Infinity, Paradoxes, Gödel Incompleteness & the Mathematical Multiverse | 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 math overflow, which I think is a legendary accomplishment. Math overflow, by the way, is like Stack Overflow, but for research mathematicians. He is also the author of several books including proof and the art of mathematics and lectures on the philosophy of mathematics and he has a great blog infinitely more.xyz. This is a super technical and super fun conversation about the foundation of modern mathematics and some mindbending ideas about infinity, 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, soulsearching, as we all do every once in a while, but mostly just 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, for the many amazing conversations I've had with people across the world. I got a little bit of hate and a whole lot of love, and I wouldn't have it any other way. I'm grateful for all of it. This is the Lex Freedman 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's Joe David Hamkins.

Some infinities are bigger than others. This idea from Caner at the end of the 19th century I think it's fair to say broke mathematics before rebuilding it and I also read that this was a devastating and transformative discovery for several reasons. So one it created a theological crisis because infinity is associated with God. How could there be multiple infinities and also Caner was deeply religious himself. Second there was a kind of mathematical civil war. The leading German mathematician chronicer called Caner a corruptor 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 don't contain themselves and those threatened to make all of mathematics inconsistent. And finally on the psychological side, on the personal side, Caner's own breakdown. He literally went mad spending his final years in and out of sanatoriums obsessed with proving the continuum hypothesis.

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

Well, that's a really great question. I would want to start talking about infinity and telling the story much earlier than caner actually because I mean you can go all the way back to ancient Greek times when Aristotle emphasized the potential aspect of infinity as opposed to the impossibility according to him of achieving an actual infinity and Archimedes method of exhaustion where he is trying to understand the the area of a region by carving it into more and more triangles say and sort of exhausting the area and thereby understanding the total area in terms of the sum of the areas of the pieces that he put into it and it proceeded on this kind of potential under this potentialist understanding of infinity for for hundreds of years 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 sort of potentialist orthodoxy in the dialogue of tunu sciences. Really lovely account there that he gave. And that the in many ways Galileo was anticipating Kandra's developments except he couldn't quite push it all the way through and ended up throwing up his hands in confusion in a sense. I mean, the Galileo paradox is the idea or the observation that if you think about the natural numbers, I would start with zero, but I think maybe he would start with one. The numbers 1 2 3 4 and so on. And you think about which of those numbers are perfect squares. So 0 squar is zero and one square is one and two squar is 4, 3 squar is 9, 16, 25 and so on.

And Galileo observed that the the perfect squares can be put into a onetoone correspondence with all of the numbers. I mean we just did it. I associated every number with its square. And so it seems like on the basis of this one to one correspondence that there should be exactly the same number of squares, perfect squares as there are numbers. And yet there's all the gaps in between the perfect squares, right? And and this suggests that you know there should be fewer perfect squares, more numbers than squares because the numbers include all the squares plus a lot more in between them, right? And Galileo was quite troubled by this observation because he took it to cause a kind of incoherence in the comparison of infinite quantities.

Right? And another example is if you take two line segments of different lengths and you can imagine drawing a kind of foliation a fan of lines that connect them. So the end points are matched from the shorter to the longer segment and the midpoints are matched and so on. So spreading out the lines as you go and so every point on the shorter line would be associated with a a unique distinct point on the longer line in a onetoone way. And so it seems like the two line segments have the same number of points on them because of that even though the longer one is longer. And so it makes again a kind of confusion of our ideas about infinity. And also with two circles, if you just place them concentrically and draw the rays from the center, then every point on the smaller circle is associated with a corresponding point on the larger circle, you know, in a one to one way. and and again that seems to show that the smaller circle has the same number of points on it as the larger one precisely because they can be put into this one to one correspondence.

Now of course the contemporary attitude about this situation is that those two infinities are are exactly the same and that Galileo was right in those observations about the equinosity and the way we would talk about it now is appeal to what I call the caner Hume principle or some people just call it Hume's principle which is the idea that if you have two collections whether they're finite or infinite then we want to say that those two collections have the same size they're equinumerous if and only if there's a onetoone correspondence between those collections. And so Galileo was observing that line segments of different lengths are equinumerous and the perfect squares are equinumerous with the whole all of the natural numbers and and two any two circles are equinumerous and so on and that the tension between the kander Hume principle and what could be called Uklid's principle which is that the whole is always greater than the part which is a principle that Uklid appealed to in in the elements. I mean many times when he's calculating area and so on he wants it's a kind of basic idea that if something is just a part of another thing then the the whole is greater than the part and so what Galileo was troubled by was this tension between what we call the canerhume principle and Uklid's principle and it really wasn't fully resolved I think until Caner he's the one who really explained so clearly about these different sizes of infinity and so on in a way that was so compelling and so he exhibited two different infinite sets and proved that they're not equinumerous they can't be put into onetoone correspondence and it's traditional to talk about the uncountability of the real numbers so Kanto's big result was that the set of all real numbers is an uncountable set so maybe if we're going to talk about countable sets then I would suggest that We talk about Hilbert's hotel which really makes that idea perfectly clear.

