation stops because the head fell off the tape.

因此计算停止，因为磁头从磁带上掉下来了。

That’s a pretty stupid reason.

这是一个相当愚蠢的理由。

Okay, but that’s half of them already, just like that.

好吧，但这已经是一半了，就这样。

And then some of them went right and they changed to a new state.

然后他们中的一些人就向右转了，他们变成了一个新的状态。

And amongst those, the new state, half of those ones are going left and half are going right from that place.

在这些新的状态中，一半是从那个地方向左走，一半是向右走。

And then most of those are changing to a new state.

然后其中大部分正在转变为新的状态。

When there are a lot of states, it’s very likely that the next state that you transition to is new.

当有很多状态时，您转换到的下一个状态很可能是新的。

And so you get this random walk behavior, if you know what that means, where half go left and half go right at each step.

所以你会得到这种随机行走行为，如果你知道这意味着什么，每一步一半向左走，一半向右走。

3:07:04 And there’s a theorem due to Pólya, which is called the Pólya recurrence theorem, which says when you have a random walk, a one-dimensional random walk, then it’s very likely to come back to where you started.

波利亚有一个定理，叫做波利亚递归定理，它说当你进行随机游走，一维随机游走时，它很可能回到你开始的地方。

And when that happens for us, then half of them from that place fall off on the next step.

当这种情况发生在我们身上时，那么那个地方的一半人就会在下一步中掉下来。

And so you can show, using this kind of analysis, that the probability one behavior of a random Turing machine is that the head falls off the tape before it repeats a state.

因此，您可以使用这种分析来证明，随机图灵机的一种行为的概率是磁头在重复某种状态之前从磁带上掉下来。

And that is the stupid proof that shows how to solve the halting problem.

这就是展示如何解决停止问题的愚蠢证明。

3:07:40 Because when that happens, we can answer the halting problem saying, “No, the computation stopped because the machine crashed, not because it halted, so therefore it doesn’t count as halting on some accounts.” Or, you know, if you want to define that as halting, crashing as halting, then…

因为当发生这种情况时，我们可以回答暂停问题，说：“不，计算停止是因为机器崩溃，而不是因为它停止，因此在某些帐户上它不算停止。”或者，你知道，如果你想将其定义为停止，将崩溃定义为停止，那么......

But in any case, however it is that you set up your formalism, you’re going to be able to answer the question for the behavior of the machine when the head falls off.

但无论如何，无论你如何设置形式主义，你都将能够回答当头部掉落时机器行为的问题。

So statistically, in the limit, you solve the halting problem.

从统计上看，在极限范围内，您可以解决停止问题。

Yes, exactly.

是的，完全正确。

3:08:08 Computably solve it.

通过计算解决它。

What do we take from that?

我们从中得到什么？

3:08:11 Because you didn’t solve the halting problem.

因为你没有解决停机问题。

No, it’s impossible to fully solve…

不，不可能完全解决……

Right

右

…the halting problem correctly in all cases.

...在所有情况下都能正确地解决停止问题。

That’s pretty cool.

这太酷了。

3:08:20 That’s kind of…

这有点……

I mean, I don’t know.

我的意思是，我不知道。

This is…

这是……

It’s a probabilistic way…

这是一种概率方式……

3:08:22 I mean, it’s probabilistic in the sense that we’re solving almost all instances…

我的意思是，从我们正在解决几乎所有实例的意义上来说，它是概率性的……

…Computably.

……可以计算。

There are versions of this that are maybe more interesting from the point of view of complexity theory and actually useful.

从复杂性理论的角度来看，有一些版本可能更有趣并且实际上很有用。

I mean, there’s the whole P-NP problem and so on.

我的意思是，这就是整个 P-NP 问题等等。

And there’s this genre of NP-complete problems, which are problems that are infeasible.

还有一类 NP 完全问题，即不可行的问题。

They would take exponential time to solve them in the ordinary way.

