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 MathOverflow, which I think is a legendary accomplishment. MathOverflow, by the way, is like StackOverflow 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, 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, soul-searching 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, uh, 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, uh, a little bit of hate and a whole lot of love, and I wouldn't have it, uh, any other way. I'm 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's Joel David Hamkins. Some infinities are bigger than others. This idea from Cantor at the end of the 19th century, I think it's fair to say broke mathematics before rebuilding it, and, uh, 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 Cantor was deeply religious himself. Second, there's a kinda mathematical civil war, the leading German mathematician, uh, Kronecker called Cantor a corrupter of youth and, uh, tried to block his career. Third, many fascinating paradoxes emerged from this, uh, like Russell's paradox about the set of all sets that don't contain themselves, and, uh, those threatened to make all of mathematics inconsistent. And finally, on the psychological side and the personal side, Cantor's own breakdown. He literally went mad spending his final years in and out of, uh, sanatoriums obsessed with proving the continuum hypothesis. So laying that all out on the table, uh, can you explain the idea of infinity that, uh, some infinities are larger than others and why was this so transformative to mathematics?

Well, that's a really great question. I would wanna start talking about infinity and telling the story much earlier than Cantor actually because, I mean, you can go all the way back to ancient Greek times when Aristotle emphasized the potential aspect of infinity as opposed to the impossibility, according to him, of achieving an actual infinity, 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 potential as understanding of infinity for, for hundreds of years, thousands of years. Uh, almost all mathematicians were potentialists only and thought that it was incoherent to speak of an actual infinity at all. Um, Galileo is an extremely prominent exception to this, though he argued against this sort of potentialist orthodoxy in The Dialogue of Two New Sciences, really lovely account there that he gave, um, and that the- i- in many ways, Galileo was anticipating Cantor's developments except he couldn't quite push it all the way through and, uh, ended up throwing up his hands in confusion i- 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 one, two, three, four, and so on, and you think about which of those numbers are perfect squares. So zero squared is zero, and one squared is one, and two squared is four, three squared is nine, 16, 25, and so on. And Galileo observed that, that the perfect squares can be put into a one-to-one 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...... uh, 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 endpoints 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 one-to-one 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 equinumerosity. And the way we would talk about it now is appeal to what, uh, what I call the Cantor Hume principle, or some people just call it Hume's principle, which is the idea that if you have two collections, whether they're finite or infinite, then we want to say that those two collections have the same size, they're equinumerous if and only if there's a one-to-one correspondence between those collections. 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 the- the tension between the Cantor Hume principle and what could be called Euclid's principle, which is that the whole is always greater than the part, which is a principle that Euclid appealed to in- in The Elements. I mean, many times when he's calculating area and so on, he wants... It- 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 Cantor Hume principle and Euclid's principle. And it really wasn't fully resolved, I think, until Cantor, 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 one-to-one correspondence. And it's traditional to talk about the uncountability of the real numbers. So Cantor's big result was that the set of all real numbers is an uncountable set. So maybe if we're gonna talk about countable sets, then I would suggest that we talk about Hilbert's Hotel, which really makes that idea perfectly clear.