Yeah, let's talk about the Hilbert's hotel. Hilbert's hotel is a hotel with infinitely many rooms. You know, each room is a full floor suite. So there's floor zero. I always start with zero because for me the natural numbers start with zero. Although that's maybe a point of contention for some mathematicians. The the other mathematicians are wrong. Like a bunch of geo programmers. So starting at zero is a wonderful place to start. Exactly. So there's floor 0, floor 1, floor two or room 0 1 2 3 and so on just like the natural numbers. So Hilbert's hotel has a room for every natural number and it's completely full. There's a person occupying room N for every N. But meanwhile, a new guest comes up to the desk and wants a room. Can I have a room, please? And the manager says, "Hang on a second. Just give me a moment." And you see, when the other guests had checked in, they had to sign an agreement with uh with the hotel that maybe there would be some changing of the rooms, you know, during the stay. And so the manager sent a message up to all the current occupants and told every person, "Hey, can you move up one room, please?" So the person in room five would move to room six and the person in room six would move to room seven and so on. And everyone moved at the same time. And of course, we never want to be placing two different guests in the same room and we want everyone to have their own private room and but when you move everyone up one room then the bottom room room zero becomes available of course and so he can put the new guest in that room.

So even when you have infinitely many things then the new guest can be accommodated and that's a way of showing how the particular infinity of the occupants of Hilbert's hotel it violates Uklid's principle. I mean, it exactly illustrates this idea because adding one more element to a set didn't make it larger because we can still have a 1:1 correspondence between the total new guests and the old guests by the room number. Right? So, to just uh say one more time, the hotel is full. the hotel is full and then you could still squeeze in one more and that breaks the uh traditional notion of mathematics and and breaks people's brains about when they try to uh 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. But one can go on with Hilbert for example.

I mean maybe the next day, you know, 20 people show up all at once, but we can easily do the same trick again. Just move everybody up 20 rooms and 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. And Hilbert's bus has, of course, infinitely many seats. There's seat zero, seat one, seat two, seat three, and so on. And so one wants to uh you know all the people on the bus want to check into the hotel but the hotel is completely full. And so what is the manager going to do? And when I talk about Hbert's hotel in when I teach Hilbert's hotel in in class I always demand that the students provide you know the explanation of of how to do it. So maybe I'll ask you can you tell me yeah what is your idea about how to fit them all in the hotel everyone on the bus and also the current occupants.

you uh you separate the hotel into even and odd rooms and you squeeze in the new Hilbert bus people into the odd rooms and uh the previous occupants go into the even rooms. That's exactly right. So I mean that's the 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 2 * n. So they're all going to get their own private room, the new room, and it will always be an even number because 2 * n is always an even number. And so all the odd rooms become empty that way. And now we can put the bus occupants into the oddnumbered rooms. And by doing so, you have now shoved in an infinity into another infinity. That's right. So what it really shows I mean another way of thinking about it is that well we can define that a set is countable if it is equinumerous with a set of natural numbers and and a kind of easy way to understand what that's saying in terms of Hilbert's hotel is that a set is countable if it fits into Hbert's hotel because Hilbert's hotel basically is the set of natural numbers in terms of the room numbers. So to be equinumerous with a set of natural numbers is just the same thing as to fit into Hilbert's hotel.

And so what we've shown is that if you have two 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, then 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 one kind of infinity, then it wouldn't be surprising at all because if you take two infinite sets and put them together, then it's still infinite. And so if there were only one kind of infinity, then it shouldn't be surprising that the union of two countable sets is countable. So there's another way to push this a bit harder. And that is when Hilbert's train arrives and Hilbert's train has infinitely many train cars. Mhm. And each train car has infinitely many seats. And so we have an infinity of infinities of the train passengers together with the current occupants of the hotel. And everybody on the train wants to check into Hilbert's hotel.