以普通方式解决这些问题需要花费指数级的时间。

And they’re not known to be polynomial time solvable, although in these cases it’s an open question whether there is a polynomial time algorithm, a feasible algorithm.

并且不知道它们是否可以用多项式时间求解，尽管在这些情况下，是否存在多项式时间算法（一种可行的算法）是一个悬而未决的问题。

And for most of the NP-complete problems, you can prove that there’s a polynomial time approximation that solves almost all instances…

对于大多数 NP 完全问题，您可以证明存在可以解决几乎所有实例的多项式时间近似……

3:09:09 …in a feasible amount of time.

……在可行的时间内。

So like the knapsack problem, you know, packing problems, and so on, other kinds of problems, the satisfaction problem when…

所以就像背包问题，你知道，包装问题等等，其他类型的问题，满意度问题......

Depending on how you set up the formalism, you can prove, and I’ve proven many instances of this, but also I think it’s widespread for almost all the NP-complete problems, the difficult problems, and these are important problems for industrial application when these are problems…

根据你如何建立形式主义，你可以证明，我已经证明了很多这样的例子，但我也认为它对于几乎所有 NP 完全问题、困难问题都是普遍存在的，当这些问题是工业应用时，这些都是重要的问题……

…That we actually want to solve.

……我们实际上想解决这个问题。

We can have feasible algorithms that solve almost every instance of them.

我们可以拥有可行的算法来解决几乎所有的问题。

The amount of fields and topics you’ve worked on is truly incredible.

您所从事的领域和主题的数量确实令人难以置信。

3:09:42 I have to ask about P versus NP.

我想问一下 P 与 NP 的关系。

This is one of the big open problems in complexity theory.

这是复杂性理论中尚未解决的重大问题之一。

So for people who don’t know, it’s about the relation between computation time and problem complexity.

所以对于不知道的人来说，这是关于计算时间和问题复杂性之间的关系。

Do you think it will ever be solved?

你认为这个问题会得到解决吗？

And is there any chance the weird counterintuitive thing might be true, that P equals NP?

P 等于 NP 这个奇怪的违反直觉的事情有可能是真的吗？

Yeah, that’s an interesting question.

是的，这是一个有趣的问题。

3:10:06 Sometimes people ask about whether it could be independent, which I think is-

有时人们会问它是否可以独立，我认为是——

3:10:11 …an interesting question for logicians.

……对于逻辑学家来说这是一个有趣的问题。

And of course, well, one has to say if you’re entertaining the idea of independence, you know, over which theory?

当然，好吧，人们必须说，如果你接受独立的想法，你知道，基于哪种理论？

Because every statement is going to be independent over an extremely weak theory.

因为每一个陈述都将独立于一个极其薄弱的理论。

So that’s, you know, it doesn’t make sense to say it’s independent all by itself.

所以，你知道，说它本身是独立的是没有意义的。

You’re only independent relative to a theory, right?

你只是相对于理论而言是独立的，对吗？

So the way I think about P-NP is that…

所以我对 P-NP 的看法是……

I mean, of course it’s a theoretical question about the asymptotic behavior of these problems.

我的意思是，这当然是关于这些问题的渐近行为的理论问题。

I mean, for a problem to be in P means that there is a computable decision procedure that runs in time bounded by some polynomial.

我的意思是，对于 P 中的问题意味着存在一个可计算的决策过程，该过程在某个多项式限制的时间内运行。

But the coefficients on that polynomial could be enormous, and the degree could be incredibly high.

但该多项式的系数可能非常大，而且次数也可能高得令人难以置信。

3:10:59 And so for small values of inputs, then it doesn’t make sense to talk about this polynomial time feasibility with respect to, say, the range of problem inputs that we will ever give it in our lifetime or in the span of human civilization or whatever.

因此，对于较小的输入值，谈论这种多项式时间可行性是没有意义的，比如说，我们在有生之年或人类文明的跨度或其他什么时间里将给出的问题输入的范围。

I mean, because it’s an asymptotic property, it’s really in the limit as the size of the inputs goes to infinity, that’s the only time that polynomial or NP becomes relevant.

