无限、悖论、哥德尔不完备性与数学多重宇宙｜Lex Fridman Podcast #488

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. He is the number-one highest-rated user on MathOverflow, which I think is a legendary accomplishment. MathOverflow, by the way, is like Stack Overflow but for research mathematicians. He is also the author of several books, including Proof in the Art of Mathematics and Lectures on the Philosophy of Mathematics. And he has a great blog, infinitelymore.xyz.

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

I have been hiding from the world a bit, reading, thinking, writing, and soul-searching, as we all do every once in a while. But mostly, I have been deeply focused on work and preparing mentally for some challenging travel I plan to take on in the new year. 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.

I want to take this moment to say thank you from the bottom of my heart for everything—for your support and for the many amazing conversations I have had with people across the world. I got a little bit of hate and a whole lot of love, and I would not have it any other way. I am grateful for all of it.

This is the Lex Fridman Podcast. 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. And now, dear friends, here is Joel David Hamkins.

Some infinities are bigger than others. This idea from Cantor, at the end of the 19th century, broke mathematics before rebuilding it. I also read that this was a devastating and transformative discovery for several reasons.

First, it created a theological crisis. Because infinity is associated with God, how could there be multiple infinities? Cantor was also deeply religious himself. Second, there was a kind of mathematical civil war. The leading German mathematician Kronecker called Cantor a “corrupter of youth” and tried to block his career.

Third, many fascinating paradoxes emerged from this, like Russell’s paradox, about the set of all sets that do not contain themselves. Those threatened to make all of mathematics inconsistent. And finally, on the psychological and personal side, there was Cantor’s own breakdown. He literally went mad, spending his final years in and out of sanatoriums, obsessed with proving the continuum hypothesis.

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

That is a really great question. I would want to start talking about infinity and telling the story much earlier than Cantor, actually. 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.

Archimedes’ method of exhaustion involved trying to understand the area of a region by carving it into more and more triangles, 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. This proceeded on the basis of a potential understanding of infinity for hundreds, even thousands, of years. Almost all mathematicians were potentialists only and thought that it was incoherent to speak of an actual infinity at all.

Galileo is an extremely prominent exception to this, though. He argued against this kind of potentialist orthodoxy in The Dialogue of Two New Sciences. He gave a really lovely account there. In many ways, Galileo was anticipating Cantor’s developments, except he could not quite push them all the way through and ended up throwing up his hands in confusion, in a sense.

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 Galileo would start with one—the numbers 1, 2, 3, 4, and so on, and you think about which of those numbers are perfect squares, then 0 squared is 0, 1 squared is 1, 2 squared is 4, 3 squared is 9, then 16, 25, and so on.

Galileo observed that the perfect squares can be put into a one-to-one correspondence with all of the numbers. We just did it: I associated every number with its square. So it seems like, on the basis of this one-to-one correspondence, there should be exactly the same number of perfect squares as there are numbers. And yet there are all these gaps in between the perfect squares, right?

This suggests that there should be fewer perfect squares and more numbers, because the numbers include all the squares plus a lot more in between them. Galileo was quite troubled by this observation because he took it to cause a kind of incoherence in the comparison of infinite quantities.

Another example is that if you take two line segments of different lengths, you can imagine drawing a kind of foliation—a fan of lines that connects them. The endpoints are matched from the shorter segment to the longer segment, the midpoints are matched, and so on, spreading out the lines as you go.

Every point on the shorter line would be associated with a unique, distinct point on the longer line in a one-to-one way. 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. Again, this creates a kind of confusion over our ideas about infinity.

The same thing happens with two circles. If you place them concentrically and draw 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. 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.

Of course, the contemporary attitude about this situation is that those two infinities are exactly the same, and Galileo was right in those observations about their equinumerosity. We would talk about it now by appealing to what I call the Cantor–Hume principle, or what some people simply call Hume’s principle.

The idea is that if you have two collections, whether they are finite or infinite, then we want to say that those two collections have the same size—they are equinumerous—if and only if there is a one-to-one correspondence between those collections. Galileo was observing that line segments of different lengths are equinumerous, that the perfect squares are equinumerous with all of the natural numbers, that any two circles are equinumerous, and so on.

The tension between the Cantor–Hume principle and what could be called Euclid’s principle—that the whole is always greater than the part—is a principle that Euclid appealed to in the Elements many times when calculating area and so on. It is a basic idea that if something is just a part of another thing, then the whole is greater than the part.

What Galileo was troubled by was this tension between what we call the Cantor–Hume principle and Euclid’s principle. It was not fully resolved, I think, until Cantor. He is the one who really explained so clearly the different sizes of infinity in a way that was so compelling.

He exhibited two different infinite sets and proved that they are not equinumerous; they cannot be put into a one-to-one correspondence. It is traditional to talk about the uncountability of the real numbers. Cantor’s big result was that the set of all real numbers is an uncountable set.

Maybe, if we are going to talk about countable sets, we should talk about Hilbert’s Hotel, which really makes that idea perfectly clear.

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

Hilbert’s Hotel is a hotel with infinitely many rooms. Each room is a full-floor suite. There is floor zero—I always start with zero because, for me, the natural numbers start with zero, although that is maybe a point of contention for some mathematicians. The other mathematicians are wrong.

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

Exactly. So there is room zero, room one, room two, room three, and so on, just like the natural numbers. Hilbert’s Hotel has a room for every natural number, and it is completely full. There is a person occupying room N for every N.

Meanwhile, a new guest comes up to the desk and wants a room. “Can I have a room, please?” The manager says, “Hang on a second. Just give me a moment.”

When the other guests checked in, they had to sign an agreement with the hotel that maybe there would be some changing of rooms during their stay. So the manager sent a message up to all the current occupants and told every person, “Hey, can you move up one room, please?”

The person in room 5 would move to room 6, the person in room 6 would move to room 7, and so on. Everyone moved at the same time. Of course, we never want to place two different guests in the same room, and we want everyone to have their own private room.

But when you move everyone up one room, the bottom room, room zero, becomes available. So the manager can put the new guest in that room. Even when you have infinitely many things, the new guest can be accommodated.

That is a way of showing how the particular infinity of the occupants of Hilbert’s Hotel violates Euclid’s principle. It exactly illustrates this idea, because adding one more element to a set did not make it larger. We can still have a one-to-one correspondence between the new total of guests and the old total of guests by the room number, right?

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

The hotel is full.

And then you could still squeeze in one more. That breaks the traditional notion of mathematics and breaks people's brains when they try to think about infinity, I suppose. This is a property of infinity.

It's a property of infinity that sometimes, when you add an element to a set, it doesn't get larger. That's what this example shows.

Timothy Gowers

But one can go on with Hilbert's Hotel. Maybe the next day, 20 people show up all at once. We can easily do the same trick again: just move everybody up 20 rooms. Then we would have 20 empty rooms at the bottom, and those new 20 guests could go in.

But on the following weekend, a giant bus pulled up—Hilbert's bus. Hilbert's bus has, of course, infinitely many seats. There's Seat 0, Seat 1, Seat 2, Seat 3, and so on. All the people on the bus want to check into the hotel, but the hotel is completely full. So what is the manager going to do?

When I talk about Hilbert's Hotel, when I teach Hilbert's Hotel in class, I always demand that the students provide the explanation of how to do it. So maybe I'll ask you: Can you tell me what your idea is about how to fit them all in the hotel—everyone on the bus and also the current occupants?

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

Timothy Gowers

That's exactly right. That's a very easy way to do it. If you just tell all the current guests to double their room number, so in Room N, you move to Room 2N, they're all going to get their own private room—the new room—and it will always be an even number, because 2N is always even. So all the odd rooms become empty that way. Now we can put the bus occupants into the odd-numbered rooms.

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

Timothy Gowers

That's right. What it really shows is that we can define a set as countable if it is equinumerous with a set of natural numbers. An 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 is basically the set of natural numbers in terms of the room numbers. To be equinumerous with a set of natural numbers is the same thing as to fit into Hilbert's Hotel.

What we've shown is that if you have 2 countably infinite sets, then their union is also countably infinite. If you put them together and form a new set with all of the elements of either of them, that union set is still only countably infinite. It didn't get bigger, and that's a remarkable property for a notion of infinity to have, I suppose.

But if you thought that there was only 1 kind of infinity, then it wouldn't be surprising at all, because if you take 2 infinite sets and put them together, then it's still infinite. If there were only 1 kind of infinity, then it shouldn't be surprising that the union of 2 countable sets is countable.

So there's another way to push this a bit harder, and that is when Hilbert's train arrives. Hilbert's train has infinitely many train cars, and each train car has infinitely many seats. We have an infinity of infinities of train passengers, together with the current occupants of the hotel, and everybody on the train wants to check in to Hilbert's Hotel.

The manager can, again, send a message up to all the rooms telling every person to double their room number. That will occupy all the even-numbered rooms again, but free up the odd-numbered rooms. So we want to put the train passengers into the odd-numbered rooms.

Every train passenger is on some car, let's say Car C, and Seat S. Somehow, we have to take these 2 coordinates—the car number and the seat number—and produce from them an odd number in a 1-to-1 way. That's actually not very difficult.

An easy way to do it is to use 3^C × 5^S: 3 to the car number, multiplied by 5 to the seat number. You multiply 3 by itself the number of times given by the train-car number, and then you multiply 5 by itself the number of times given by the seat number. Then you multiply those 2 numbers together.

That's always an odd number, because the prime factorization has only 3s and 5s in it. There's no 2 there, so it's definitely an odd number. It's always different because of the uniqueness of prime factorization. Every number can be factored uniquely into primes. 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. So you know exactly which person it was, which car they came from, and which seat they came from.

Prime factorization is that every single number can be decomposed into the atoms of mathematics, which are the prime numbers. You can multiply them together to achieve that number, and that's prime factorization. You're showing that 3 and 5 are both prime numbers and odd. Through this magical formula, you can deal with this train: an infinite number of cars, with each car having an infinite number of seats.

Timothy Gowers

Exactly right. We've proved that if you have countably many countable sets, then the union of those sets—putting all those sets together into 1 giant set—is still countable. The train cars are each countable, plus the current hotel. It's another train car, if you want to think about it that way. The current occupants of the hotel could have the same number as any of the train cars.

Putting countably many countable sets together to make 1 big union set is still countable. It's quite remarkable, I think. When I first learned this many years ago, I was completely shocked by it and transfixed by it. 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.

The new set that we built has many more elements than the old set in the sense that there are 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.

Have you been able to internalize a good intuition about countable infinity? That's a pretty weird thing. You can have a countably infinite set of countably infinite sets, and you can shove it all in and it still is a countably infinite set.

Timothy Gowers

Yeah, that's exactly right. When you work with these notions, the argument of Hilbert's Hotel becomes clear. There are many other ways to talk about it, too.

For example, let's think about the integer lattice—the grid of points that you get by taking pairs of natural numbers, the upper-right quadrant of the integer lattice. There's Row 0, Row 1, Row 2, and so on; Column 0, Column 1, Column 2, and so on. Each row and column has a countable infinity of points on it.

Those dots, if you think about them as dots, are really the same as the train cars. Each column in that integer lattice is a countable infinity, like 1 train car. Then there's the next train car next to it, and the next column next to that—the next train car.

If we think about it in this grid manner, then I can imagine a winding path through these grid points, like up and down the diagonals, winding back and forth. I start at the corner point, and then I go down, up and to the left, 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 going to hit every grid point on this path.

This gives me a way of assigning room numbers to the points, because every grid point is going to be the Nth point on that path for some N. That gives a correspondence between the grid points and the natural numbers themselves.

It's a different picture. Before, we used 3^C × 5^S, which is an overly arithmetic way to think about it. But there's a direct 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. As long as you give each of the infinite collections a chance to add 1 more person to the list, you're going to accommodate everyone in any of the sets in 1 list.

Yeah, it's a really nice visual way to think about it. You just zigzag your way across the grid to make sure everybody's included. That gives you an algorithm for including everybody.

Can you speak to the uncountable infinities?

Timothy Gowers

Yeah, absolutely.

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

Timothy Gowers

Maybe there's 1 more step I want to insert before doing that, which is the rational numbers. We did pairs of natural numbers, basically the train car. But maybe it's informative to think about the rational numbers—the fractions—because a lot of people have an expectation that maybe this is a bigger infinity because the rational numbers are densely ordered.

Between any 2 fractions, you can find another fraction, right? The average of 2 fractions is another fraction. Sometimes it seems to be a different character than the integers, which are discretely ordered. From any integer, there's a next one and a previous one, and so on. But that's not true in the rational numbers.

And yet, the rational numbers are also still only a countable infinity. The way to see that is actually exactly the same as Hilbert’s train again, because every fraction consists of 2 integers: the numerator and the denominator. If I tell you 2 natural numbers, then you know what fraction I’m talking about. There’s also the sign issue, whether it’s positive or negative, but if you just think about the positive fractions, then you have numbers of the form P over Q, where Q is not zero.

You can still do 3 to the P times 5 to the Q. The same idea works with the rational numbers, so this is still a countable set. You might think, “Well, every set is going to be countable because there’s only one infinity,” if that’s the perspective you’re adopting. But it’s not true, and that’s the profound achievement that Cantor made: proving that the set of real numbers is not a countable infinity. It’s a strictly larger infinity, and therefore there’s more than one concept of infinity, more than one size of infinity.

So let’s talk about the real numbers. What are the real numbers? Why do they break the countable infinity? 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. 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 2 integers.

John Baez

That’s right. With the real numbers, we have the algebraic numbers. We have, of course, all the rational numbers. 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. Numbers that solve an algebraic equation over the integers are known as algebraic numbers.

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. The transcendental numbers are real numbers that are not algebraic.

And we won’t even go to the surreal numbers, about which you have a wonderful blog post. We’ll talk about that a little bit later.

John Baez

Oh, great. Liouville was the first to prove 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. Cantor also famously proved that there are many, many transcendental numbers. In fact, it follows from his argument on the uncountability of the real numbers that there are uncountably many transcendental numbers. So most real numbers are transcendental.

And again, going to Perplexity: “Transcendental numbers are real or complex numbers; they are not the root of any nonzero polynomial with integer or rational coefficients. This means they cannot be expressed as solutions to algebraic equations with integer coefficients, setting them apart from algebraic numbers.”

John Baez

Some of the famous transcendental numbers would include the number pi, 3.14159265 and so on. That’s a transcendental number. Also, Euler's constant, the e, like e to the x, the exponential function.

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

John Baez

Absolutely. That’s true. Although, I don’t know, the square root of 2 is pretty—

Square root. All right. So it depends. Let’s not—beauty can be found in all the different kinds of sets.

John Baez

That’s right. If you have a kind of simplicity attitude, then zero and 1 are looking pretty good, too.

Sorry to take that tangent, but what is your favorite number? Do you have one?

John Baez

Oh, gosh.

Is it zero?

John Baez

Did you know there’s a proof that every number is interesting? You can prove it.

Yeah? What’s that proof look like? How do you even begin?

John Baez

I’m going to prove to you that every natural number is interesting.

Okay.

John Baez

Zero’s interesting because it’s the additive identity, right? That’s pretty interesting. One is the multiplicative identity, so when you multiply it by any other number, you just get that number back, right? Two is the first prime number. That’s super interesting, right?

One can go on this way and give specific reasons, but I want to prove as a general principle that every number is interesting. This is the proof: suppose, toward contradiction, that there were some boring numbers.

Oh, okay.

John Baez

But if there was an uninteresting number—

Yes.

John Baez

—then there would have to be a smallest uninteresting number.

Yes.