So the manager can again of course send a message up to all the rooms telling every person to double their room number again. And so that will occupy all the even-umbered rooms again and but free up again the oddnumbered rooms. So somehow we want to put the train passengers into the oddnumbered rooms. And so well every train passenger is on some car let's say car C and seat S. So somehow we have to take these two coordinates you know C comma S the car number and the seat number and produce from it an odd number in a one to one way you know and that's that's actually not very difficult. In fact one can just use say an easy way to do it is to just use the number 3 to the C * 5 to the S 3 to the C 3 to the car number. So 3 * 3 * 3 you know the number of the car you multiply three by itself the number of the train car and then you multiply five by itself the seat number times and then you multiply those two numbers together so 3 to the c * 5 to the s that's always an odd number because the prime factorization has only threes and fives in it there's no two there so therefore it's definitely an odd number and it's always different because of the uniqueness a prime factoriization.

So every number can be factored uniquely into prime. So if you have a number of that form, then you can just factor it and that tells you the exponent on three and the exponent on five. And so you know exactly which person it was, which car they came from and which seat they came from. And prime factorization is every single number can be uh decomposed into the atoms of mathematics which is the prime numbers. You can multiply them together to achieve that number. And that's prime factorization. You're showing three and five are both prime numbers odd. So through this magical formula, you can deal with this train infinite number of cars with each car having infinite number of seats. 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 one giant set is still countable.

Yeah, cuz the the train cars are each countable. Plus, the current hotel, it's sort of like another train car if you want to think about it that way. The current occupants of the hotel could, you know, have the same number as as any of the train cars. So putting countably many countable sets together to make one big union set is still countable. It's quite remarkable I think. I mean when I first learned this many 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 Uklid's principle once again right so the new set that we built is has many more elements than the old set in the sense that there's additional elements but it doesn't have many more elements in terms of its size because it's still just to countable infinity and it fits into Hbert's hotel.

have you been able to sort of internalize a good intuition about countable infinity? Cuz that is a pretty weird thing that you can have a countably infinite set of countably infinite sets. You can shove it all in and it still is countable infinite set. Yeah, that's that's exactly right. Right. I mean, I guess of course when you when you work with these notions that the the argument of of Hilbert Sortell becomes kind of clear. There's many many other ways to talk about it too. For example, let's think about say the the integer lattice the grid of points that you get by taking pairs of natural numbers say so the the upper right quadrant of the integer lattice. Yeah. So there's the, you know, row 0, row one, row two, and so on, column 0, column 1, column 2, and so on. And each each row and column has an countable infinity of points on it, right? So those dots, if you think about them as dots, are really the same as the train cars. If you think about each column of in the in that integer lattice, it's a countable infinity. It's like one train car and then there's the next train car next to it and then the next column next to that the next train car and so but if we think about it in this grid manner then I can imagine a kind of winding path winding through these grid points like up and down the diagonals winding back and forth.

So I start at the corner point and then I go down up and to the left and then down and to the right up and to the left down and to the right and so on in such a way that I'm going to hit every grid point in on this path. So this gives me a way of assigning room numbers to the points because every every grid point is going to be the nth point on that path for some n and that that gives a correspondence between the grid points and the natural numbers themselves. So it's a kind of different picture. I mean before we used this 3 to the C 5 * 5 to the S which is a kind of you know overly arithmetic way to think about it but there's a kind of direct you know way to understand that it's still a countable infinity when you have countably many countable sets because you can just start putting them on this list and as long as you give each of the infinite collections a chance to add one more person to the list then you're going to accommodate everyone in any of the sets in one list. 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 kind of an algorithm for including everybody.

So, can you speak to the uncountable infinities? So, what are the integers and the real numbers and what is the line that Caner was able to find? So, maybe there's there's one more step I want to insert before doing that which is the rational numbers. So, we did we did pairs of natural numbers, right? that that's the train car basically. But maybe it's a little bit informative to think about the rational the fractions the set of fractions or rational numbers because a lot of people maybe have an expectation that maybe this is a bigger infinity because the rational numbers are are densely ordered between any two fractions you can find another fraction. Right? The average of two fractions is another fraction. And so so sometimes people it seems to be a different character than um than the integers which are discreetly ordered right from every any integer there's a next one and a previous one and so on. But that's not true in the rational numbers and yet the rational numbers are also still only accountable infinity.