我的意思是，因为它是渐近性质，所以当输入的大小趋于无穷大时，它确实处于极限，这是多项式或 NP 唯一相关的时候。

And so maybe it’s important to keep that in mind when…

因此，也许在……时牢记这一点很重要。

Sometimes you find kind of overblown remarks made about, you know, if P equals NP, then this will be incredibly important for human civilization because it means that we’ll have feasible algorithms for solving these incredibly important…

有时你会发现一些过分夸大的言论，你知道，如果 P 等于 NP，那么这对人类文明将非常重要，因为这意味着我们将拥有可行的算法来解决这些极其重要的问题……

3:11:45 …problems in NP.

…NP 中的问题。

You know, that it would cause immense wealth for human societies and so on because we would be able to solve these otherwise intractable problems, and that would be the basis of new technology and industry and so forth.

你知道，它将为人类社会带来巨大的财富等等，因为我们将能够解决这些原本棘手的问题，这将成为新技术和工业等的基础。

I mean, people make these kinds of remarks, but…

我的意思是，人们会发表这样的言论，但是……

Of course.

当然。

…you have to temper those remarks by the realization that P and P equal NP or P not equal NP are not about these practical things at all because of the asymptotic nature of the question itself.

…你必须认识到 P 和 P 等于 NP 或 P 不等于 NP 根本与这些实际问题无关，因为问题本身的渐近性质，从而缓和这些言论。

3:12:04 Okay, that’s on the one hand.

好吧，这是一方面。

But on the second hand, we already have the algorithm, so we could use it already, except it’s a terrible algorithm because it involves this incredible amount of coding and so on.

但另一方面，我们已经有了算法，所以我们已经可以使用它了，只是它是一个糟糕的算法，因为它涉及大量的编码等等。

And on the third hand, like you said, we already have approximation algorithms that…

第三方面，就像你说的，我们已经有了近似算法……

Yes.

是的。

…that from a pragmatic perspective, solve all the actual real engineering problems of human civilization.

……从务实的角度，解决人类文明的所有实际工程问题。

Like the SAT solvers work amazingly well, you know, in lots and lots of cases, even though we can prove we don’t expect…

就像 SAT 求解器在很多很多情况下都工作得非常好一样，尽管我们可以证明我们不期望......

3:12:42 If P is not equal to NP, then there won’t be a polynomial time SAT solver.

如果 P 不等于 NP，则不会有多项式时间 SAT 求解器。

But actually, the SAT solver approximations are really quite amazing.

但实际上，SAT 求解器的近似值确实非常惊人。

Sorry to ask the ridiculous question, but who is the greatest mathematician of all time?

很抱歉问这个荒谬的问题，但是谁是有史以来最伟大的数学家？

3:12:59 Who are the possible candidates?

谁是可能的候选人？

Euler, Gauss, Newton, Ramanujan, Hilbert.

欧拉、高斯、牛顿、拉马努金、希尔伯特。

We mentioned Gödel, Turing, if you throw him into the bucket.

我们提到了哥德尔、图灵，如果你把他扔进桶里。

So this is, I think, an incredibly difficult question to answer.

所以我认为这是一个非常难以回答的问题。

3:13:14 Personally, I don’t really think this way about ranking mathematicians by greatness.

就我个人而言，我并不真的这么认为按照伟大程度对数学家进行排名。

Um…

嗯……

So you don’t have, like…

所以你没有，比如……

3:13:28 You know, some people have a Taylor Swift poster in their dorm room.

你知道，有些人的宿舍里有一张泰勒·斯威夫特的海报。

You don’t have it.

你没有。

I mean, if you forced me to pick someone, it would probably be Archimedes because…

我的意思是，如果你强迫我选一个人，那可能是阿基米德，因为……

Archimedes

阿基米德

…he had such incredible achievements in such an early era, which totally transcended the work of the other people in his era.