John Baez

But that’s a contradiction, because the smallest uninteresting number is a super-interesting property to have. Therefore, there cannot be any boring numbers.

Ah, that’s good. I’m going to have to try to find a hole in that proof, because there’s a lot baked into the word “interesting,” but that’s beautiful.

John Baez

Right.

That doesn’t say anything about the transcendental numbers, about the real numbers that you just proved for just natural numbers.

John Baez

That’s right.

Okay, should we get back to Cantor’s argument?

John Baez

Sure. You’ve masterfully avoided the question. You basically said, “I love all numbers.”

Yeah, basically.

John Baez

Is that what you said?

Yeah. That was my intention.

John Baez

Back to Cantor’s argument. Let’s go.

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. The natural numbers are the numbers that start with 0 and add 1 successively: 0, 1, 2, 3, and so on. 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—all of those numbers altogether.

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. The claim that we want to prove is that they’re strictly larger. So suppose that they weren’t strictly larger. Then they would have the same size.

But to have the same size, remember, means by definition that there’s a one-to-one correspondence between them. So we suppose that the real numbers can be put into one-to-one correspondence with the natural numbers. Therefore, for every natural number N, we have a real number; let’s call it R sub N. R sub N is the Nth real number on the list. Basically, our assumption allows us to think of the real numbers as having been placed on a list: R1, R2, and so on.

I’m going to define the number Z, and it’s going to be—its integer part is going to be 0. Then I’m going to put a decimal place, and I’m going to start specifying the digits of this number Z: D1, D2, D3, and so on. What I’m going to make sure of is that the Nth digit after the decimal point of Z is different from the Nth digit of the Nth number on the list.

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. Whatever that digit is, I make sure that my digit is different from it. I want to do something a little bit more: I’m going to make it different in a way that means I never use the digits 0 or 9. I’m always using the other digits, not 0 or 9.

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

We’ve made our number Z so that the Nth digit of Z is different from the Nth digit of the Nth number. It now follows that Z is not on the list, because Z is different from R1: the first digit after the decimal point of Z is different from the first digit of R1 after the decimal point. That’s exactly how we built it.

The second digit of Z is different from the second digit of R2, and so on. The Nth digit of Z is different from the Nth digit of R sub N for every N. Therefore, Z is not equal to any of these numbers R sub N. But that’s a contradiction, because we had assumed that we had every real number on the list. Yet here is a real number Z that’s not on the list.

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

John Baez

Exactly. Given a list of numbers, Cantor is proving—actually, it’s interesting that you say that, because there’s a kind of philosophical controversy that occurs in connection with this observation about whether Cantor’s construction is constructive or not. Given a list of numbers, Cantor gives us a specific means of constructing a real number that’s not on the list. That’s one way of thinking about it.

There’s one aspect that I alluded to earlier: some real numbers have more than one decimal representation, and it causes this slight problem in the argument. For example, the number 1 can be written as 1.0000 forever, but it can also be written as 0.999 forever. Those are 2 different decimal representations of exactly the same number.

You beautifully got rid of the 0s and the 9s. Therefore, we don’t even need to consider that, and the proof still works.

John Baez

Exactly, because the only kind of case where that phenomenon occurs is when the number is eventually 0 or eventually 9. Since our number Z never had any 0s or 9s in it, it wasn’t one of those numbers. In those cases, we didn’t need to do anything special to diagonalize. The mere fact that our number has a unique representation already means that it’s not equal to those numbers.

Maybe it was controversial in Cantor’s day, more than 100 years ago, but I think it’s most commonly looked at today as one of the initial main results in set theory. It’s profound and amazing and insightful, and the beginning point of so many later arguments.

And this diagonalization idea has proved to be an extremely fruitful proof method. Almost every major result in mathematical logic uses, in an abstract way, the idea of diagonalization. It was really the start of so many other observations that were made, including Russell's paradox, the halting problem, and the recursion theorem. So many other principles use diagonalization at their core.

Can we just step back a little bit?

John Baez

Sure.

This infinity crisis led to a kind of rebuilding of mathematics. It would be nice if you laid out the things it resulted in. So one is that set theory became the foundation of mathematics. All mathematics could now be built from sets, giving math its first truly rigorous foundation. The axiomatization of mathematics—the paradoxes forced mathematicians to develop ZFC and other axiomatic systems—and mathematical logic emerged. Gödel, Turing, and others created entire new fields. 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?

That's a great question. Set theory really has two roles that it's serving. There are two ways that set theory emerges. On the one hand, set theory is its own subject of mathematics, with its own problems and questions and answers and proof methods. From this point of view, set theory is about transfinite recursive constructions or well-founded definitions and constructions. Those ideas have been enormously fruitful, and set theorists have looked into them and developed so many ideas coming out of that.

But set theory has also happened to serve in this other foundational role. 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. It's its own subject, but it's also serving as a foundation of mathematics.

So, in its foundational role, set theory provides a way to think of a collection of things as one thing. That's the central idea of set theory. A set is a collection of things, but you think of the set itself as one abstract thing. So when you form the set of real numbers, then that is a set. It's one thing. It's a set, and it has elements inside of it. So it's sort of like a bag of objects. A set is kind of like a bag of objects. And so we have a lot of different axioms that describe the nature of this idea of thinking of a collection of things as one thing itself, one abstract thing.

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

Yeah, I think that's right. 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. The history is quite fascinating because Zermelo, in 1904, offered a proof that what's called the axiom of choice implies the well-ordering principle. He described his proof, and that was extremely controversial at the time.

There was no theory; there weren't any axioms there. Cantor was not working in an axiomatic framework. He didn't have a list of axioms in the way that we have for set theory now, and Zermelo didn't either. His ideas were challenged so much with regard to the well-ordering theorem that he was pressed to produce the theory in which his argument could be formalized. That was the origin of what's known as Zermelo set theory.

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

That's right. On the one hand, the axiom of choice principle is completely obvious: we want this to be true, that it is true. A lot of people take it as a law of logic. If you have a bunch of sets, then there's a way of picking an element from each of them. There's a function. 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.

It's a completely natural principle. It's called the axiom of choice, which is a way of anthropomorphizing the mathematical idea. It's not like the function is choosing something. It's just that if you were to make such choices, there would be a function that consisted of the choices that you made.

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. You want to have the view that there is a way of choosing. I don't have an easy way to say what the function is, but there definitely is one. This is the way of thinking about the axiom of choice.

So we're going to say the three letters of ZFC may come up a lot in this conversation. You already mentioned Zermelo–Fraenkel set theory. That's the Z and the F, and the C in that comes from this axiom of choice.

That's right.

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

Yeah, absolutely. One should be aware also that there are huge parts of mathematics that don't 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. That's quite a vibrant amount of work in that area.

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

The butler can do this easily because, for any pair of shoes, he can just always pick the left shoe. There's a way of picking that we can describe. 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. We can invent rules that would result in these kinds of choice functions, so we can describe explicit choice functions. For those cases, you don't need the axiom of choice to know that there's a choice function.

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. But the problematic case occurs when you think about the infinite collection of socks that the person has in their closet. If we assume that socks are sort of indistinguishable within each pair—they match each other, indiscernible—then the butler wouldn't have any kind of rule for which sock in each pair to pick.

So it's not so clear that he has a way of producing one sock from each pair. That's what's at stake: the question of whether you can specify a rule by which the choice function operates, a rule that defines the choice function, or whether there's this arbitrary choosing aspect to it. That's when you need the axiom of choice to know that there is such a function.

But, of course, as a matter of mathematical ontology, we might find attractive the idea that not every way of choosing the socks has to be defined by a rule. Why should everything that exists in mathematical reality follow a rule or a procedure of that sort? 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. 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.

Yes, there is a way of choosing, but I can't necessarily tell you what it is. But in a mathematical argument, I can assume that I fix the choice function because I know that there is one. The philosophical difference between working when you have the axiom of choice and when you don't is the question of the constructive nature of the argument.

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. You're not necessarily going to be able to say specific things about them. But if you're just making an existence claim, that's totally fine.

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.

Can we maybe speak to the axioms that underlie ZFC? ZFC, or Zermelo–Fraenkel set theory with the axiom of choice, as we mentioned, is the standard foundation for most modern mathematics. It consists of the following main axioms: the axiom of extensionality, the axiom of the empty set, the axiom of pairing, the axiom of union, the axiom of power set, the axiom of infinity, the axiom of separation, the axiom of replacement, the axiom of regularity, and the axiom of choice.

Some of these are quite basic, but it would be nice to give people a sense of what it means to be an axiom. What kind of basic facts can we lay on the table on which we can build some beautiful mathematics?

Yeah, the history of it is really quite fascinating. 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-ordering theorem, which was an extremely controversial result. In 1904, he gave the proof without the theory, and then he was challenged to provide the theory. So in 1908, he produced Zermelo set theory and gave the proof that, in that theory, you can prove that every set admits a well-ordering.

The axioms on the list, things like extensionality, express the most fundamental principles of the understanding of sets that he wanted to be talking about. For example, extensionality says that if two sets have the same members, then they’re equal. It’s the idea that sets consist of the collection of their members, and that’s it. There’s nothing else going on in the set. So if two sets have the same members, then they are the same set. It’s maybe the most primitive axiom in some respect.

Well, there’s also, just to give a flavor, that there exists a set with no elements, called the empty set. For any two sets, there’s a set that contains exactly those two sets as elements. For any set, there’s a set that contains exactly the elements of the elements of that set, so the union set.

And then there’s the power set. For any set, there’s a set whose elements are exactly the subsets of the original set—the power set. In the Axiom of Infinity, there exists an infinite set, typically a set that contains the empty set and is closed under the operation of adding one more element. Back to our hotel example.

That’s more, but this is kind of fascinating. Just put yourself in the mindset of people at the beginning of this, trying to formalize set theory. It’s fascinating that humans can do that.

I read some historical accounts by historians about that time period, specifically about Zermelo’s axioms and his proof of the well-ordering theorem. The historians were saying that never before in the history of mathematics had a mathematical theorem been argued about so publicly and so vociferously as that theorem of Zermelo’s.

It’s fascinating also because the Axiom of Choice was widely regarded as a basic principle at first. But people were very suspicious of the well-ordering theorem because no one could imagine a well-ordering, say, of the real numbers. This was a case when Zermelo seemed to be proving, from principles that seemed quite reasonable, this obvious untruth. So mathematicians were objecting.

But then Zermelo and others 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. It’s so natural to use because it’s such an obvious principle in a way. It’s easy to use it by accident if you’re not critical enough and don’t even realize that you’re using the Axiom of Choice.

That’s true even now. People like to pay attention to when the Axiom of Choice is used or not used in mathematical arguments, up until this day. It used to be more important. 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. So there was a worry about the consistency of the axioms.

But then, of course, eventually, with the results of Gödel and Cohen and so on, this consistency question specifically about the Axiom of Choice sort of fell away. We know that the Axiom of Choice itself will never be the source of inconsistency in set theory. If there’s an inconsistency with the Axiom of Choice, then it’s already inconsistent without the Axiom of Choice. So it’s not the cause of inconsistency.

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

And we should say that, in set theory, consistency means that it is impossible to derive a contradiction from the axioms of the theory. It means that there are no contradictions.

That’s right.

A consistent axiomatic system is one in which there are no contradictions.

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

Maybe a quick bathroom break. You mentioned to me offline—we were talking about Russell’s paradox—and that there’s another kind of anthropomorphizable proof of uncountability. I was wondering if you can lay that out.

Oh yeah, sure. Absolutely.

Both Russell’s paradox and the proof.

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

The power set is the set containing all the subsets of the original set. If you have a set and look at the collection of all of its subsets, then Cantor proved that this is a bigger set. They’re not equinumerous. Of course, there are 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. So there are always at least as many subsets as elements. But the question is whether there are strictly more.

Cantor reasoned like this. It’s very simple. It’s a kind of distillation of the abstract diagonalization idea without being encumbered by the complexity of the real numbers. We have a set X, and we’re looking at all of its subsets. That’s the power set of X. Suppose, toward a contradiction, that X and the power set of X have the same size. That means we can associate every individual of X with a subset.

Now let me define another set. Let’s call it D. D is the subset of X that contains all the individuals that are not in their set. Every individual was associated with a subset of X, and I’m looking at the individuals that are not in their set. Maybe nobody’s like that, 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. It doesn’t really matter for the argument.

I defined a subset D consisting of the individuals that are not in the set attached to them, but that’s a perfectly good subset. Because of the equinumerosity, it would have to be attached to a particular individual. Let’s call that person—she should have a name starting with D—Diana.

Now we ask, is Diana an element of D or not? If Diana is an element of D, then she is in her set, so she shouldn’t be, because the set D was the set of individuals who are not in their set. But if she isn’t in D, then she wouldn’t be in her set, and so she should be in D. That’s a contradiction. Therefore, the number of subsets is always greater than the number of elements for any set.

The anthropomorphizing idea is the following. I’d like to talk about it this way: for any collection of people, you can form more committees from them than there are people, even if you have infinitely many people. Suppose you have an infinite set of people. What’s a committee? A committee is just a list of who’s on the committee—the members of the committee.

So there are all the two-person committees, all the one-person committees, and the universal—the worst committee—the one that everyone is on. The best committee is the empty committee, with no members, and it never meets and so on. Or is the empty committee meeting all the time? I’m not sure.

Yeah, that’s a profound question. And does a committee with just one member meet?

Yeah, maybe it’s always in session. I don’t know. So the claim is that there are more committees than people.

Suppose not. Then we could make an association between the people and the committees. Every committee could be named after a person in a one-to-one way. I’m not saying whether the person is on the committee that’s named after them or not. Maybe sometimes that happens, sometimes it doesn’t. It doesn’t matter.

Let’s form what I call committee D, which consists of all the people who are not on the committee that’s named after them. Maybe that’s everyone, maybe it’s no one, maybe it’s half the people. It doesn’t matter. That’s a committee; it’s a set of people. So it has to be named after someone. Let’s call that person Daniella.

Now we ask, is Daniella on the committee that’s named after her? If she is, then she shouldn’t be, because it was the committee of people who aren’t on their own committee. And if she isn’t, then she should be. Again, it’s a contradiction.

When I was teaching at Oxford, one of my students came up with the following different anthropomorphization of Cantor’s argument. Let’s consider all possible fruit salads. We have a given collection of fruits—apples, oranges, and grapes. A fruit salad consists of some collection of those fruits. So there’s the banana, pear, grape salad and so on. There are a lot of different kinds of salad. Every set of fruits makes a salad, a fruit salad.

Okay. We want to prove that for any collection of fruits, even if there are infinitely many different kinds of fruit, there are more possible fruit salads than there are fruits. If not, then you could put a one-to-one correspondence between the fruits and the fruit salads, so you could name every fruit salad after a fruit. That fruit might not be in that salad; it doesn't matter. We're just naming them with a one-to-one correspondence.

Then, of course, we form the diagonal salad, which consists of all the fruits that are not in the salad that's named after them. That's a perfectly good salad. It might be the kind of diet salad if it was the empty salad, or it might be the universal salad, which had all fruits in it, if all the fruits were in it. Or it might have some and not all.

So that diagonal salad would have to be named after some fruit. Let's suppose it's named after durian, meaning that it was associated with durian in the one-to-one correspondence. Then we ask, well, is durian in the salad that it's named after? If it is, then it shouldn't be. And if it isn't, then it should be. So it's again the same contradiction.

All of those arguments are just the same as Cantor's proof that the power set of any set is bigger than the set. 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. If it were, then we could form the set of all sets that are not elements of themselves.

