Theories of Everything with Curt Jaimungal - Curt Jaimungal: What Is Infinity, Actually?

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Starting point is 00:00:58 It's time to turn those what-ifs into with Shopify today. Sign up for your $1 per month trial at Shopify.com slash to. That's Shopify.com slash T-O-E. For over 2,000 years, humanity insisted that infinity was only potential. So roughly this means that you can add one more, but you can never actually arrive. Aristotle thought so, Gauss just said so, and then Cantor showed up like a boss

Starting point is 00:01:31 with a heresy, and we're still fighting about it to this day. So what is infinity, and why the heck is it so controversial? Cantor treated infinities as objects that are completed in and of themselves that you can grab. Now, not only is that nuts,

Starting point is 00:01:50 but he then proved that there's strictly more of these infinities than anyone imagined. chronicer called Cantor a corruptor of the youth, Poincerey called Cantor's work a disease, Cantor then died in a sanatorium. So what the heck is going on? Now, to understand this,

Starting point is 00:02:08 we have to talk about potentiality versus actuality. And no, this isn't Deepak Tropa mixed with set theory. The distinction is sharper than it sounds. A potential infinity is a process. Like, you keep counting, you do one, two, three, four, etc. You don't stop. and at no point do you say that you've had a complete collection, you just say that there's always something else I can do. That's called a potential infinity. Now, an actual infinity says that that whole

Starting point is 00:02:35 collection exists right there as a single object. Now, what does that mean that you can handle it as a single object? Well, you can examine its properties, you can compare it to other collections, you can then ask difficult questions about its size, you get the idea. Now, Cantor's heresy was insisting on the latter that you can actually do some math with this concrete object. His first discovery was quite peculiar. There are as many even numbers as there are natural numbers. Now look, over here you have to be specific by what you mean as there's just as many. You also have to be specific as to what you mean by size, and mathematicians call that cardinality. Two sets are of the same size if you can pair them up exactly. You can think of it like how

Starting point is 00:03:21 kindergartners are in pairs, they hold their pinkies together. Now on screen, I'll show you what it looks like with all the even numbers and the full natural numbers. You can see there's a one-to-one pairing, a bi-ejection is what mathematicians call it, meaning you can go from one set to the next and back and not lose anything. Nothing's left over. It turns out that the same goes for the integers and the same goes for the rationales, which is quite absurd since the rationales are dense in the real line. But Cantor found a bijection between n, so the natural numbers, and cue the rational numbers. The exact way that you form these pairings is quite clever, and it's too much for the margins of this video, but I'll link a video to Trevor Bassett on this topic.

Starting point is 00:04:05 When I'm wrestling with a guest's argument about, say, the hard problem with consciousness or quantum foundations, I refuse to let even a scintilla of confusion remain unexamined. Claude is my thinking partner here. Actually, they just released something major, which is Claude Opus, 4.6, a state-of-the-art model. Claude is the AI for minds that don't stop at good enough. It's the collaborator that actually understands your entire workflow thinks with you, not for you.

Starting point is 00:04:33 Whether you're debugging code at midnight or strategizing your next business move, Claude extends your thinking to tackle problems that matter to you. I use Claude actually live right here during this interview with Eva Miranda. That's actually a feature called Artifacts and none of the other LLM providers have something that even comes close to rivaling it. Claude handles, interalia, technical philosophy, mathematical rigor, and deep research synthesis, all without producing slovenly reasoning. The responses are decorous, precise, well-structured, never sycophantic, unlike some other models,

Starting point is 00:05:08 and it doesn't just hand me the answers. The way that I prompted it is that it helps me think through problems. Ready to tackle bigger problems? Get started with Claude today at clod.a.I. slash theories of everything. That's clod.a.ai slash theories of everything and check out Claude Pro,

Starting point is 00:05:27 which includes access to all of the features mentioned in today's episode. Let's imagine that you take all of these here on screen, you put them in a bag, you throw them at Cantor. He'll tell you these are all countably infinite sets. He would say that these are all the same size. Now, because he's a mathematician and wants no ambiguity, he discarded that eight symbol,

Starting point is 00:05:51 that upside down or sideways 8, instead called it alif no. The reason is, well, Kurt, if all infinities are equal, then why do you have to invent a new symbol? The answer is that not all infinities are equal, even though intuitively they are. So let's think about this. If you add 157 to infinity, you just get infinity. Same if you do infinity minus 157. So what number outside of infinity itself can you add or take away from infinity to give you something other than infinity? It turns out this is the defining property of infinity. This is actually my favorite definition at all of math. I remember hearing it for the first time and it just blew my mind in that it makes it so unintuitive, but it's so darn brilliant at the same time.