And the the way to see that is actually it's just exactly the same as Hilbert's train again because every fraction consists of two integers the numerator and the denominator and so if I tell you two natural numbers then you know what fraction I'm talking about I mean plus the sign issue I mean if it's positive or negative but if you just think about the positive fractions then you know you have the numbers of the form p over q where q is not zero. So you can still do 3 to the P * 5 to the Q. The same idea works with the rational numbers. So this is still a countable set. And you might think, well, every every set is going to be countable because there's only one infinity. I mean, if that's a kind of perspective maybe that you're adopting, but it's not true. And that's the profound achievement that Caner made is proving that the set of real numbers is not a countable infinity. It's a strictly larger infinity and therefore there are 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 infinity? The countable infinity right? Looking it up on perplexity. Uh 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 two integers. That's right. So 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 two or the cube root of five and so on. Numbers that solve an algebraic equation over the integers. Those are known as algebraic numbers. It was an open question for a long time whether that was all of the um 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.

Oh great. So it was Louisville who first proved that there are transcendental numbers and he exhibited a very specific number that's now known as the Louisville constant which is a transcendental number. Caner 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 that are not the root of any nonzero polomial 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. That's right. So some of the famous transcendental numbers would include the number pi you know the the uh 3.14159265 and so on. Uh so that's a transcendental number also oilers'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. Absolutely. That's true. Yeah. Although you know I don't know square of two is pretty square. All right. So it depends. Let's not beauty can be found in in all the different kinds of sets. But yeah, and if you have a kind of simplicity attitude, then you know zero and one are looking pretty good, too. So, and they're definitely not. Sorry to take that tangent, but what is your favorite number? Do you have one?

Oh, gosh. You know, is it zero? Did you know there's a proof that every number is interesting? You can prove it because Yeah. What's that proof look like? How do you even begin? I'm going to prove to you that every natural number is interesting. Okay? Yeah. I mean zero is interesting because you know it's the additive identity, right? That's pretty interesting. And one is the multiplicative identity. So when you multiply it by any other number, you just get that number back, right? And two is, you know, the the first prime number that's super interesting, right? And okay so one can go on this way and and give specific reasons but I want to prove as a general principle that every number is interesting and and this is the proof. Suppose toward contradiction that there were some boring numbers. Okay. Uh but if if there was an uninteresting number Yes. then there would have to be a smallest uninteresting number. Mhm. Yes. But that's a contradiction because the smallest uninteresting number is a super interesting graphology to have. So therefore there cannot be good there cannot be any boring numbers.

I'm going to have to try to find a hole in that proof cuz there's a lot of big tin in the word interesting. But yeah that's a be that's beautiful. That doesn't say anything about the transcendental numbers about the real numbers. You just prove from just for natural numbers. Okay. Should we get back to Caner's argument or Sure. You you've masterfully avoided the question. You basically said I love all numbers. Yeah, basically. Okay, that's what I my back to Caner's argument. Let's go. Okay. So, Caner wants to prove that the infinity of the real numbers is different and strictly larger than the infinity of the natural numbers. So, the natural numbers are the numbers that start with zero and and add one successively. So, 0 1 2 3 and so on. And the real numbers as we said are the the numbers that come from the number line including all the integers and the rationals and the algebraic numbers and the transcendental numbers and all of those numbers altogether. 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 and so the claim that we want to prove is that it's strictly larger.

So suppose that it wasn't strictly larger. So that then they would have the same size. But to have the same size remember means by definition that there's a 1 to1 correspondence between them. So we suppose that the real numbers can be put into one toone correspondence with the natural numbers. So therefore for every natural number n we have a real number. Let's call it r subn. R subn 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. Okay. And now, now I'm going to define the number Z and it's going to be the integer part is going to be a zero. And then I'm going to have put a decimal place. And then I'm going to start specifying the digits of this number Z. D1, D2, D3, and so. And what I'm going to make sure is that the nth digit after the decimal point of Z is different from the nth digit of the nth number on the list. Okay. So, so to specify the nth digit of Z, I go to the nth number on the list, R subn, and I look at its nth digit after the decimal point. And whatever that digit is, I make sure that my digit is different from it. Okay? And then I want to do something a little bit more and that is I'm going to make it different in a way that I I'm never using the digits zero or nine. I'm just always using the the other digits and not zero. There's a certain technical reason to do that. But the main thing is that I make the digits of Z different in the nth place from the nth digit of the nth number.