Basically, what Russell is proving is that there are more collections of sets than elements, because we can form the diagonal class—the class of all sets that are not elements of themselves. If that were a set, then it would be an element of itself if and only if it was not an element of itself. It's exactly the same logic in all four of those arguments.

Yeah. 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. But that set would be an element of itself if and only if it's not an element of itself, which is a contradiction. So this is the essence of the Russell paradox. I don't call it the Russell paradox. Actually, when I teach it, I call it Russell's theorem. There's no universal set. And it's not really confusing anymore. 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. 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. 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. And those principles happened to imply that for any property whatsoever, you could form the set of objects with that property. This is known as the general comprehension principle. And he was appealing to the principles that support that axiom, throughout his work. I mean, it wasn't just an incidental thing. He was really using this principle. 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. That's just an instance of the general comprehension principle. And 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. And so Russell wrote this letter to Frege, and it was just at the moment when Frege was finishing his work. It was already at the publishers and, you know, in press basically. But it's completely devastating. 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. 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. This is the position into which I was put by a letter from Mr. 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, and...

It's heartbreaking. There's nothing more traumatic to a person who dreams of constructing all of mathematics from logic than to get a very clean, simple contradiction. That's just—

You devote your life to this work, and then it's shown to be contradictory. That must have been heartbreaking.

What do you think about Frege's project—the philosophy of logicism, the dream of the power of logic to construct a mathematical universe?

The project of logicism, of course, did not die with Frege. It was continued, and there's a whole movement—the neo-logicists and so on—even in contemporary times. But my view is that the main goals of logicism are basically completely fulfilled in the rise of set-theoretic foundationalism.

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

But I think if you adopt the view that the principles of ZFC have to do with the principles of abstract set formation, which is fundamentally logical in character, then it's a complete success for logicism. The fact that set theory is able to serve as a foundation means that mathematics can be founded on logic.

I think this is a good moment to talk about Gödel's incompleteness theorems. Can you explain them, and what do they teach us about the nature of mathematical truth?

Absolutely. They're one of the most profound developments in mathematical logic. The incompleteness theorems are when mathematical logic, in my view, first became sophisticated. It's a kind of birth of the subject of mathematical logic.

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

This minefield of paradoxes.

Right. A minefield. That's a really good way of describing the situation. Hilbert was famously supportive of set theory. There's this quote of his: “No one shall cast us from the paradise that Cantor has created for us.”

What I take him to mean by that is that he was so captured by the idea of using set theory as a foundation of mathematics. It was so powerful, convenient, and unifying in a way that was extremely important. He didn't want to give that up, despite the danger of these paradoxes—these contradictions, basically—is how some people viewed them.

Hilbert said, “Well, look, we have to fix this problem. We want to use set-theoretic foundations, but we want to do it in a way that is trustworthy and reliable. We can't allow the foundations of mathematics to be in question.” That's the kind of attitude, I think, that underlies Hilbert and the Hilbert program.

He proposed, “Look, we're going to have this strong theory, this set theory that we want to be proving our theorems in. On the one hand, we want it to be as strong as possible. We would like it to answer all the questions.”

There's another famous quote from Hilbert in his retirement address where he proclaims, “Wir müssen wissen, wir werden wissen”—“We must know, we will know”—in which he's very optimistic about the ability of mathematics to answer all the questions of mathematics that we have posed. We have all these problems we want to solve, and he is saying, “We're going to do it. We're going to solve all these problems.”

So we want to propose this strong theory, and one has the sense that he had set theory in mind, in which all the questions are going to be answered. But secondly, we want to combine that with a very weak arithmetic, a purely finitistic theory, in which we want to prove that the reasoning process of the strong theory is safe.

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 and what the nature of mathematical reasoning is. On Hilbert's way of thinking about this, a proof is basically itself a finitistic kind of object.

If you think about the nature of what a proof is, it's a sequence of assertions, which can be viewed as sequences of symbols that conform with certain rules of logical reasoning. This is a formalist way of understanding the nature of proof. We think about a proof in a kind of syntactic, formal way.

Even though the contents of those statements might be referring to infinite, uncountable objects, the statements themselves are not infinite, uncountable objects. The statements themselves are just finite sequences of symbols.

So when you think of proof as—maybe it's fair to say—almost like something outside of math...

It's like tools operating on math. And then, for Hilbert, he thought proof is inside the axiomatic system. Something like this.

Yeah, that's helpful.

That's wild.

The main thing about formalism is that you think of the process of doing mathematics. You divorce it from the meaning of the mathematical assertions, right? 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. And that's a very uncertain realm, maybe, and the source of the paradoxes and so on in some people's minds.

But the reasoning process itself consists of writing down sequences of symbols on your page and undertaking an argument with them, which is following these finitary rules. 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 the meaning may be totally absent. I don't think it's necessarily part of the formalist view that there is no meaning behind it, but rather it's emphasizing that we can divorce the meaning of the sentences from the process of manipulating those sentences.

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. So those were the 2 aims of the Hilbert program: to found the strong infinitary theory, probably set theory, which is going to answer all the questions, and then, secondly, to prove in the finitary theory that the strong theory is safe—in other words, consistent.

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

Yeah. Well, this is, of course, philosophically contentious, and people have different ideas about what exactly it should mean. So there are hundreds of papers on exactly that question. But I like to take it just informally. I mean, it means that we're talking about finite sequences of symbols, and we're going to have a theory of finite strings of symbols. 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.

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. The last line on the list is the theorem that you're proving. So that's what a proof is in this kind of way of thinking.

To take a specific example, the most natural finitary theory that one would be called upon to exhibit would be Peano arithmetic, which is a first-order theory of the nature of arithmetic. But some people say, “Well, Peano arithmetic has these strong first-order induction axioms, and there are much, much weaker versions of arithmetic, like IΣ₀ or IΣ₁ and so on, which are even more finitary than Peano arithmetic.” So different philosophical positions take different attitudes about what it takes to be finitary. How finitary do you have to be to be truly finitary?

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. Peano arithmetic provides a formal language and axioms for arithmetic operations, such as addition and multiplication over the natural numbers. The axioms define the existence of a first natural number, usually 0 or 1; the concept of a successor function, which generates the next natural number; rules for addition and multiplication built from these concepts; the principle of induction, allowing proofs about all natural numbers; and so on. So it's a very particular kind of arithmetic that is finitary.

I view it as finitary, but this is a contentious view. Not everyone agrees with that. That's what I was trying to hint at.

Okay. I got it. All right.

Peano arithmetic is one of the hugely successful theories of the natural numbers and elementary number theory. Essentially, all of classical number theory—whatever kind of theorems you want to prove 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. I mean, that's the basic situation. 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 can be developed almost entirely inside Peano arithmetic.

So if we go back to the Hilbert program, Hilbert has these 2 goals: produce the strong theory which is going to answer all the questions, and then prove by purely finitary means that that theory will never lead to a contradiction. One can think about whether the incompleteness theorem should be viewed as a decisive refutation of the Hilbert program. It defeats both of those goals decisively, completely. But before explaining that, maybe one should think about what it would be like if Hilbert had been right. What would be the nature of mathematics in the world that Hilbert is telling us to search for?

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. 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.

Right. Exactly right. Let's imagine what it would be like if he had been right. So we would have this finitary theory, and it would prove that the strong theory was free of contradiction. So we could start enumerating proofs from the strong theory. Right now, we can write a computer program that would systematically generate all possible proofs from a given theory.

And so we could have this theorem-enumeration machine that would just spit out theorems all day long in such a manner that every single theorem would eventually be produced by this device. 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 from the machine or the answer to come out no. So the nature of mathematical investigation in Hilbert's world is one of just turning the crank of the theorem-enumeration machine, devoid of creative thinking or imagination; it's just getting the answer from this by-rote procedure.

So Hilbert, in effect, is telling us, with his program, that the fundamental nature of mathematics is rote computation. 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 on—how can we block them? 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 known. They didn't know anything like that had ever happened.

And so it's natural to think that it wouldn't happen, and also that they would be able to guard against this inconsistency. So it seems like the goals of the Hilbert program are quite natural in that historical context. But when you think a little more about what the nature of it would be like, it shows you this kind of rote procedure. And now you're saying, well, that doesn't seem so unlikely, maybe. In light of the 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 in a way that could be alarming.

To talk about the alternative to the Hilbert point of view, if he's wrong, then what is the nature of mathematical reality? Well, for the first goal, it would mean that we could never write down a theory that answered all the questions. 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. Independence would occur.

But 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. And the fact of Gödel's incompleteness theorem shows that that is exactly the nature of mathematical reality, actually. Those are the 2 incompleteness theorems.

So the first incompleteness theorem says you cannot write down a computably axiomatizable theory that answers all the questions. Every such theory will be incomplete, assuming it includes a certain amount of arithmetic. And secondly, no such theory can ever prove its own consistency. 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? That's the second incompleteness theorem.

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. It's quite amazing. There's another aspect that's kind of easy to think about. 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?

That's like the used-car salesman telling you, “Oh, I'm trustworthy.” It's not a reason to trust the used-car salesman, is it? Just because he says that. So similarly, if you have a theory that proves its own consistency, well, even an inconsistent theory would prove its own consistency.

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.

Just for clarification, you used the word “theory.” Is it, in this context, synonymous with “axiomatic system”?

Right. So in mathematical logic, “theory” is a technical term, and it means any set of sentences in a formal language. If you say “axiomatic system,” it's basically synonymous with my usage of “theory.” A theory means the consequences of a set of axioms. People are sometimes unclear on whether they just mean the axioms or the consequences of the axioms.

So theory includes both the axioms and the consequences of the axioms, and you use it interchangeably, with the context supposed to help you figure out which of the two you're talking about—the axioms or the consequences? Or maybe to you, they're basically the same?

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

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. So that's one difference between the list of axioms of the theory and the theory itself. 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.

Usually, you only get to decide the positive instances. 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.”

And that's, of course, connected to the halting problem. All of these contradictions and paradoxes are nicely, beautifully interconnected. So can we just linger on Gödel's incompleteness theorem? You mentioned the 2 components there. There are so many questions to ask, like, what is the difference between provability and truth? What is true and what is provable? Maybe that's a good line to draw.

Yeah, this is a really core distinction. 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. It wasn't clear at all until Gödel, basically. Although even as late as Bourbaki, there's this kind of confusion in their foundational work.

There is a standard graduate-level textbook used in France for the presentation of logic, and they are conflating truth and proof. To be true for them means to be provable. 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. But because of the incompleteness theorem, we realized that there are actually quite subtle things happening.

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: this distinction between truth and proof. Truth is on the semantic side of the syntax-semantics dichotomy. Truth has to do with the nature of reality. When I talk about reality, I'm not talking about physical reality. I'm talking about mathematical reality.

So, we have a concept of something being true in a structure—a statement being true in a mathematical structure. Maybe you have the real field, 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. 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?

Tarski gave this absolutely wonderful account of the nature of truth in what's now known as the disquotational theory of truth. What Tarski says is, the sentence “Snow is white” is true if and only if snow is white. What he means by that is, to say that truth is a property of an assertion, we can think of the assertion syntactically. The sentence is true if and only if the content of the sentence is the case.

The sentence “Snow is white,” in quotations, is true—that just means that snow is white. That's why it's called the disquotational theory, because we remove the quotation marks from the assertion.

You can use this idea of disquotation to give a formal definition of truth in a mathematical structure for a statement in a formal language. For example, if I have a formal language that allows me to make atomic statements about the objects and relations of the structure, I can build up a formal language with the logical connectives of “and,” “or,” “implies,” “not,” and so on, and maybe I have quantifiers.

For example, to say that the structure satisfies phi and psi—that single statement, “phi and psi,” I'm thinking of that as one statement—just means that it satisfies phi and it satisfies psi. If you notice what happened there, 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 2 conditions.

Yeah, it has the disquotation.

And so this idea can be applied to all the logical connectors and quantifiers and everything. 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.

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. Maybe we have in mind the standard model of arithmetic, with the natural numbers and the arithmetic structure, and I want to know whether a given statement is true in that structure. Then we have a formal definition of what that means according to Tarski's recursive definition of truth.

That's truth. Proof, on the other hand, is, in this Hilbert way of thinking, something for which we can develop proof theory. What is a proof for a mathematician, for a mathematical logician? A proof is a certain sequence or arrangement of sentences in the formal language that accords with the logical rules of a proof system.

There are certain modes of reasoning that are allowed. 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. If you know A and you know A implies B, those are both statements that are known, then you can deduce B as a consequence according to the rule of modus ponens. This is the rule of modus ponens. Some people would call this implication elimination.

There are different kinds of proof systems. There are a lot of different formal proof systems that exist that are studied by 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. So that's what it means to be sound. Proofs preserve truth. They're truth-preserving arguments.

But the proof systems are also generally complete. 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, then there is a proof of it. The proof systems generally have both of those properties: they're sound and complete.

There's a third property. A lot of logicians talk about “sound and complete,” “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. So there's a computable aspect to the proof systems.

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. We don't want a proof system in which someone claims to have a proof, but we can't check whether it's a proof or not. We want to be able to correctly adjudicate all claims to having a proof.

Yeah. A mathematician comes to mind who said he had a proof, but the margins were too small—

That's right.

—to continue.

Exactly.

So that doesn't count as a proof.

Yeah. 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.

So what is, again, the tension between truth and proof? Which is more powerful, and how do the 2 interplay with the contradictions that we've been discussing?

Right. So the incompleteness theorem asks whether we could, say, write down a theory for arithmetic—for the standard model of arithmetic, where we have the natural numbers, plus and times, 0, 1, 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. So the question would be, can we write down a computable list of axioms that will answer all those questions by proof?

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. Hilbert would love that. That would support Hilbert's program: to have such a complete theory of arithmetic. Gödel proved that this is impossible. You cannot write down a computable list of axioms that is complete in that sense.

There will always be statements that you cannot prove and cannot refute, if the theory is consistent. So they are independent of that theory.

How traumatic is that—that there are statements that are independent of the theory?

My view is that this isn't traumatic at all. This is completely eye-opening in terms of our understanding of the nature of mathematical reality. 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 answer all the questions. It's impossible. I don't think of it as trauma. I just think, look, this is the nature of mathematical reality, and it's good that we know it. Now we need to move on from that and do what we can in light of that.

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. In general, you cannot. The provability problem can be formulated as a decision problem: Given a theory and a statement, is that statement a consequence of that theory?

This is one of the most famous decision problems. In fact, it's the very first one, because it's equivalent to the Hilbert–Ackermann Entscheidungsproblem, which also appears in the title of Turing's 1936 paper that was so important for computability theory. So it's a formulation of the Entscheidungsproblem: Does a given theory have a given statement as a logical consequence?

Because of Gödel's completeness theorem—not his incompleteness theorem, but his earlier completeness theorem—Gödel had proved that the proof systems they studied did have this completeness property that I mentioned. So provability is the same as logical consequence. This is an undecidable decision problem. Turing proved, and we now know, that it's equivalent to the Halting Problem.

Can you describe the Halting Problem? It's a thing that shows up in a very useful and, again, traumatic way through a lot of computer science and a lot of mathematics.

The Halting Problem expresses a fundamental property of computational processes. Given a program—or perhaps we think of it as a program together with its input, but let me just call it a program—we could run that program, but I want to pose it as a decision problem: Will this program ever complete its task? Will it ever halt?

The Halting Problem is the question: Given a program, will it halt? Yes or no? For any one instance, the answer is either yes or no. That's not what we're talking about. We're talking about whether there's a computable procedure to answer all instances of this question.

It's a decision problem given as a scheme of instances for all possible programs that you could ask about. What I want to know is whether there is a computable procedure that will answer those questions. It turns out the answer is no. The Halting Problem is computably undecidable. There is no computable procedure that will correctly answer all instances of whether a given program will halt.