……他在这么早的时代就取得了如此令人难以置信的成就，完全超越了他那个时代其他人的工作。

3:13:37 But I also have the view that I want to learn mathematics and gain mathematical insight from whoever can provide it and wherever I can find it.

但我也有这样的观点：我想学习数学，并从任何可以提供数学见解的人和我能找到的地方获得数学见解。

And this isn’t always just coming from the greats.

这并不总是来自伟人。

Sometimes the greats are doing things that are just first and not…

有时，伟人所做的事情只是第一，而不是……

You know, somebody else could have easily been first.

你知道，其他人很容易成为第一。

So there’s a kind of luck aspect to it when you go back and look at the achievements.

所以当你回头看看所取得的成就时，有一种运气的成分。

And because of this progress issue in mathematics that we talked about earlier, namely we really do understand things much better now than they used to.

由于我们之前谈到的数学进步问题，即我们现在确实比以前更好地理解事物。

3:14:22 And when you look back at the achievements that had been made, then maybe you can imagine thinking, “Well, somebody else could’ve had that insight also.” And maybe they would have…

当你回顾已经取得的成就时，也许你可以想象这样的想法：“好吧，其他人也可以有这样的洞察力。”也许他们会……

It’s already a known phenomenon that disparate mathematicians end up proving essentially similar results at approximately the same time.

不同的数学家最终在大约同一时间证明了本质上相似的结果，这已经是一个众所周知的现象。

But, okay, the person who did it first is getting the credit and so on.

但是，好吧，第一个做这件事的人会得到荣誉，等等。

What do you make of that?

你对此有何看法？

3:14:48 Because I see that sometimes when mathematicians…

因为我有时会看到数学家......

This also applies in physics and science, where completely separately, discoveries are made…

这也适用于物理学和科学，它们的发现是完全分开的……

Right.

对。

3:14:58 Yeah.

是的。

…maybe at a very similar time.

……也许是在非常相似的时间。

What does that mean?

这意味着什么？

It’s relatively common.

这是比较常见的。

3:15:01 I mean, I think it’s like certain ideas are in the air and being thought about but not fully articulated, and so this is the nature of growth in knowledge.

我的意思是，我认为这就像某些想法已经存在并被思考但没有完全阐明，所以这就是知识增长的本质。

Do you understand where ideas come from?

你知道想法从何而来吗？

Not really.

并不真地。

I mean, what’s your own process when you’re thinking through a problem?

我的意思是，当你思考问题时，你自己的过程是什么？

Yeah, that’s another difficult question.

是的，这是另一个难题。

3:15:22 I suppose it has to do with…

我想这与……有关

My mathematical style, my style as a mathematician, is that I don’t really like difficult mathematics.

我的数学风格，我作为数学家的风格，是我不太喜欢困难的数学。

What I love is simple, clear, easy-to-understand arguments that prove a surprising result.

我喜欢的是简单、清晰、易于理解的论点，这些论点证明了令人惊讶的结果。

That’s my favorite situation.

这是我最喜欢的情况。

And actually, the question of whether it’s a new result or not is somehow less important to me.

事实上，这是否是一个新结果的问题对我来说并不那么重要。

And so that has to do with this question of the greats and so on, whoever does it first.

所以这与伟人的问题有关，等等，无论谁先做。

Because I think, for example, if you prove a new result with a bad argument or a complicated argument, that’s great because you proved something new.

因为我认为，例如，如果你用一个糟糕的论证或一个复杂的论证证明了一个新的结果，那很好，因为你证明了一些新的东西。

But I still want to see the beautiful, simple, because that’s what I can understand.

但我还是想看到美丽的、简单的，因为那是我能理解的。

3:16:16 Also, I’m kind of naturally skeptical about any complicated argument because it might be wrong.

另外，我对任何复杂的论点都自然地持怀疑态度，因为它可能是错误的。

And…

还有……

…If I can’t really understand it fully, like every single step all at once in my head, then I’m just worried maybe it’s wrong.