If you had ma if you had drawn out the numbers on the original list R1, R2, R3 and so on and you made it you know and they were each filling a whole row and you thought about the nth digit of the nth number it would form a kind of diagonal going down and to the right and that for that reason this argument is called the diagonal argument because we're looking at the nth digit of the nth number and those exist on a kind of diagonal going down and we've made our number D so that the nth digit of Z is different from the nth digit of the nth number. But now it follows that Z is not on the list because Z is different from R1 because well the 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. And 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 subn for every N. So therefore Z is not equal to any of these numbers R subn and but that's a contradiction because we had assumed that we had every real number on the list but yet here is a real number Z that's not on the list. Okay. And so that's the main contradiction.

And so it's a kind of proof by construction. Exactly. So given a list of numbers Caner is proving it's interesting that you say that actually because there's a kind of philosophical controversy that occurs as a uh in connection with this observation about whether Caner's construction is constructive or not. Given a list of numbers, Caner gives us a specific means of constructing a real number that's not on the list is a way of thinking about it. There's this one aspect which I alluded to earlier but some real numbers have more than one decimal representation and it causes this slight problem in the argument. For example the number one you can write it as 1.0000 forever but you can also write it as 0.999 forever. Those two those are two different decimal representations of exactly the same number. And then you beautifully got rid of the zeros and the nines. Therefore, we don't need to even consider that. And the proof still works. Exactly. Because the only kind of case where that phenomenon occurs is when the number is eventually zero or eventually nine. And so since our number Z never had any zeros or nines in it, it wasn't one of those numbers. And so actually in those cases, we didn't need to do anything special to diagonalize. Just the mere fact that our number has a unique representation already means that it's not equal to those numbers.

So maybe it was controversial in Caner's day more than 100 years ago, but I think it's most commonly looked at today as you know one of the initial main results in set theory and it's profound and amazing and insightful and the beginning point of so many later arguments. And this diagonalization idea has proved to be an extremely fruitful proof method. And almost every major result in mathematical logic is using in an abstract way the idea of diagonalization. It was really um the start of so many other observations that were made including Russell's paradox and the halting problem and the recursion theorem and so many other principles are using diagonalization at their core.

So can we uh can we just step back a little bit? This infinity crisis led to a kind of uh rebuilding of mathematics. So uh it would be nice if you lay out the things it resulted in. So one is set theory became the foundation of mathematics. All mathematics could now be built from sets giving math its first truly rigorous foundation. The exiomatization of mathematics. The paradoxes forced mathematicians to develop ZFC and other aimatic systems and uh mathematical logic emerged. Geredo Touring and others created entire new fields. So can you uh explain what set theory is and uh 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. this kind of two ways that set theory emerges. On the one hand, set theory is the is its own subject of mathematics which with its own problems and questions and answers and proof methods. And so really from this point of view, set theory is about the transfinite recursive constructions or wellfounded definitions and constructions. And 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 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 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. So I mean the history of how of the current set theory aims known as the Zera Franco axioms came out in the early 20th century with with Zermelo's idea. I mean the history is quite fascinating because um Zerlo in 1904 offered a proof that the what's called the axim of choice implies the well-order principle. So he described his proof and that was extremely controversial at the time and there was no theory there weren't any axioms there. Caner was not working in an aimatic framework. He didn't have a list of axioms in the way that we have for set theory now and Zerlo didn't either. Um and his ideas were challenged so much with regard to the well-order theorem that he was pressed to produce the theory that in which his argument could be formalized and that was the origin of what's known as their melo set theory.

and going to perplexity the axiom of choice is a fundamental principle in set theory which states that for any collection of non-mpty sets it is possible to select exactly one element from each set even if no explicit rule to make the choices 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 uh this was described before there was even a language for exiomatic systems. That's right. So on the one hand I mean the exo choice principle is completely obvious that we want this to be true that it is true. I mean 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. And 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 it's a completely natural principle. I mean, it's called the eximma choice, which is a way of sort of anthropomorphizing the mathematical idea. It's not like the function is choosing something. I mean, it's just that if you were to make such choices, there would be a function that consisted of the choices that you made.

And the difficulty is that when you when you can't specify a rule or a procedure by which you're making choices, then it's difficult to say what the function is that you're asserting exists. You know, you want to have the view that well there is a way of choosing. I don't have an easy way to say what the function is, but there definitely is one. Yeah, this is the way of thinking about the XMA choice. So we're going to say the the the three letters of ZFC maybe a lot in this conversation. You