Of course, we can get half the answers in the sense that you give me a program and say, "Will this halt?" I could take that program and run it. I could keep running it, and maybe in a week it would halt. At that time, I could say, "Yes, it halted." So I can get all the yes answers correctly for halting.

But the problem is, if it hasn't halted yet—maybe I waited 1,000 years and it still hasn't halted—I don't seem entitled to say, "No, it's not going to halt." Maybe in 1,001 years it'll halt. At no point can I seem to say, "No." 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.

Giving the yes answers is trivial. You don't have to understand the program; you just need to run it, which is a kind of rote task. 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."

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. It turns out to be impossible to have a computable procedure that gives the no answers. The argument is not very difficult. Should we do it?

Yes, let's do it.

Okay. Suppose, toward a 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, the Cantor argument, and Gödel's argument, which we haven't talked about yet. So many diagonal arguments come in.

Suppose, toward a contradiction, that we had a procedure for determining whether a given program halted on a given input. Let me describe how I'm going to use that procedure as a subroutine in the following process. Let's call my process Q, and it takes as input a program P.

The first thing it does is ask that subroutine, "Would P halt if I ran it on P itself?" That's the diagonal part, because we're applying P to P, right? So I'm describing program Q, and program Q takes as input P, which is itself a program.

The first thing it does is ask the halting subroutine, "Would P halt on P?" If the answer comes back from the subroutine, "Yes, that would halt," then what I do in program Q is immediately jump into an infinite loop, so I don't halt. If P halts on P, I don't halt.

But if the answer comes back, "No, P is never going to halt on P," then I halt immediately. That's it. I've described what Q does. The thing about Q is that Q's behavior on P is the opposite of P's behavior on P. That's how we designed Q specifically, so that Q on P had the opposite behavior from P on P.

So now, of course, what do we do? Well, the same thing that Russell and Cantor did: We ask, "What would Q do on Q?" Because of this opposite behavior, Q would halt on Q if and only if Q does not halt on Q, which is a contradiction. Q has to have the opposite behavior on Q from what Q does, but that's contradictory.

What a beautiful proof. Simple.

It's absolutely beautiful. I agree. It's following the same logic as Russell and Cantor—going back to Cantor, basically, because Russell is also quoting Cantor in his letter to Frege.

Therefore, the conclusion is that the Halting Problem is not computably decidable. Now we can immediately prove Gödel's theorem using this, actually. I view this as the simplest proof of Gödel's theorem. You don't need the Gödel sentence to prove Gödel's theorem; you can do it with the Halting Problem.

Suppose that we could write down a computable axiomatization of all the true facts of elementary mathematics, meaning arithmetic and finite combinatorial things such as Turing machine computations and so on. In fact, all those finite combinatorial processes are formalizable inside arithmetic with the standard arithmetization coding process.

But let me just be a little bit informal and say, suppose we could write down a complete theory of elementary finite mathematics. We have an axiomatization of that theory. Then we could produce all possible theorems from those axioms in the way that I was describing earlier with Hilbert's program.

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. 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.

You give me a program and input that you want to run that program on, and I'm going to answer the Halting Problem. 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 that P does not halt on that input.

One of them is going to happen because it was a complete theory that was enumerating all the true statements of elementary mathematics. 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. That proves Gödel's theorem.

Maybe to take a little bit of a tangent, can you speak... You've written a wonderful book about proofs and the art of mathematics. So what can you say about proving things in mathematics? What is the process of proof? What are the tools? What is the art? What is the science of proving things in mathematics?

This is something that I find so wonderful to teach young mathematicians who are learning how to become mathematicians and learning about proof. I wrote that book when I was teaching a proof-writing class in New York.

Many universities have such a course, usually taken by students who have learned some mathematics. Usually, they've completed the calculus sequence and are making the transition to higher mathematics, which tends to involve much more proof, and it's a challenging step for them. Many math departments have this kind of course on proof-writing, where the students are exposed to how to write proofs.

I wasn't happy with most of the other books that exist for those kinds of courses. The reason was that they were often so dull because they concentrated on totally uninteresting parts of what it's like to write a proof—these mechanistic procedures about how to write a proof. If you're going to prove an implication, then you assume the hypothesis and argue for the conclusion, and so on. All of that is true and fine, and it'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.

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 of elementary proof. So I wrote this book and tried to fill it with compelling mathematical statements with very elementary proofs that exhibited lots of different proof styles. I found that the students appreciated it a lot.

We should say, you dedicate the book to your students: “May all their theorems be true, proved by elegant arguments that flow effortlessly from hypothesis to conclusion while revealing fantastical mathematical beauty.” Are there some interesting proofs that might illustrate this for people outside of mathematics, or for people who just take math classes in high school and so on?

Yeah, let's do a proof. There's one in the book that we can talk about. I think it's a nice problem. It's in the discrete math section, 5.1, “More Pointed At Than Pointing.”

Suppose you're gathered with some friends in a circle, and you can point at each other however you want, or at yourself—it doesn't matter—and you can point at more than 1 person. You can use all your fingers or your feet or whatever you want. Maybe you point at 3 of your friends, and they point at 2 or 3 of their friends. One person is pointing at 10 people, somebody isn't pointing at anybody, and various people are being pointed at as well.

The question is, could we arrange a pattern of pointing so that everyone was pointed at more than they were pointing at others? In other words, maybe there are 7 people pointing at me, but I'm only pointing at 5 people. Maybe there are 20 people pointing at you, but you're only pointing at 15 people.

There's a similar question on Twitter. For a group of people on Twitter, could you arrange it so that everyone has more followers than people they're following? It's the same question mathematically—it's identical. Although, I don't know, it's not identical because I said you could point at yourself. Can you follow yourself?

No, I don't think so.

I don't think you can. Okay. So can you arrange it so that everyone is pointed at more than they're pointing? In my book, I give a couple of different proofs of this. I think I give an induction proof, and then there's another proof. I think there are 3 different proofs in there.

Why don't we just talk about my favorite proof? Suppose it were possible to arrange that we're all pointed at more than we're pointing, okay? What we're going to do is agree to give a dollar to everyone we're pointing at.

What happens? Everybody made money, because I was pointed at by more people than I'm pointing at, so I got $10, but I only paid out $7. Similarly, you got paid $20, but you only paid out $15. If everyone is pointed at more than they're pointing, then everyone makes money.

But it's obviously impossible for us to make money as a group by just trading money with ourselves. Therefore, it can't be possible that we're all pointed at more than we're pointing.

This proof illustrates one of my habits that I suggest in the book: anthropomorphizing your mathematical ideas. You should imagine that the mathematical objects playing a role in your question are people, or active animals, or something that might have a will and a goal. This process of anthropomorphizing often makes problems easier to understand, because we're all familiar with the fact that it's difficult to make money.

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. But that by itself is actually a difficult mathematical claim. If someone had to prove that you can't make money by trading within a group—that it can't be that everyone in the group makes money just by shifting money around in the group—you might think that's obvious, and it is obvious if you think about money.

But if you had asked the question about mathematical functions of a certain kind, then maybe it wouldn't be as clear as it is when you're talking about money, because we can build on our human experience about the difficulty of getting money or other resources. It doesn't have to be money; it could be candy, whatever. We just know that you can't easily get more things of that kind just by trading within a group.

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. In the context of money or social systems, there's a bunch of things that are non-obvious. The whole point is that proof can guide us to the truth, to an accurate description of reality. We just proved a property of money.

It's interesting to think about what if there were infinitely many people in your group. Then it's not true anymore. The theorem fails. In fact, you can arrange it so that everyone is strictly more pointed at than pointing.

Also, if everyone has even just 1 dollar bill, you can arrange it so that afterward everyone has infinitely many dollar bills, because in terms of cardinality, that's the same. It's just countable infinity in each case. If you had countably many friends and everyone had 1 dollar bill, then you could arrange a pattern of passing those dollar bills among each other so that afterward everyone had infinitely many dollar bills.

What you need is for each person to be attached to one of the train cars or something. Think of everyone as coming from Hilbert's train, but also think of them as fitting into Hilbert's Hotel. Have everyone on the Nth car give all their money to the person who ends up in the Nth room. They each give 1 dollar to that person.

Afterward, that person has infinitely many dollars, but everyone only paid out 1 dollar. So it's a way of making it happen.

To what degree, sticking with the topic of infinity, should we think of infinity as something real?

That's an excellent question. A huge part of the philosophy of mathematics is about this kind of question: What is the nature of the existence of mathematical objects, including infinity?

But I think asking about infinity specifically is not that different from asking about the number 5. What does it mean for the number 5 to exist? What are the numbers, really? This is perhaps one of the fundamental questions of mathematical ontology. There are many different positions to take on the question of the nature of the existence of mathematical objects, or abstract objects in general.

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 wish that we could give an account of the existence of numbers, or other mathematical objects or abstract objects, that was more like the existence of tables and chairs and rocks.

There seems to be this desire to reduce mathematical existence to something that we can experience physically in the real world. But my attitude about this attempt is that it's very backward, because I don't think we have such a clear understanding of the nature of physical objects, actually.

We all have experience of 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. If I ask you to imagine a certain kind of steam locomotive, and I describe its engineering, its weight, and the nature of the gear linkages, and I show you schematic drawings of the whole design, and we talk in detail about every single aspect of this steam locomotive...

But then suppose, after all that conversation, I say, “Okay, now I would like you to tell me what it would mean for it to exist physically, as opposed to just being an imaginary steam locomotive.” What could you possibly say about it? I mean, except by saying, “I just mean that it exists in the physical world.” But what does that mean? That’s the question, right? It’s not an answer to the question. That is the question.

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. In fact, it becomes more and more mysterious the more physics we know. Back in, say, Newtonian physics, one had a picture of the nature of physical objects as little billiard balls, or maybe as things that are infinitely divisible.

But then this picture is upset with the atomic theory of matter. That picture is upset when we realize that atoms actually can be split and consist of electrons, protons, neutrons, and so on. Then that picture is upset when we realize that those things themselves are built out of quarks and leptons, and who knows what’s coming.

Furthermore, all of those things have a nature of existence that is actually as wave functions in some cloud of probability and so on. It just becomes more and more mysterious the more we learn, and not at all clarifying. 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.

Whereas we do seem to have a much more satisfactory account of the nature of abstract existence. I can talk about the nature of the empty set: this is the predicate which is never true, or something like that. I can talk about those kinds of logical properties, or the singleton of the empty set, and so on. 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. The more that you say, it becomes only more and more clear.

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 in the abstract world where existence is much more clear.

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. Again, we anthropomorphize, and that actually gets us into trouble sometimes, because I’m not feeling the quantum mechanics when I’m touching it.

That’s right.

Therefore, it’s easy to forget and feel like this is real and mathematical objects are not, but you’re making the opposite argument. When you draw a distinction between numerals and numbers, where numerals are the representation of the number on the page and so on, could you say that a number is real? Do numbers exist?

I happen to think so. 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 the way I just tried to describe.

So, you would describe it as the size of a set with 4 elements in it?

There are different ways to understand the nature of 4. Actually, this gets into the question of structuralism, which is maybe a good place to talk about it.

What is structuralism?

Structuralism is a philosophical position in mathematics, or in 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.

What I call the structuralist attitude in mathematics is that we should only care about our mathematical structures up to isomorphism. 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. There’s no important mathematical difference that would ever arise from working with this isomorphic copy instead of the original structure.

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. To ask a question like, “What is the number 4 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 4 with something else. This bottle of water could play the role of the number 4 in that structure, and it would be isomorphic.

It wouldn’t matter at all for any mathematical purpose to use this alternative mathematical system. That’s to say that we don’t care what the number 4 is really. That is irrelevant. The only thing that matters is: What are the properties of the number 4 in a given mathematical system?

We recognize that there are other isomorphic copies of that system, and the properties of that other system’s number 4 are going to be identical to the properties of this system’s number 4 with regard to any question that’s important about the number 4. But those questions won’t be about essence. In a sense, structuralism is an anti-essentialist position in mathematics.

Is it fair to think of numbers as a kind of pointer to a deep underlying structure?

I think so, because part of the point of structuralism is that it doesn’t make sense to consider mathematical objects, or individuals, in isolation. What’s interesting and important about mathematical objects is how they interact with each other and how they behave in a system. One wants to think about the structural role that the objects play in a larger system, a larger structure.

There’s a famous question that Frege had asked when he was looking into the nature of numbers. In his logicist program, he was trying to reduce all of mathematics to logic. In that process, he was referring to the Cantor–Hume principle: whenever 2 sets are equinumerous, they have the same number of elements, and vice versa.

He founded his theory of number on this principle, but he recognized that there was something that dissatisfied him about that situation. The Cantor–Hume principle does not seem to give you criteria for which things are numbers. It only tells you a kind of identity criterion for when 2 numbers are equal to each other.

Two numbers are equal just in case the sets of those sizes are equinumerous, so that’s the criterion for number identity. But it is not a criterion for what is a number. 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.

He’s asking about the essence of number, and whether—of course, one has a sense that he picked what he was trying to present as a ridiculous example, because maybe you have the idea that, obviously, Julius Caesar is not a number. There’s a lot of philosophical writing that seems to take that line also: obviously, the answer is that Julius Caesar is not a number.

But the structuralists disagree with that position. The structuralist attitude is: Look, you give me a number system. If Julius Caesar isn’t a number, then I can just take the number 17 out of that system and plug in Julius Caesar for that role. Now I’ve got a new number system, and Julius Caesar happens to be the number 17. That’s totally fine.

The point of structuralism is that the question of whether Julius Caesar is a number or not is irrelevant to mathematics. It is irrelevant because it is not about structure; it’s about the essence of the mathematical objects. So that’s the structuralist criticism of Frege’s point.

You’ve 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. So what’s more real: the reality we see with our eyes, or the reality we can express in mathematical theorems?

I’m not quite sure. I live entirely in the Platonic realm, and I don’t really understand the physical universe at all, so I don’t have strong views.

Let’s talk about the Platonic realm. Because you live there, is it real?

Oh, yeah, totally, yeah. This is the realist position in mathematics: abstract objects have a real existence. What’s meant by that is that there’s some sense of existence in which those objects can be regarded as real.

How should we think about that? How should we try to visualize that? What does it mean to live among abstract objects?

Right.

Because life is finite, we’re all afraid of death. We fall in love with other physical manifestations of objects. You’re telling me that maybe reality actually exists elsewhere, and this is all just a projection from the abstract realm.

Do abstract objects exist in a place and at a time? That’s very debatable, I think.

Right. And what does place and time mean?

So what’s more real: physics or the mathematical Platonic space?

The mathematical Platonic realm is—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 convincing way. I don’t think we understand the nature of physical reality very well at all.

I think most people aren’t even scratching the surface of the question as I intend to be asking it. Obviously, we understand physical reality.

I knock on the table, and so on, and we know all about what it's like to have a birthday party or to drink a martini or whatever. So we have a deep understanding of existing in the physical world. But maybe understanding is the wrong word. We have an experience of living in the world—

Yeah, experience.

—and riding bicycles and all those things, but I don't think we actually have an understanding at all—very, very little of the nature of physical existence. I think it's a profound mystery. Whereas I think we have something a little better: an understanding of the nature of mathematical existence and abstract existence. 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.

Maybe I could hope that someone will give the convincing account, but it seems to be a profound mystery to me. I can't even imagine what it would be like to give an account of physical existence.

Yeah, I wonder, a thousand years from now, as physics progresses—

Right.

—what this same conversation would look like.