……如果我不能真正完全理解它，就像每一步都在我的脑海中同时出现，那么我只是担心它可能是错误的。

And so these different styles, sometimes mathematicians get involved with these enormous research projects that involve huge numbers of working parts and…

因此，这些不同风格的数学家有时会参与这些庞大的研究项目，其中涉及大量的工作部件......

…Different technology coming together.

……不同的技术融合在一起。

I mean, mathematical technology, not physical technology.

我的意思是，数学技术，而不是物理技术。

And sometimes it actually involves now more and more something like the Lean programming language where some parts are automated, so you have this gigantic…

有时它实际上涉及越来越多的东西，比如精益编程语言，其中某些部分是自动化的，所以你有这个巨大的......

Yeah, yeah, I see.

是的，是的，我明白了。

3:16:54 Well, that’s another issue because maybe those things are less subject to skepticism when it’s validated…

好吧，这是另一个问题，因为当这些事情得到验证时，也许不会受到怀疑……

Sure

当然可以

…by Lean.

……精益。

3:17:02 But I’m thinking about the case where the arguments are just extremely complicated, and so I sort of worry whether it’s right or not, whereas you know, I like the simple thing.

但我正在考虑论点极其复杂的情况，所以我有点担心它是否正确，而你知道，我喜欢简单的事情。

So I tend to have often worked on things that are a little bit off the beaten path from what other people are working on from that point of view.

因此，从这个角度来看，我往往会做一些与其他人正在做的事情有点偏离的事情。

Your curiosity draws you towards simplicity.

你的好奇心将你引向简单。

Yeah.

是的。

3:17:25 I want to work on the things that I can understand and that are simple.

我想做一些我能理解并且简单的事情。

Luckily, I’ve found that I’ve been able to make contributions that other people seem to like, in this way, in this style.

幸运的是，我发现我能够以这种方式、这种风格做出其他人似乎喜欢的贡献。

So I’ve been fortunate from that point of view.

所以从这个角度来看我很幸运。

My process always, though, and I’ve recommended this always to my students, is just a kind of playful curiosity.

不过，我的过程始终只是一种好玩的好奇心，而且我总是向我的学生推荐这一点。

So whenever I have…

所以每当我有...

3:17:55 Whenever there’s an idea or a topic then I just play around with it and change little things or understand a basic case and then make it more complicated or press things a little bit on this side or apply the idea to my favorite example that’s relevant, and see what happens, or you just play around with ideas, and this often leads to insights that then lead to more methods or more, then pretty soon you’re making progress on the problem.

每当有一个想法或一个主题时，我就会摆弄它，改变一些小事情或理解一个基本情况，然后让它变得更复杂，或者把事情稍微压在这一边，或者将这个想法应用到我最喜欢的相关示例中，看看会发生什么，或者你只是摆弄想法，这通常会带来洞察力，然后带来更多的方法或更多的方法，然后很快你就在问题上取得了进展。

So this is basically my method, is I just fool around with the ideas until I can see a path through towards something interesting…

所以这基本上就是我的方法，我只是胡乱思考这些想法，直到我能看到一条通向有趣事物的道路……

…And then prove that, and that’s worked extremely well for me.

......然后证明这一点，这对我来说非常有效。

So I’m pretty pleased with that method.

所以我对这个方法非常满意。

You do like thought experiments where you anthropomorphize like you mentioned?

你确实喜欢像你提到的那样进行拟人化的思想实验吗？

Yeah, yeah.

是啊是啊。

3:18:51 So this is a basic tool.

所以这是一个基本工具。

I mean, I use this all the time.

我的意思是，我一直用这个。

You imagine a set-theoretic model, a model of ZFC, as like a place where you’re living, and you might travel to distant lands by forcing.

你想象一个集合论模型，一个ZFC模型，就像你居住的地方，你可能会通过强迫去遥远的地方旅行。

This is a kind of metaphor for what’s going on.