Right. That would be quite interesting.

Do you think there'll be breakthroughs a thousand years from now on the mathematics side? Because we've just discussed, and we'll return to, a lot of turmoil a century ago.

Right. It's interesting to me because I have my feet in two worlds, mathematics and philosophy, and to compare the differences between these subjects. There are many cultural differences, but one of the big cultural differences is toward the idea of progress in the subject.

Mathematics has huge progress. We simply understand mathematical ideas much, much better. We're continually improving our understanding, and there's growth in knowledge. We understand the nature of infinity now better than they did 100 years ago—definitely better. And they understood it better 100 years ago than they did for the previous thousands of years, and so on.

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 moves on to more difficult, interesting questions.

Whereas in philosophy, there's a little bit of progress. But meanwhile, it's also true that there are these eternal questions that have been with us for thousands of years, in fact, so much so that you can find a lot of philosophers arguing that the important contribution of philosophy is in asking the questions rather than answering them because it's hopeless to answer them. The nature of these deep philosophical questions is so difficult. Less of a sense of progress is what I'm trying to say.

I don't see any reason to think that progress in mathematics—the growth in our mathematical understanding and knowledge—won't simply continue. So, a thousand years from now, maybe the mathematics they will be doing will probably be completely unrecognizable to me. I might not even begin to understand what they're talking about, even without witnessing the intervening developments.

If you bring someone from ancient times to today, they might not even understand what we're talking about with some of the questions. But I feel that 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. Or anyone from that time, I mean. So I think it is possible to have this kind of progress even when the subject shifts away from the earlier concerns as a result of the progress, basically.

To take a tangent on a tangent, since you mentioned philosophy—maybe it's 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. You're ranked number 1 of all time on there, with currently over 246,000 reputation points. How do you approach answering difficult questions on there?

Well, MathOverflow has really been one of the great pleasures of my life. I've really enjoyed it, and I've learned so much from interacting on MathOverflow. I've been on there since 2009, which was shortly after it started. It wasn't exactly at the start, but a little bit later.

I think it gives you the stats for how many characters I typed. I don't know how many million it is, but this enormous amount of time that I've spent thinking about those questions has really just been amazing to me.

How do you find the questions that grab you, and how do you go about answering them?

I'm interested in any question that I find interesting. It's not all questions. Sometimes certain kinds of questions just don't appeal to me that much.

So you go outside of set theory as well?

I think when I first joined MathOverflow, I was basically one of the only, one of the few people in logic who was answering. There were other people who knew 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.

So I really found myself able to make a contribution in those very early days by engaging with the logic-related questions. But there weren't many people in logic asking questions either.

What I found was that there was an enormous amount of interest in topics that were logic-adjacent. So a question would arise in group theory, but it had a logic aspect, or in analysis or whatever, and there would be some logic angle on it. What I found was that I was often able to figure out an answer by learning enough about that other subject matter.

This is what was so rewarding for me, because I had to learn enough. My main expertise was logic, but someone would ask a question 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. 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.

I was quite happy to do that, and I also learned a lot by doing that because I had to learn about these other problem areas. So it really allowed me to grow enormously as a mathematician.

To give some examples of questions you've answered: What are some reasonable-sounding statements that are independent of ZFC? What are the most misleading alternate definitions in taught mathematics? Is the analysis as taught in universities, in fact, the analysis of definable numbers? Solutions to the continuum hypothesis? Most unintuitive application of the axiom of choice? Nontrivial theorems with trivial proofs? Reductio ad absurdum or the contrapositive? What is a chess piece mathematically?

We should say you worked quite a bit on infinite chess, which we should definitely talk about. It's awesome. You've worked on so many fascinating things. Has philosophy ever clarified mathematics? Why do we have two theorems when one implies the other?

And, of course, as an example, you've given a really great, almost historical answer on the topic of the continuum hypothesis. Maybe that's a good place to go. We've touched on it a little bit, but it would be nice to lay out what the continuum hypothesis is that Cantor struggled with. I would also love to speak to the psychology of his own life story, his own struggle with it. The human side of mathematics is also fascinating. So what is the continuum hypothesis?

The continuum hypothesis is the question that arises so naturally whenever you prove that there's more than one size of infinity. Cantor proved that the infinity of the real numbers is strictly larger than the infinity of the natural numbers. But immediately when you prove that, one wants to know: Is there anything in between? What could be a more natural question to ask immediately after that? And so Cantor did ask it, and he spent his whole life thinking about this question.

The continuum hypothesis is the assertion that there is no infinity in between the natural numbers and the real numbers. And, of course, Cantor knew many sets of real numbers. Everything in between—I mean, everything that's in that interval—would be equinumerous with some set of real numbers. But we know lots of sets of real numbers. There are all these various closed sets, Cantor sets, and so on. There are Vitali sets. We have all kinds of sets of real numbers.

So you might think, well, if the continuum hypothesis is false, then we've probably seen the set already. We just have to prove that it's strictly in between. But it turned out that for all the sets that anyone could ever define, pick out, or observe, it was always the case that either they were countable, in which case they're equinumerous with the natural numbers, or else they were fully equinumerous with the whole real line. So they were never strictly in between.

You're in this situation and you have hundreds or 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. And so in every situation where you're able to figure out whether it's in between or not, it's never strictly in between.

Now, Cantor was obsessed with this.

I think he was. I'm not a historian, so I don't know the exact history.

Well, from everything I've seen, it seems to be the question that broke him, huh? I mean, just struggling with different opinions on the hypothesis within himself and...

Desperately chasing, trying to prove it.

So he had a program for proving it, which has been affirmed in a certain respect. Of course, the Continuum Hypothesis holds for open sets. That's easy to see. If you have an open interval, then this is fully equinumerous with the whole real line. Any interval is equinumerous with the whole line because all you would need is a function, like the arctangent function, that maps the whole real line into an interval.

That's a one-to-one function, so we know the open sets have the property that their nontrivial open sets are all fully equinumerous with the whole real line. So they are never strictly in between. But remarkably, Cantor proved it also for the closed sets, using what's called the Cantor–Bendixson theorem. It's quite a remarkable result. It's definitely not obvious.

This theorem was actually the origin of the ordinals. Cantor had to invent the ordinals in order to make sense of his Cantor–Bendixson process.

Can you define the open and the closed set in this context?

Oh, yeah. Sure. 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. But that tiny little interval is already, just by itself, equinumerous with the whole line. So that's why that question is sort of easy for open sets.

A closed set is the complement of an open set, and there are a lot of closed sets that are really complicated, of varying sizes. Of course, any closed interval is a closed set, but it's not only those. There's also things like the Cantor set, which you get by omitting middle thirds. Maybe some people have seen this construction.

Or you can imagine randomly taking a lot of little tiny open intervals all over the line. That altogether would be an open set, and the complement of it would be a closed set. So you can imagine just tossing down these open intervals, and what's left over is the closed set.

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. 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. That would be a closed set also.

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 goes in between?

Right. 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.” Every closed set is either countable or equinumerous with the whole real line.

The Cantor program for solving the Continuum Hypothesis was a sort of working up. You did it for open sets and for closed sets, and you worked up. Maybe he wants to go into what are called the Borel sets, which are combinations of open and closed sets. There's a vast hierarchy of Borel complexity, and it turns out that the Continuum Hypothesis has been proved also for the Borel sets in this hierarchy.

But then one wants to go beyond. What about more complicated sets? So there's this hierarchy of complexity for sets of real numbers. Cantor's idea was to 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. That has been carried out to a remarkable degree.

It turns out that one begins to need large cardinal assumptions, though, in order to get to the higher realms, even at the level of the projective hierarchy. These are sets that you can define by using quantifiers over the real numbers themselves. So you get this hierarchy on top of the Borel hierarchy, the hierarchy of projectively definable sets.

It turns out that if you have enough large cardinals, then the projective sets are also always either countable or equinumerous with the whole real line. Then one can try to go beyond this and so on. I view all of those results that came in the past 50 years—the later ones—as fulfilling this Cantor idea that goes back 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.

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. Some of them transcend this hierarchy completely, in a way. So the program can't ever be fully successful, especially in light of the independence result.

Yeah. Well, spoiler alert, can you go to the independence result?

Sure.

So what does that mean? The Continuum Hypothesis was shown to be independent from the ZFC axioms of mathematics?

Right. The ZFC axioms were the axioms that were put forth first by Zermelo in 1908 in regard to his proof of the well-ordering theorem using the axiom of choice. That wasn't fully ZFC. At that time, it was just Zermelo theory because there was a kind of missing axiom: the replacement axiom. The foundation axiom was added later, and that's what makes the Zermelo–Fraenkel axiomatization, which became standard.

Actually, there's another aspect, which is that Zermelo's original theory allowed for the existence of ur-elements, or 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.

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 the ZFC axioms with ur-elements—ZFCU, it's called, or ZFA—then 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.

You don't need the atoms if you're a structuralist, because you only care about the structures up to isomorphism anyway. The theory is simply more elegant and clear without the atoms. They're just not needed. 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.

We formulate the ZFC axioms of set theory. These express the main principles and ideas that we have about the nature of sets and set existence. Cantor had asked about the Continuum Hypothesis in the late 19th century, and it remained open—totally open—until 1938.

We should mention—I apologize—that it was the number-one problem in Hilbert's set of 23 problems, formulated at the beginning of the century.

That's right.

Maybe you can comment on why he put that as number one.

Right. 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. That's how people talk about it now, although I'm not sure at all. Of course, I can't really speak for Hilbert, 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.

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. But at the time, Hilbert would have had no reason to think that would be true. He was just giving a lecture and had a list of problems that he thought were very important.

So I would find it more reasonable to think that he was just making a list of problems that he thought were extremely interesting, important, and fundamental, without the heavy burden of guiding 20th-century research. Although it turns out that, in fact, that's exactly what they did.

We already discussed Hilbert's views on the nature of set theory and its fundamental character—the 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 of the importance and fundamental nature of the Continuum Hypothesis for the foundations of mathematics, which was a critically important development for the unity of mathematics.

Before set theory emerged as a foundation of mathematics, there were different subjects in mathematics. There's algebra, and there's analysis—real analysis—and topology and geometry, and so on. There are all these disparate subjects with their own separate axioms.

Sometimes this happens, like when you're proving the fundamental theorem of algebra: that the complex numbers are an algebraically closed field in which you can solve any polynomial equation. But the proof methods for that theorem come from other parts of mathematics—those topological proofs and so on.

How does that work? 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. So the unity of mathematics was provided by the existence of a mathematical foundation like set theory.

At the time, it was set theory. 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 transfer and borrowing phenomenon that was definitely happening. That must have been part of Hilbert's thinking about why it was so important to have a uniform foundation, and set theory was playing that role at the time.

Now, of course, we have other possible foundations coming from category theory or type theory, and there's univalent foundations now.

So there are competing foundations now. There's no need to just use one foundation, one set-theoretic foundation. Although set theory continues to, in my view, have an extremely successful metamathematical analysis as a foundation—I think it's much more successful than any of those other foundations—it's much less amenable to things like computer proof and so on, which is part of the motivation to find these alternative foundations.

So, yeah, just to talk about Hilbert, I think he was motivated by the need for a unifying foundation of mathematics. 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.

There were a few other logic-related questions on the list, though. Hilbert's tenth problem is also related to logic. This is the question about Diophantine equations, and he asked for an algorithm to decide whether a given Diophantine equation has a solution in the integers.

A Diophantine equation is just—I mean, it's maybe a fancy way of talking about something that's easy to understand—a polynomial equation, except it's not just one variable; it has many variables. So you have polynomials in several variables over the integers, and you want to know: Can you solve it?

The problem, as stated by Hilbert, was to provide an algorithm for answering the question of whether a given polynomial equation has a solution in the integers. So he's presuming that there is an algorithm, but he wants to know what it is. What is the algorithm?

But the problem was solved by proving that there is no algorithm. It's an undecidable problem, like the halting problem. There is no computable procedure that will correctly decide whether a given polynomial equation has a solution in the integers. That's quite a remarkable development, I think.

And so eventually, the continuum hypothesis was shown to be independent of the ZFC axioms, as we've mentioned. How does that make you feel? What is independence, and what does that mean?

But once you tell the historical story—

Yes.

—the story is really quite dramatic.

Yeah, that's great.

I think so, because Cantor posed the question in the late 19th century, and then it was totally open. Hilbert asked about it at the turn of the 20th century, and nobody had any clue. There was no answer until 1938. This is 4 decades later, right? So it's a long time, and Gödel—Kurt Gödel—proved half of it.

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. What he's doing is showing what is called the constructible universe, Gödel's L. He solved this. This is the same result where he answers the consistency question of the axiom of choice, but also for the continuum hypothesis. If ZF, without the axiom of choice, is consistent, then so is ZFC plus the continuum hypothesis.

That was the result in 1938. It's really such a beautiful argument. It's incredible, I think, because he's building an alternative mathematical reality. That's the structure of the proof: 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. Maybe it's the same as the original one; it could be. If we started already in the one that he built, then it would be the same. But there's no reason to assume it was the same.

So he has this kind of model-construction method to build this alternative set-theoretic reality, the constructible universe. Then he proves that the axiom of choice is true there, and also that the continuum hypothesis is true there. It's just amazing. It's a really beautiful argument.

Okay, so then for the other part of the independence, that's only half of it, because Gödel shows basically that you can't refute the continuum hypothesis, but that's not the same thing as proving that it's true. He showed that if set theory is consistent without the continuum hypothesis, then it's consistent with the continuum hypothesis. That's not the same thing as proving that it's true.

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. So Cohen also gives us this extremely powerful tool for building alternative mathematical realities. That's how I think about it. He's explained to us how to take any set-theoretic world and build another, different one in which the continuum hypothesis is false: the forcing extension.

It's just such a fascinating technique, this tool of forcing. Maybe I'm anthropomorphizing it, but it seems like a way to escape one mathematical universe into another, or to expand it or alter it. So you travel between mathematical universes. Can you explain the technique of forcing?

Yeah, exactly. It's all those things. It's so wonderful. That's exactly how I think about it.

And we should mention, maybe this is a good place to give a bigger picture. One of your more controversial ideas in mathematics, as laid out in the paper “The Set-Theoretic Multiverse,” is that there may not be one true mathematics, but rather multiple mathematical universes. Forcing is one of the techniques that gets you from one to the other. Can you explain the whole shebang? The whole—

Yeah, sure. Let's get into it.

The lesson of Cohen's result and Gödel's result, and so on, is that they produce these alternative set-theoretic universes. 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. We have thousands of independence results.

Practically every nontrivial statement of infinite combinatorics is independent of ZFC. This is the fact. It's not universally true. There are some extremely difficult, prominent results where people proved things in ZFC, but for the most part, if you ask a nontrivial question about infinite cardinalities, then it's very likely to be independent of ZFC. We have these thousands of arguments, these forcing arguments that are used to establish that.

How should we take that? On the one hand, if you have a theory and it doesn't answer any of the questions that you're interested in, what does that mean? 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. We need a better theory because the current theory isn't answering the questions. Everything's independent.”

That seems like a quite reasonable thing to think if you believe that every set-theoretic question has a definite answer and there's a unique set-theoretic truth, or a unique fact of the matter. That's the universe view.

And by the way, to reiterate, independent means it cannot be proved or disproved within this axiomatic system, within this theory.

Right, exactly. To be independent means you can't prove it, and you also can't prove that it's false. You can't refute it.

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 of ZFC.

But that's an interesting way to put it, I think, because it reminds me of this: When I was a graduate student at Berkeley, there was another graduate student who was working with a non-logic professor in C*-algebras or something like this. 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.

The attitude of this other professor was, “Oh, I guess I asked the wrong question.” But my attitude, and the attitude of all the set theorists, was that when you ask a question that turns out to be independent, then you asked exactly the right question. This is the one that is carving nature at its joints. You're adjudicating the nature of set-theoretic reality by finding these two realms. You find one of these dichotomies: There are the worlds where it's true and the worlds where it's false.

When you ask that question, that's to be celebrated. It means you asked exactly the right, interesting, fascinating question. So it's not a bleak thing that you can't prove it and you can't refute it, and that it's such a disaster. Rather, it means that you found this cleavage in reality, in mathematical reality, and it's good to know about those when they happen.

“Carving nature at its joints.” So what can you do about the things that are shown to be independent from ZFC?

Right. One thing is that because of the incompleteness theorem, we know that for any theory that we can write down, there are going to be things that we can't prove—true things that we can't prove in it. Those things are going to be independent.

We're already aware of the fact that there will always be these independent phenomena for any theory that we write. Furthermore, some of those theories we won't even be able to prove are consistent, such as the consistency of the theory itself. So that's called the consistency-strength hierarchy.

It's a direct consequence of Gödel's second incompleteness theorem that for any theory we can write down, towering over it is this incredibly tall tower of consistency strength. The stronger theories aren't just adding another axiom; they're adding another axiom whose consistency was not provable in the previous layers of the hierarchy.

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. It exactly fulfills the prediction that Gödel's theorem makes about that kind of thing. Except the axioms in the large cardinal hierarchy aren't metamathematical, self-referential statements of the form that sometimes arise in the Gödel analysis, but rather profess the existence of big infinities—these large cardinal axioms. 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.

None of the large cardinal axioms we can prove can settle the continuum hypothesis. So the independence phenomenon is still there for things like the continuum hypothesis and the cardinal combinatorics that I mentioned.

So you're building this incredible hierarchy of axiomatic systems that are more powerful than ZFC.

More powerful than ZFC, and then more powerful than that, more powerful than that, and so on. It keeps going forever, and it will never be finished.

And still, to this day, the continuum hypothesis does not...

It's not settled by any of the large cardinal axioms.

Wow. Wow. What does that mean? How does that make you feel? Will it ever be settled?

Yeah, well, it's part of my multiverse view. We started by describing the universe view, which is the view that there are facts of the matter about all of these questions and that 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. What we should be aiming to do is figure out this one true set theory.

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. 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. 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.

You can sort of turn it on and off like a light switch. The fundamental nature of the continuum hypothesis is that you can have it or you can have the negation as you like within a very closely related set-theoretic world. 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.

That itself is a kind of answer. It's not a singularist answer, a universe-view answer. It's a pluralist answer. 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.

So, what does the multiverse view of mathematics enable you to do? What does it empower you to do, and what are the limitations? What are the things it breaks about mathematics as a field, as a space of knowledge, and what does it enable?

First of all, 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, don't ever disagree about the mathematics. We're all agreeing on what the theorems are. It's a question of philosophical perspective on the underlying meaning or the context, or really, what is a philosophy of mathematics for?

If you look back in history, for example, to the time of calculus with Newton and Leibniz, they famously developed the ideas of calculus using their concepts of infinitesimals. Those foundations were roundly mocked by Bishop Berkeley, who talked about “these same evanescent increments,” and asked, “Shall we not call them the ghosts of departed quantities?” But the foundations really were completely suspect, I think, at the time.

That foundation of infinitesimal calculus really only became rigorous in the 1950s or so with the development of nonstandard analysis and Robinson's work. The point I'm trying to make is this: Do you need a robust, rigorous foundation of mathematics to make enduring insights in mathematics? 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 had all the main insights in those early days with that extremely bad foundation.

That shows you something about the relevance of foundational views on mathematics, and how important they are for mathematical developments, progress, and insight. I view those early mathematical developments in calculus as genuinely mathematical and extremely important and insightful, even though the foundations weren't any good from contemporary perspectives.

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. Rather, it tells us, “Where should set theory go? What kind of set theory should we be looking at? What kind of questions should we be asking?”

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. 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. He's really striving—

Who was also your advisor.

He was also my supervisor, my graduate supervisor.

Which is a personal story as well.

This fundamental dispute on this question—he has a very strong and successful research program, sort of trying to give legs to finding the nature of the one true set-theoretic universe. It's driving the questions that he's asking and the mathematical programs that he's pursuing.

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 how the models of set theory are related to their forcing extensions, and so on.

This led to things that I call set-theoretic potentialism, where you think about a set-theoretic universe in a potentialist way. 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. The universe could be wider and taller, and so on, by forcing or by extending upward.

We want to understand the nature of this realm of set-theoretic universes, and that's quite exciting work. Benedikt Löwe and I proved some theorems on the modal logic of forcing and set-theoretic potentialism under end-extension. I've done a bunch of work on this topic.

Also, together with Günter Fuchs and Jonas Reitz, who was one of my own PhD students, I developed the topic of set-theoretic geology, which studies the idea of taking the metaphor of forcing. In forcing, you have the ground model and the forcing extension. When I was first working with Jonas, he said, “I want to undo forcing. I want to go backwards.”

At first I said, “But Jonas, it doesn't work that way. You start in the model, in the ground model, and you go out to the bigger one. That's how forcing works.”

He said, “No, no, I want to go backwards.” He was quite persistent, actually. Finally, I said, “Okay, let's do it. Let's take it seriously.” We sat down and started thinking more precisely, carefully, and deeply about the nature of taking a set-theoretic universe and seeing where it came from by forcing, which was a new way of thinking about forcing at the time.

Like reverse engineering the forcing?

Yeah, something like that. Forcing is a way of producing a new universe. 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.” So we defined the notions of the bedrock model and ground, sort of undoing the forcing. It was quite fruitful.

I view this as part of the pluralist perspective, except the difference is that set-theoretic geology is amenable to the universe view. Even though the work was inspired by this philosophical view of the multiverse, nevertheless, the central ideas of geology have now been picked up by the people with the research program in the universe view.

It turns out that set-theoretic geology is helping them, or us, to discover how the one true universe relates to its mantle. There's this concept of the set-theoretic mantle that I had introduced, which is extremely interesting.

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-view research program in a way that is quite important.

Can you prove something in the world that you arrived at through forcing and then take some of that back to the ground model?

Yeah, absolutely. That’s a really powerful argument method, actually. People often want to do that. Suppose you’re in some set-theoretic context. You could think about it as living in a set-theoretic universe, and you want to prove something in that universe only. But maybe one way to do it is to first construct this forcing extension and then use the features of this forcing extension to realize that certain things must have already been true in the ground model. And then you throw the forcing extensions away and you…

Oh, cool.

Yeah, so this can happen. 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, they would have these algebraic equations that they were trying to solve. They would have the tools and methods for doing it, and in the course of it, they would have to do things to the polynomial, change the factors, produce other polynomials, solve them, and so on.

Sometimes they could produce solutions. In the middle of their construction, they were led to the square root of minus 5 or something. They didn’t have any meaning for that, but they would just do it symbolically. Eventually, because of the methods that they had, the terms would combine and cancel, and all the imaginary parts would cancel out. They’d end up with an actual answer, 3 plus the square root of 17 or whatever, and they could check it, and it worked. It was a solution of the original equation.

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 proceed through the land of nonsense with these square roots of negative numbers, then end up with an answer that was real again and that they could verify was correct. I view this kind of forcing argument that I was just describing in a similar way. You start in set theory, and you go to this land of nonsense in the forcing extension, this imaginary world. You argue, and you come back. You make a consequence in the ground model, and it’s such a beautiful way of arguing.

So, speaking of the land of nonsense, I have to ask you about surreal numbers, but first, I need another bathroom break.

All right, we’re back, and there’s this aforementioned wonderful blog post on the surreal numbers. There’s quite a simple surreal number generation process that can basically construct all numbers. So maybe this is a good spot to ask: What are surreal numbers, and what is the way we can generate all numbers?

The surreal number system is an amazingly beautiful mathematical system that was introduced by John Conway.

Rest in peace, one of the greatest mathematicians ever on this earth.

Yes, absolutely. I really admire his style of mathematical thinking and working in mathematics, and the surreal number system is a good instance of this. The way I think about the surreal number system is that it provides us with a number system that unifies all the other number systems.

It extends not only the natural numbers, the integers, the rational numbers, and the real numbers, but also the ordinals and the infinitesimals. They’re all sitting there inside the surreal numbers. It’s this colossal system of numbers. It’s not even a set; it’s a proper class, it turns out, because it contains all the ordinal numbers.

But it’s generated from nothing by a single rule. We’re going to generate the numbers in stages, in a transfinite sequence of stages. At every stage, we take the numbers that we have so far and, in all possible ways, divide them into 2 sets: a lower set and an upper set, or a left set and a right set.

We divide them into these 2 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. That’s it. That’s all we do.

Let me say it again. The rule is that we proceed in stages, and at any stage, in all possible ways, we divide the numbers we have into 2 collections: the left set and the right set. Everything in the left set has to be less than everything in the right set, and we create a new number, a new surreal number, that will fit in that gap.

For example, we could start at the beginning. We don’t have any numbers. We haven’t created anything yet, so we could take nothing and divide it into 2 sets: the empty lower set and the empty upper set. Everything in the empty set is less than everything in the empty set because that’s a vacuous statement. We satisfy the conditions, and we apply the number-generation rule, which says we should create a new number.

This is what I call the Big Bang of numbers, the surreal genesis, when the number 0 is born. 0 is the firstborn number that is bigger than everything in the empty set and less than everything in the empty set.

Now we have this number 0, and therefore we can define new gaps. If we put 0 into the left set and have an empty right set, then we should create a new number that’s bigger than 0 and less than everything in the empty set. That number is called 1.

Similarly, at that same stage, we could have put 0 into the right set, so that would be the firstborn number that’s less than 0, which is called −1. Now we have 3 numbers: −1, 0, and 1. They have 4 gaps, because there could be a number below −1, between −1 and 0, between 0 and 1, or above 1.

So we create those 4 new numbers. The first number above 1 is called 2. The first number between 0 and 1 is called 1/2. On the negative side, we have −1/2 and −2, and so on. Now we have 7 numbers, so there are 8 gaps between them.

At the next birthday—the next stage, as they call it—all the numbers between those gaps, and then between those, and between those, and so on, will be born. As the days progress, we get more and more numbers, but those are just the finite birthdays, because, as I said, it’s a transfinite process.

At day omega, the first infinite day, we’re going to create a lot of new surreal numbers. 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. It’s not all the rationals, because the rational numbers that are born at the finite stages are just the rationals whose denominator is a power of 2, it turns out. Those are called the dyadic rationals.

The real numbers are all born on day omega, but some other numbers are born on day omega as well. 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.

We also have the number epsilon, which is the firstborn number that’s strictly bigger than 0 and strictly less than all the positive rational numbers. So that’s going to be an infinitesimal number in that gap, and so on. On day omega plus 1, we get more numbers, then omega plus 2, and so on. The numbers just keep coming forever.

This is how you build the surreal number system. It turns out you can define the arithmetic operations of addition and multiplication in a natural way that engages with this recursive definition. So we have recursive definitions of plus and times for the surreal numbers.

It turns out you can prove that they make the surreal numbers into what’s called an ordered field. They satisfy the field axioms, which means that you have distributivity and commutativity of addition and multiplication. You also have reciprocals for every nonzero number, so you can divide by the number. You can add, multiply, divide, and subtract.

Furthermore, you can take square roots. 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, it’s going to cross the axis. It has opposite behaviors at the 2 infinities: On the positive side, it’s going to positive infinity, and on the negative side, it’s going to minus infinity. So it has to cross.

We know that in the real numbers, every odd-degree polynomial has a root, and that’s also true in the surreal numbers. So that makes it what’s called a real closed field, which is a very nice mathematical theory. It’s really quite interesting how we can find copies of all these other number systems inside the surreal numbers.

But the surreal numbers are fundamentally discontinuous, as you were worried about. What are the consequences of this?

The surreal numbers have a property that they form a nonstandard 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 nonstandard theory that I had mentioned earlier.

But they don’t have the least-upper-bound property for subcollections. There’s no nontrivial set of surreal numbers that has a least upper bound, and there are no convergent sequences in the surreal numbers. So for ordinary use in calculus based on limits and convergence, that method does not work in the surreal numbers at all.

So that’s what I mean when I say the surreal numbers are fundamentally discontinuous. They have a fundamental discontinuity going on. But you can still do calculus with them, because you have infinitesimals if you use these nonstandard methods—the infinitesimal-based methods of calculus. And people do that.

I once organized a conference in New York, and we had John Conway as a speaker at that conference. During the question session, someone asked him—I think it’s a bit rude, but they asked it anyway—“What is your greatest disappointment in life?” I would never ask a question like that at a conference in a very public setting.

But Conway was extremely graceful, and he answered by saying, “The surreal numbers...” Not the numbers themselves, but the reception of the surreal numbers, because he had the ambition that the surreal numbers would become a fundamental number system used throughout mathematics and science. It was able to do nonstandard analysis, it was able to do calculus, it unified the ordinals, and so on. It’s such a unifying, amazing structure—a beautiful structure with elegant proofs and sophisticated ideas all around it.

He was disappointed that it never really achieved the unifying status that he had hoped for. He mentioned this as his greatest disappointment.

Yeah, Donald Knuth tried to celebrate it. It never quite took hold.

Gregory Chaitin

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 studying them. 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. In the Conway style, everything is a game. He treated the surreal numbers as a kind of plaything, a toy, and maybe that makes people not take it seriously.

Although my view is that it is extremely serious, useful, and profound. I’ve been writing a whole series of essays on the surreal numbers for my Substack, Infinitely More, and I just find the whole subject so fascinating and beautiful. It’s true, I’m not applying it in engineering, which maybe was part of this Conway ambition.

I wanted to mention, before I forget, Conway’s turning everything into a game. It’s a fascinating point that I didn’t quite think about. I think the Game of Life is just an example of the exploration of cellular automata.

I think cellular automata are one of the most incredible, complicated, fascinating things. It feels like an open door into a world we have not quite yet explored. The Game of Life is such a beautiful illustration of that world. Calling it a game—maybe “life” balances it, because that’s your powerful word—but it’s not quite a game. It’s a fascinating invitation to an incredibly complicated and fascinating mathematical world.

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. Speaking of a thousand years from now, it feels like that is a world we might make some progress on.

Gregory Chaitin

The Game of Life is a sort of playground for computably undecidable questions. In fact, you can prove that the question of whether a given cell will ever become alive is computably undecidable. In other words, given a configuration, you can ask, “Will this particular cell ever be alive in the evolution?” You can prove that this question is equivalent to the Halting Problem. It’s computably undecidable.

It’s semidecidable 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. 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.

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.

Nevertheless, when you sit back and look at 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.

Maybe that speaks to a thing I wanted to ask about the Halting Problem and decidability. You’ve mentioned this thing where, if you understand the program deeply, you might be able to say something. Can we say something interesting, maybe statistically, about how many programs we know something about in terms of whether they halt or not? What does it mean to understand a program deeply enough to be able to make a prediction?

Gregory Chaitin

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. The content of what you learn from a program, in the most general case, is always obtained just by running it and looking at the behavior. The proof of that is a theorem called Rice’s Theorem, which makes that idea completely robust.

But I want to take a little detour toward another question, riffing on something that you just said. Namely, one can ask the question: What is the behavior of a random program? You have some formal computing language, and you want to look at the collection of all programs of a certain size. Maybe there are only finitely many. Can you say something about the behavior of a randomly chosen one? With a certain likelihood, will it have a certain behavior?