这是对正在发生的事情的一种隐喻。

Of course, the actual arguments aren’t anything like that because there’s not land and you’re not traveling and you’re not…

当然，实际的争论并不是这样的，因为没有土地，你没有旅行，你也没有……

But you allow your mind to visualize that kind of thing-

但你让你的头脑想象那种事情——

Yeah

是啊

…

……

3:19:15 in the natural real world.

在自然现实世界中。

And it helps you to understand.

它可以帮助你理解。

3:19:16 Particularly when there are parts of the argument that are in tension with one another, then you can imagine that people are fighting or something.

特别是当争论的某些部分彼此之间存在紧张关系时，你可以想象人们正在打架或发生其他事情。

And those kinds of metaphors, or you imagine it in terms of a game theoretic, you know, two players trying to win.

这些隐喻，或者你用博弈论来想象，你知道，两个玩家试图获胜。

So that’s kind of tension.

这就是一种紧张。

And those kinds of metaphorical ways of understanding a mathematical problem often are extremely helpful in realizing, aha, the enemy is going to pick this thing to be like that because, you know, it makes it more continuous or whatever, and then we should do this other thing in order to…

这些理解数学问题的隐喻方式通常非常有助于认识到，啊哈，敌人会选择这个东西，因为，你知道，它使它更加连续或其他什么，然后我们应该做另一件事，以便......

So it makes you realize mathematical strategies for finding the answer and proving the theorem that you want to prove because of the ideas that come out of that anthropomorphization.

因此，它让你意识到寻找答案并证明你想要证明的定理的数学策略，因为拟人化产生的想法。

What do you think of somebody like Andrew Wiles, who spent seven years grinding at one of the hardest problems in the history of mathematics?

你如何看待像安德鲁·怀尔斯这样的人，他花了七年时间研究数学史上最难的问题之一？

3:20:01 And maybe contrasting that a little bit with somebody who’s also brilliant, Terence Tao, who basically says if he hits a wall, he just switches to a different problem and he comes back and so on.

也许可以将这一点与同样才华横溢的陶哲轩进行对比，陶哲轩基本上说，如果他碰壁了，他就会转向另一个问题，然后再回来，等等。

So it’s less of a focused grind for many years without any guarantee that you’ll get there, which is what Andrew Wiles went through.

因此，这并不是多年专注的磨练，也没有任何保证你能达到目标，这就是安德鲁·威尔斯所经历的。

Right.

正确的。

Maybe Grigori Perelman did the same.

也许格里戈里·佩雷尔曼也做了同样的事。

I mean, Wiles proved an amazing theorem, Fermat’s Last Theorem result is incredible.

我的意思是，怀尔斯证明了一个惊人的定理，费马大定理的结果令人难以置信。

3:20:32 This is a totally different style than my own practice, though, of working in isolation.

不过，这与我自己的孤立工作方式完全不同。

For me, mathematics is often a kind of social activity.

对我来说，数学往往是一种社交活动。

I have…

我有……

I counted, I mean, it’s pushing towards a hundred collaborators, co-authors on various papers and so on.

我数了一下，我的意思是，它正在推动一百个合作者、各种论文的合著者等等。

And, you know, if anybody has an idea they want to talk about with me, if I’m interested in it, then I’m going to want to collaborate with them and we might solve the problem and have a joint paper or whatever.

而且，你知道，如果有人有一个想法想与我讨论，如果我对此感兴趣，那么我会想与他们合作，我们可能会解决问题并发表联合论文或其他什么。

You want to have a joint paper?

你想要联合论文吗？

Let me-

让我-

Yeah, exactly.

是的，完全正确。

3:21:06 Let’s go.

我们走吧。

So my approach to making mathematical progress tends to involve working with other people quite a lot rather than just working on my…

因此，我取得数学进步的方法往往涉及与其他人大量合作，而不仅仅是致力于我的……

3:21:17 …own, and I enjoy that aspect very much.

……拥有，我非常喜欢这一点。

So I, personally, I couldn’t ever do what Wiles did.

所以就我个人而言，我永远无法做怀尔斯所做的事情。