The answer turns out to be extremely interesting. Years ago, Alexei Miasnikov asked me a question. He had this concept of a decision problem with a black hole, and what that means is that it’s a decision problem that is possibly difficult in the worst case, but the difficulty is concentrated in a very tiny region called the black hole.

For example, this kind of problem is a terrible problem to use if you’re basing your encryption scheme on it. You don’t want to use a black-hole problem because if someone can rob the bank 95% of the time, that’s not what you want. Even any nontrivial percentage of the time is too dangerous. You don’t want to use problems where almost every case is easily solved as the basis of your encryption.

The question Alexei asked me was, “Does the Halting Problem have a black hole?” If we take, say, the standard model of Turing machines—it’s a one-way infinite tape with zeros and ones on it, 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.

What that means is that there’s a computer procedure that decides correctly almost every instance of the Halting Problem. Even though the Halting Problem is not decidable, we can decide almost every instance. More precisely, there’s a collection of Turing-machine programs such that we can easily decide whether a program is in that collection or not. For the programs in the collection, we can easily decide the Halting Problem for those programs.

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%. The asymptotic density of the programs is 1.

The proof was quite fascinating because it’s one of these situations where the theorem sounds really surprising to many people—at least to computability experts—when I first tell it to them. It’s intriguing to think that you can solve almost every instance of the Halting Problem. But then, when they hear the proof, it’s a complete letdown. Unfortunately, nobody likes the theorem after hearing the proof.

The proof is so simple, though. 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. 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, change to this new state, and either move left or right as specified.

So a program consists of instructions like that. If you look at a program, one of the states is the halt state, and that’s when the program halts. But you can calculate how many programs don’t have any instruction that transitions to the halt state. You can easily calculate the proportion. In the limit, it goes to 1/e², or 13.5%.

If you calculate the limit, the proportion of programs with n 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. They don’t have any instruction that says “halt.”

So 13% of programs, you could say—

Gregory Chaitin

13%, you can say they don’t halt, because you just look at them and you can understand them.

There’s no halt state.

Gregory Chaitin

There’s no halt state. They never change to the halt state, so they can’t halt.

I mean, that nevertheless is beautiful to know.

Gregory Chaitin

So that’s a kind of trivial reason for nonhalting. When I first made that observation, I thought, “Okay, this is the proof strategy.” The goal at first was: Look, that’s a stupid reason for a program not to halt, 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.”

That was brilliant.

Gregory Chaitin

Yeah. That was my goal.

I love this.

Gregory Chaitin

Yeah. So we thought more about it, though, and we hit the jackpot because we found one gigantic stupid reason that converged to 100%—I mean, in the limit. The stupid reason for a program not to halt is that, well, if you think about the behavior, the head is sitting there. It's on the leftmost cell of the tape at the very beginning. It's in the start state, and the head is following an instruction.

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. If you move left and you are already at the end, then the head falls off, and the computation stops because the head fell off the tape.

That's a pretty stupid reason. Okay, but that's half of them already, just like that. Some of them went right, and they changed to a new state. Amongst those, the new state—half of those are going left and half are going right from that place, and then most of those are changing to a new state. When there's a lot of states, it's very likely that the next state that you transition to is new.

So you get this random-walk behavior, if you know what that means, where half go left and half go right at each step. There's a theorem due to Pólya called Pólya's recurrence theorem, which says that when you have a one-dimensional random walk, then it's very likely to come back to where you started.

When that happens for us, half of them from that place fall off on the next step. You can show using this kind of analysis that, with probability 1, the behavior of a random Turing machine is that the head falls off the tape before it repeats a state.

That is the stupid proof that shows how to solve the halting problem. When that happens, we can answer the halting problem by 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, if you want to define crashing as halting, then— In any case, however you set up your formalism, you're going to be able to answer the question about the behavior of the machine when the head falls off.

So statistically, in the limit, you solve the halting problem.

Gregory Chaitin

Yes, exactly. Computably solve it.

What do we take from that? Because you didn't solve the halting problem.

Gregory Chaitin

No, it's impossible to fully solve the halting problem correctly in all cases.

That's pretty cool.

Gregory Chaitin

It's a probabilistic way—I mean, it's probabilistic in the sense that we're solving almost all instances computably. There are versions of this that are maybe more interesting from the point of view of complexity theory and actually useful.

There's the whole P versus NP problem and so on. There's this genre of NP-complete problems, which are problems that are infeasible. They would take exponential time to solve in the ordinary way, and they're not known to be solvable in polynomial time. Although, in these cases, it's an open question whether there is a polynomial-time algorithm—a feasible algorithm.

For most of the NP-complete problems, you can prove that there's a polynomial-time approximation that solves almost all instances in a feasible amount of time. The knapsack problem, packing problems, and other kinds of problems, satisfiability problems—depending on how you set up the formalism, you can prove this.

I've proven many instances of this, but I also think it's widespread for almost all the NP-complete problems, the difficult problems. These are important problems for industrial applications, and they're problems that we actually want to solve. We can have feasible algorithms that solve almost every instance of them.

The amount of fields and topics you've worked on is truly incredible. I have to ask about P versus NP. This is one of the big open problems in complexity theory. For people who don't know, it's about the relation between computation time and problem complexity.

Do you think it will ever be solved? And is there any chance the weird, counterintuitive thing might be true—that P equals NP?

Gregory Chaitin

Yeah, that's an interesting question. Sometimes people ask whether it could be independent, which I think is an interesting question for logicians. Of course, one has to say: If you're entertaining the idea of independence, over which theory? Every statement is going to be independent over an extremely weak theory. So it doesn't make sense to say it's independent all by itself. It's only independent relative to a theory.

The way I think about P versus NP is that, of course, it's a theoretical question about the asymptotic behavior of these problems. For a problem to be in P means that there is a computable decision procedure that runs in time bounded by some polynomial. But the coefficients on that polynomial could be enormous, and the degree could be incredibly high.

For small values of the inputs, it doesn't make sense to talk about 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. It's an asymptotic property. Polynomial time or NP becomes relevant only in the limit, as the size of the inputs goes to infinity.

Maybe it's important to keep that in mind when you find overblown remarks about, “If P equals NP, then this will be incredibly important for human civilization,” because it would mean that we have feasible algorithms for solving these incredibly important problems in NP. People say that it would cause immense wealth for human societies 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.

You have to temper those remarks with the realization that P equals NP or P does not equal NP are not about these practical things at all, because of the asymptotic nature of the question itself.

On the other hand, we already have the algorithm, so we could use it already, except it's a terrible algorithm because it involves all this incredible amount of coding and so on.

And on the third hand, like you said, we already have approximation algorithms that, from a pragmatic perspective, solve all the actual real engineering problems of human civilization.

Gregory Chaitin

Yes.

Like the SAT solvers work amazingly well in lots and lots of cases, even though we can prove we don't expect—if P is not equal to NP, then there won't be a polynomial-time SAT solver. But the SAT solver approximations are really quite amazing.

Sorry to ask the ridiculous question, but who is the greatest mathematician of all time? Who are the possible candidates? Euler, Gauss, Newton, Ramanujan, Hilbert. We mentioned Gödel and Turing, if you throw him into the bucket.

Gregory Chaitin

This is an incredibly difficult question to answer. Personally, I don't really think this way about ranking mathematicians by greatness.

So you don't have, like—you know, some people have a Taylor Swift poster in their dorm room. You don't have one?

Gregory Chaitin

If you forced me to pick someone, it would probably be Archimedes, because he had such incredible achievements in such an early era, which totally transcended the work of the other people in his era.

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. This isn't always just coming from the greats. Sometimes the greats are doing things that are just first, and somebody else could have easily been first.

So there's a kind of luck aspect to it when you go back and look at the achievements. Because of this progress issue in mathematics that we talked about earlier—namely, we really do understand things much better now than we used to—when you look back at the achievements that had been made, maybe you can imagine thinking, “Well, somebody else could have had that insight also.” And maybe they would have.

It's already a known phenomenon that disparate mathematicians end up proving essentially similar results at approximately the same time. But the person who did it first is getting the credit, and so on.

What do you make of that? I see that sometimes when mathematicians—and this also applies in physics and science—completely separately make discoveries at a very similar time. What does that mean?

Gregory Chaitin

It's relatively common. I think certain ideas are in the air and being thought about but not fully articulated, and so this is the nature of growth in knowledge.

Do you understand where ideas come from?

Gregory Chaitin

Not really.

What's your own process when you're thinking through a problem?

Gregory Chaitin

Yeah, that's another difficult question. I suppose it has to do with— My mathematical style, my style as a mathematician, is that I don't really like difficult mathematics. What I love is simple, clear, easy-to-understand arguments that prove a surprising result. That's my favorite situation.

Actually, the question of whether it's a new result or not is somehow less important to me. That has to do with this question of the greats and so on, whoever does it first. For example, if you prove a new result with a bad argument or a complicated argument, that's great because you proved something new.

But I still want to see the beautiful, simple argument, because that's what I can understand. I'm naturally skeptical about any complicated argument because it might be wrong. If I can't really understand it fully—every single step all at once in my head—then I'm just worried that maybe it's wrong.

These different styles sometimes lead mathematicians to get involved with enormous research projects that involve huge numbers of working parts and different technology coming together. I mean mathematical technology, not physical technology. Sometimes it actually involves, more and more, something like the Lean programming language, where some parts are automated, so you have this gigantic—

Yeah, yeah, I see. Well, that's another issue, because maybe those things are less subject to skepticism when they're validated by Lean. But I'm thinking about the case where the arguments are just extremely complicated, and so I worry whether they're right or not, whereas I like the simple thing.

Gregory Chaitin

And so I have often tended to work on things that are a little bit off the beaten path from what other people are working on, from that point of view.

Your curiosity draws you toward simplicity.

Gregory Chaitin

Yeah. I want to work on the things that I can understand. Luckily, I've found that I've been able to make contributions that other people seem to like in this way, in this style. I've been fortunate from that point of view.

My process, though, and I've always recommended this to my students, is just a kind of playful curiosity. Whenever there's an idea or a topic, I play around with it: change little things, understand a basic case and then make it more complicated, press things a little bit on this side, or apply the idea to my favorite relevant example and see what happens.

You just play around with ideas, and this often leads to insights that then lead to more methods, and pretty soon you're making progress on the problem. This is basically my method: I fool around with the ideas until I can see a path through toward something interesting. Then I prove that, and that's worked extremely well for me. I'm pretty pleased with that method.

You do like thought experiments where you anthropomorphize, as you mentioned?

Yeah, yeah. This is a basic tool. I use this all the time. You imagine a set-theoretic model, a model of ZFC, as a place where you're living, and you might travel to distant lands by forcing. This is a kind of metaphor for what's going on.

Of course, the actual arguments aren't anything like that, because there's no land, you're not traveling, and you're not—

But you allow your mind to visualize that kind of thing in the natural, real world.

And it helps you to understand, particularly when there are parts of the argument that are in tension with one another. You can imagine that people are fighting or something. Those kinds of metaphors are helpful, or you imagine it in terms of a game-theoretic situation, with 2 players trying to win. So there's that kind of tension.

Those metaphorical ways of understanding a mathematical problem are often extremely helpful in realizing, “Aha, the enemy is going to pick this thing to be like that because it makes it more continuous or whatever.” 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.

What do you think of somebody like Andrew Wiles, who spent 7 years grinding at one of the hardest problems in the history of mathematics? Maybe contrast that a little bit with somebody who's also brilliant, Terence Tao, who basically says that if he hits a wall, he just switches to a different problem and comes back to it. 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. Maybe Grigori Perelman did the same.

I mean, Wiles proved an amazing theorem. The result on Fermat's Last Theorem is incredible. This is a totally different style from my own practice, though, of working in isolation.

For me, mathematics is often a kind of social activity. I've counted—it's pushing toward 100 collaborators and co-authors on various papers and so on. If anybody has an idea they want to talk about with me, and 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. You want to have a joint paper? Let me—

Yeah, exactly. Let's go.

My approach to making mathematical progress tends to involve working with other people quite a lot rather than just working on my own, and I enjoy that aspect very much. Personally, I couldn't ever do what Wiles did. Maybe I'm missing out. Maybe if I locked myself in the bedroom and just worked on whatever, then I would solve it.

But I tend to think that being on MathOverflow so much has given me so many ideas. So many papers have grown out of the MathOverflow conversations and back-and-forth. Someone posts a question and I post an answer to part of it, and then someone else has an idea and it turns into a full solution, and then we have a 3-way paper coming out of that. That's happened many times.

For me, I enjoy this social aspect to it. It's not just the social part. Rather, that's the nature of mathematical investigation as I see it: putting forth mathematical ideas to other people, and they respond to them in a way that helps me learn, helps them learn, and I think that's a very productive way of undertaking mathematics.

I think when you work solo on mathematics, from my outsider perspective, it seems terrifyingly lonely. Especially if you stick to a single problem—especially if that problem has broken many brilliant mathematicians in the past—you’re really putting all your chips in. And then there's the torment, the roller coaster of day-to-day work.

I imagine you have these moments of hopeful breakthroughs, and then you have to deal with the occasional realization that, no, it wasn't a breakthrough, and that disappointment. Then you have a weekly, maybe daily, disappointment where you hit a wall, and you have no other person to brainstorm with or any other avenue to pursue. I don't know—the mental fortitude it takes to go through that.

But everybody's different. Some people are reclusive and find solace in that lone grind. I have to ask about Grisha, Grigori Perelman. What do you think of him famously declining the Fields Medal and the Millennium Prize? He stated, “I'm not interested in money or fame. The prize is completely irrelevant to me. If the proof is correct, then no other recognition is needed.” What do you think of him turning down the prize?

I guess what I think is that mathematics is full of a lot of different kinds of people. My attitude is that, hey, it doesn't matter. Maybe they have a good math idea, and so I want to talk to them and interact with them.

I think the Perelman case is maybe an instance where he's such a brilliant mind, and he solved this extremely famous and difficult problem. That is a huge achievement. But he also had these views about prizes, and I don't really fully understand why he would turn it down.

I do think I have a similar reaction, just observing Olympic athletes who, in many cases, don't get paid very much, and nevertheless dedicate their entire lives to the pursuit of the gold medal. I think his case is a reminder that some of the greatest mathematicians, some of the greatest scientists, and some of the greatest human beings do what they do and take on these problems for the love of it, not for the prizes, the money, or any of that.

Now, as you're saying, if the money comes, you could use it for stuff. If the prizes come, and the fame, and so on, that might be useful. But fundamentally, the reason the greats do it is because of the art itself.

Sure, I totally agree with that. I share the view. That's why I'm a mathematician: because I find the questions so compelling, and I've spent my whole life thinking about these problems. But if I won an award—

Yeah, it's great. I'm pretty sure you don't contribute to MathOverflow for the wealth and power that you gain. I mean, it's genuine curiosity.

Well, you asked who the greatest mathematician is, and of course, if we want to be truly objective about it, we would need a kind of objective criteria.

Criteria, yeah.

About how to evaluate the relative strength and reputation of various mathematicians. So, of course, we should use the MathOverflow score.

I mean, nobody's objectively the greatest mathematician of all time.

Yes, that's true. I've also argued that tenure and promotion decisions should be based—

Based on MathOverflow.

Yeah. My daughter introduced me to her boyfriend and told me that she had a boyfriend. I wanted to know, first of all, what his chess rating was, and secondly, what his MathOverflow score was.

Oh, man. Well, that's the only way to judge a person, I think. That's objectively correct. Since you bring up chess, I've got to ask you about infinite chess. I can't let you go. You've worked on a million things, but infinite chess is one of them. Somebody asked on MathOverflow for the mathematical definition of chess.

Right.

So can we talk about the math of chess and the math of infinite chess? What is infinite chess?

Oh, yeah, absolutely. Infinite chess is fantastic. Chess ordinarily is played on this tiny, tiny board. It's an 8-by-8 board, right? So when you play chess, normally it's on the 8-by-8 board. But we want to play infinite chess on the integer chessboard. It's infinite in all 4 directions, but it still has the chessboard pattern, and maybe there are pieces on this board—maybe infinitely many pieces; we allow that.

But one difference from finite, ordinary chess is that in infinite chess, we don't play from a standard starting position. Rather, the interesting situation is that you present a position where there are already a lot of pieces on the board in a complicated way, and you say, “What would it be like to start from this position or from that one?” We want to produce positions that have interesting features—mathematically interesting features.

I can tell you, for example, that probably a lot of people are familiar with the mate-in-2 genre of chess problem. You have a chess problem, and it's White to mate in 2, which means that White is going to make 2 moves, but the second move is going to be checkmate. Or maybe mate in 3 or mate in 5 or whatever. We can have mate-in-N positions for any N.

In infinite chess, you can create a position that is not mate in N for any N, but White has a winning strategy that will win in finitely many moves. In other words, let me say it again: There are positions in infinite chess that White can definitely win. In finitely many moves, White is going to make checkmate. But there's no particular N for which White can guarantee to win in N moves.

There's no N?

No N. So it's not mate in N for any N, but it's a White win in finitely many moves. The way to think about it is that White is going to win, but Black controls how long it takes.

Ah, got it.

But it's doomed. Black can say, “Well, I know you're going to win, but this time it's going to take at least 1,000 moves.” Or maybe, in a different way of playing, Black can say, “Well, I know you're going to win, but this time you're going to have to take 1 million moves.” Black can say that for any number. So these are really interesting positions.

There's a position in my first infinite chess paper where it's Black to play. If Black doesn't move that rook there, then White is going to checkmate pretty quickly.

By the way, can we describe the rules of infinite chess?

Right. The rules of infinite chess are that there are just the ordinary pieces, and they move on this infinite board, which is just a chessboard extended infinitely in all directions, with no edge. So there's no boundary, but the pieces move just like you'd expect. The knights move the same way, and the rooks move on the ranks and files. The bishops move on the same-color diagonals, just like you would expect, except they can move as far as they want if there's no intervening piece in the way.

The one thing is that the White pawns always move upwards and the Black pawns always move downwards, but when they're capturing, the pawns capture on the diagonal. So I think the piece movement is pretty clear.

There are a couple of differences that you have to pay attention to from ordinary chess. For example, there's this threefold-repetition rule in ordinary chess, but we get rid of it for infinite chess because, of course, threefold repetition is just a proxy for infinite play. The real rule is that infinite play is a draw, not that threefold repetition is a draw. That's just a convenient approximation to what I view as the actual rule, which is that infinite play is a draw.

The only way to win is to make checkmate on the board at a finite stage of play. If you play infinitely, you haven't done that, and so it's a draw.

And the pawns can't be converted into—?

And there's no promotion because there's no edge.

Right, exactly.

This position that we were just talking about is a position with game value ω, which means that because it has an ordinal value, White is going to win, but Black can play as though counting down from ω.

What is the nature of counting down from ω? If you're Black and you need to count down from ω, then you have to say a finite number, and after that, it's going to be at most that many moves to count down. The nature of counting down from ω is that you take this giant step on the first count, and then after that, you subtract 1 each time. You can't subtract 1 from ω because that's not an ordinal. So if you count down from ω, you have to go to some finite number, and then if you just subtract 1 each time, that's how many more moves you get.

That's the sense in which Black can make it take as long as he wants, because he can pick his initial number to be whatever he wants.

By the way, I just noticed that you were citing a MathOverflow question, which is really cool.

That's right, yeah. My interest in infinite chess was born on MathOverflow because someone asked this question.

Noam Elkies asked this question. That's so cool to see a MathOverflow citation in an arXiv paper. How do you construct the position that satisfies this? Is there an algorithm for construction?

No. This is an act of mathematical creativity, really, to come up with it.

I had a co-author, Cory Evans. He's a U.S. National Master, a very strong chess player. He's also a philosophy professor of law.

Your collaborations are wonderful. That's great.

I met him because he was a graduate student at CUNY, where I was at the time in New York. He was also my son's chess coach when my son was playing chess competitively in elementary school. Cory was the coach, so we knew him that way.

That was right around the time when I was getting interested in infinite chess, and I knew I needed a chess-knowledgeable partner. Cory was invaluable for the paper because the proofs in infinite chess are extremely finicky. You create these positions, but the details of the argument have to do with chess reasoning. My chess reasoning wasn't quite up to it because I would create the positions—almost all the positions are ones that I made—but this was after many generations of being corrected by Cory.

Cory would come and say, “Hey, this pawn is hanging, and it breaks your argument,” or, “This bishop can leak out of the cage,” or whatever. The process was that I knew, in terms of these ordinals, what we needed to create with the position. I would struggle to do it and create something that sort of had the features that I wanted, and then I would show it to Cory, and he would say, “Look, it doesn't work because of this and that,” and so on.

This back and forth was extremely helpful to me, and eventually we converged on arguments that were correct. So, yeah, it's quite interesting.

Another thing to say is that the follow-up paper to this one was a three-way paper with Cory, myself, and my PhD student, Norman Perlmutter, in which we improved the bound. We were aiming to produce more and more chess positions with higher and higher ordinal values. The initial position was value ω, and then we made ω² and ω³ in the first paper. In this three-way collaboration, we made ω⁴.

The title of the paper: The Position in Infinite Chess with Game Value Omega to the 4th.

Right. At the time, this was the best-known result, the state of the art, but since then, it's been improved dramatically. In fact, we now know that every countable ordinal arises as the game value of a position in infinite chess, so it's a fantastic result.

Before I forget, let me ask about your views on AI and LLMs that are getting better and better at mathematics. We've spoken about collaborators, and you have so many collaborators. Do you see AI as a potential great collaborator for you as a mathematician, and what do you think the future role of those kinds of AI systems is?

I guess I would draw a distinction between what we have currently and what might come in future years. I've played around with it and tried experimenting, but I haven't found it helpful at all—basically zero.

I've used various systems, including the paid models. My typical experience interacting with AI on a mathematical question is that it gives me garbage answers that are not mathematically correct. I find that not helpful and also frustrating. If I were interacting with a person, the frustrating thing would be having to argue about whether the argument they gave you is right. You point out exactly the error, and the AI says, “Oh, it's totally fine.”

If I were having such an experience with a person, I would simply refuse to talk to that person again. But, okay, one has to overlook these kinds of flaws. I tend to be a skeptic about the value of the current AI systems as far as mathematical reasoning is concerned. It seems unreliable.

I know for a fact that there are several prominent mathematicians whom I have enormous respect for who are saying that they are using it in a way that's helpful. I'm often very surprised to hear that based on my own experience, which is quite the opposite. So maybe my process isn't any good, although I use it for other things, like programming and image generation.

It's amazingly powerful and helpful. But for mathematical arguments, I haven't found it helpful, and maybe I'm not interacting with it in the right way yet, or it could be that I just need to improve my skill. I also wonder whether these examples provided by other people involved a huge amount of interaction, and whether the mathematical ideas are really coming from the person—these great mathematicians who are doing it—rather than the AI. So I tend to be skeptical.

But I'm also skeptical for another reason, and that is because of the nature of the large language model approach to AI doing mathematics. I recognize that the AI is trying to give me an argument that sounds like a proof rather than an argument that is a proof. The motivation is misplaced. And so I worry that this is a very dangerous source of error.

It often happens in mathematics that, if I think back to when I was an undergraduate here at Caltech, and I eventually became a math major, LaTeX was a pretty new thing. I was learning LaTeX, so I was typing up my homework in LaTeX, and it looked beautiful. Actually, it looked like garbage by my current standards. I'm sure it was terrible, but at the time, I didn't know anything. I was an undergraduate, and LaTeX was sort of unheard of.

I was producing these beautifully typeset problem sets, solutions, and so on. I would print them and submit them, and the grades would come back—terrible grades. I realized what was happening: the copy was so beautifully typeset mathematically that it looked like the kind of mathematics you find in a book. Basically, the only time you saw that kind of mathematical typesetting was in a professional published book, and that mathematics was almost always correct in a book, right?

Because it was so beautiful, and I was used to seeing that kind of typesetting only when an argument was totally right, I wasn't critical enough and made these bonehead mistakes in the proofs. And so I corrected this, of course.

But this kind of effect is very much real with modern LLM systems.

That's right. I think the chat programs and so on are producing these arguments. That's what they're striving to do; that's what they're designed to do. They're not designed to make a logically correct argument. They're designed to make something that looks like a logically correct argument.

It's easy to get fooled if you're not skeptical, and so that's why I worry a bit when people rely on AI for mathematical arguments. Using them—tying them to Lean in formal proof-verification systems—is a totally different way of operating. But for the ordinary person sitting down and using chat to come up with a mathematical argument, I think it's a dangerous source of error if you're not especially attuned to this very issue: the AI is going to produce something that's not grounded in mathematical understanding, but rather something that's trying to look like something grounded in mathematical understanding. Those are not the same thing at all.

Furthermore, I really wonder if one can make a kind of system for producing genuine mathematical insight that isn't based in what I would view as mathematical understanding, as opposed to the text-generation systems. The methods that are used don't seem close enough to being grounded in an understanding of the underlying mathematical concepts, but rather grounded in the way words appear on a page in arguments about those concepts, which are not the same.

So there are a couple of things to say there. One, I think there is a real skill in providing the LLM system with enough information to be a good collaborator. You really are dealing with something different. It's not a human being. You really have to load in everything you possibly can from your body of work and from the way you're thinking, and that's a real skill.

For me, if it's anything like programming—because I have a lot of colleagues and friends who are programmers who feel similarly to you—I've gotten better and better at giving as much information as possible to the systems in a really structured way, maybe because I just like natural language as a way to express my thinking.

The benefit comes from the inspiration that the system can provide through its ability to know a lot of things and make connections between disparate fields and disparate concepts. In that way, it provides not the answer but the inspiration, the handholding, the camaraderie that helps me get to the answer, because it knows a lot more than I do.

If you give it a lot of information and ask the broader questions, it can make some really beautiful connections. But I do find that I have to be extremely patient, like you said. The number of times I'll do something dumb where I feel like, “You don't get this at all, do you?”—that's a source of a lot of frustration for us humans. It's like, “Wait, this thing doesn't understand at all.”

If you can have the patience to look past that, there might be some brilliant little insights that it can provide.

Right.

At least for me, in the realm of programming—I should say programming—there's just so much training data. There's so much there. And at least I see the light at the end of the tunnel of promising possibilities of it being a good collaborator, versus something that gives you really true genius-level insights.

Right. It's probably true.

As far as mathematical training data is concerned, I just have to assume that MathOverflow answers are part of the training data.

Yes, of course. You're talking to yourself, essentially.

Yeah, maybe. Sorry for the ridiculously big question, but what idea in mathematics is most beautiful to you? We've talked about so many.

The most beautiful idea in mathematics is the transfinite ordinals. This was the number system invented by Georg Cantor for counting beyond infinity—the idea of counting beyond infinity.

You count through the ordinary numbers, the natural numbers: 0, 1, 2, 3, and so on. And then you're not done, because after that comes omega, then omega plus 1, omega plus 2, and so on. You can always add 1. And so, of course, after you count through all those numbers of the form omega plus n, then you get to omega plus omega, the first number after all those. Then comes omega plus omega plus 1, and so on. You can always add 1.

You can just keep counting through the ordinals. It never ends. Eventually, you get to omega times 3, omega times 4, and so on. Then the limit of those numbers—the first number that comes after all those numbers—is omega squared.

This is the first compound limit ordinal because it's a limit of limit ordinals. A limit ordinal is one of these numbers, an ordinal that doesn't have an immediate predecessor, like omega, omega times 2, and omega times 3. Those are all limit ordinals. But omega squared is a limit ordinal, and it's also a limit of limit ordinals, because omega times 3, omega times 4, and so on, are all limit ordinals whose limit is omega squared.

Then, of course, you form omega squared plus 1, omega squared plus 2, and so on, and it never stops. It's absolutely beautiful and amazing. Furthermore, it forms the foundation for these transfinite recursive constructions that came later.

Starting with the Cantor-Bendixson theorem that I mentioned, and continuing with the construction of the V hierarchy, Gödel's constructible universe is built this way, and Zermelo's proof of the well-ordering principle using the axiom of choice is a transfinite recursive construction. The idea of just counting past infinity is so simple and elegant, and has led to so much fascinating mathematics.

Yeah, infinity's not the end. What, to you, is the most beautiful idea in philosophy?

I have a foot in both fields: philosophy and mathematics. In some contexts, I seem to be required to choose whether I'm a mathematician or a philosopher. My training is in mathematics. My PhD and all my degrees are in mathematics. But somehow I turned myself into a philosopher over the years, because my mathematical work was engaging with these philosophical issues.

When I went to New York, I had appointments first in mathematics only, but eventually I also joined the philosophy faculty at the Graduate Center. When I went to Oxford for the first time, my main appointment was in philosophy, and that's also true now at Notre Dame, although I'm also a concurrent professor in mathematics. I still have math PhD students and philosophy PhD students.

I don't really care to decide whether I'm a mathematician or a philosopher. My work engages with mathematics, philosophical issues in mathematics, and plain philosophy, and there's this ample region between these two subjects. So it's not necessary to choose.

I remember when I first went to Oxford and told my daughter that I was going to become a professor of philosophy in Oxford. She looked at me plaintively and said, “But, Papa, you're not a philosopher.” In her mind, her father was the mathematician and her mother was the philosopher, because my wife, Barbara, is a philosopher. She's now also at Notre Dame. We're together there.

Fortunately, I don't really have to choose between them.

So you ask about the most beautiful idea in philosophy, and I would have to say that I think it's the distinction between truth and proof, the one that we discussed already. It's so profound and gets at the heart of so many philosophical issues. Of course, this is a distinction that's maybe born in mathematics or mathematical logic, but that's already philosophical to a degree, and it's fundamentally a philosophical distinction.

The truth is about the nature of the world and the way things are. It's about objective reality, in a sense. Whereas proof is about our understanding of the world and about how we come to know the things that we know about the world. And so to focus on proof is to focus on the interaction that we have with objective reality.

I'm talking about the reality of mathematics, not the physical world, because, as I said, I live in the Platonic realm and I interact with mathematical reality. And so proof is about the interaction and how we come to know the facts that are true in this mathematical reality, whereas truth is about what's really the case, apart from our knowledge of it. And this is, I think, such a core way that I have of understanding the world and the nature of logic and reasoning.

And the gap between the two is full of fascinating mysteries, both in the Platonic realm, but also in the realm of physics. And I would even say in human psychology, sociology, politics, geopolitics—all of it, if you think about proof more generally, which is the process of discovery versus the truth itself. And that's our journey, whatever field we're in.

I, for one, am grateful for how marvelous a philosopher, mathematician, and human being you are. It's truly an honor to speak with you today.

Well, thank you so much. It's such a pleasure to be here, and thank you for inviting me.

Thanks for listening to this conversation with Joel David Hamkins. To support this podcast, please check out our sponsors in the description where you can also find links to contact me, ask questions, get feedback, and so on. Thank you for listening. As always, happy New Year. I love you all.