Starting point is 00:06:43 Something is infinite. If you can take a finite amount of away from it and it doesn't change sides. I subscribe to the economist. Their science and their AI coverage is among the best I've found anywhere. And I say that as someone who reads plenty of it. I'll give you some examples. They just ran an analysis on how attitudes towards science are changing in American politics and what this means for research and funding in scientific institutions moving forward. This sort of high quality reporting is fantastic. They even covered how dark energy may be weakening over time. Now, if that holds up,

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Starting point is 00:08:14 So R is not countably infinite. The argument is a proof by contradiction and uses something called diagonalization. You may have heard this, and I'll show on screen here where you just list all of the real numbers that you think are all of the reels in a column, and then via this diagonal construction, you'll find a way to build another number that wasn't in your initial list. Thus, your list wasn't complete, thus you have a contradiction by assuming that the reels were countable to begin with. I know I glossed over that, but I'll put more links in the description.

Starting point is 00:08:42 The cardinality of the reels is written as two to the power of alif not. Now, that's quite wonky. Like, why two to the power of alif not? Why not call it alif one? Now, it turns out that's precisely what the continuum hypothesis is. Is it the case that two to the power of alif not is equal to alif one? We'll get to that. So each number can be expressed as a binary.

Starting point is 00:09:10 I'm sure you've heard this. and real numbers are no different. You can think of them as an infinite sequence of zeros and ones. The collection of all such functions has cardinality two to the power of alif not. And the reason is just, I can ask you, well, what's in the first digit? And you could say either zero one. I could say what's in the second digit? You could say either zero and ones and so forth. So you have two options for the first one, two options for the second one. And since real numbers go on to infinitum, then you have two to the power of alf not. But there's nothing deep about two here. sorry, duelists. The base doesn't matter. You can go to 3 to the power of alif not. You could say

Starting point is 00:09:45 7, we could say 157 to the power of alif not. You can even do alif not to the power of alif not. It's still the same cardinality of the reels. Here's something that genuinely disturbed me. It still disturbs me, if I'm honest. The real line has exactly as many points as the complex plane. And r squared is the same cardinality as r. And r to the 100 is. it's the same cardinalities are, Cantor himself wrote to Dedekin, I see it, but I don't believe it. So how do you get more infinities?

Starting point is 00:10:19 Remember earlier I said there was some method of infinitely producing infinities. Cantor, along with someone else whose last name is Hartog, discovered that there's at least two machines for manufacturing larger and larger infinities. And they work by entirely different mechanisms. So machine number one is something called

Starting point is 00:10:41 the power set construction. And that's actually what I used earlier in a sense. I mean, I didn't make it explicit, but that method of going from the natural numbers to the reels and showing the cardinalities two to the power of alif not, that's a power set construction. It just means you count all of the possible subsets of a set. And it turns out that all the subsets of a set equals two to the power of alif not. Don't worry about exactly why. The point is that that's machine number one. Machine number two is, well, just define alif one. Remember, you were asking, well, why is it that we said two to the power of alif not

Starting point is 00:11:16 and instead of calling it, say, alif one? The answer is that we're going to call the next possible infinity alif one from alif not. And then the next one after that, we're going to call alif two. Now, of course, you may be thinking, well, why am I making it into these discrete alf not, alif one, alif two?

Starting point is 00:11:37 How do we not know there's a contingent? continuum of a smeared amount of infinities. And that's a great question. It's a tricky question, and Cantor asked it as well. He said, how do you know if the first step of machine two equals the first step of machine one? In other words, is two to the power of alif not equal to alif one. And that's the continuum hypothesis. You've worked your way across thousands of years of human history in something like 1,200 words so far. I think that's the cardinality of the word count. Now, also, I should have said that Hartog's machine number two produces a concrete way of getting discrete infinities. It's not a continuum of infinities. There's a theorem. But either way,

Starting point is 00:12:24 it's quite cool that a 1,200 rung ladder can lead you to infinity. Now, there's so much more to say, like how the continuum hypothesis was the first problem in Hilbert's famous 1900 list and how Gertl showed that you can't prove the continuum hypothesis from the standard axioms of set theory of ZFC. And then in 1963, Cohen showed that you can't prove it either. So you can't disprove it, you can't prove it. So this means that the continuum hypothesis is actually independent of ZFC. It's super wild that there exists statements that are independent. And this is worth a video of its own, and by the way, I talked about what independence means in the context of Gerdels' work. See here in my video about misinterpretations of Gerdels and Completeness theorem.

Starting point is 00:13:09 And by the way, many people don't realize that Gerdels theorem implicitly relies on infinity, and the reason is that you have to model piano arithmetic. Anyhow, all of this is extremely abstract, yes, but paradoxically, it's extremely concrete. Infinity is where the abstract and the concrete meet. Every construction here is specific and finite and checkable. Cantor's diagonal argument from earlier actually fits on a napkin, and Hartog's construction is an explicit recipe that you can follow and crank out infinities, step by step. Cohen and his forcing method is a technique that you can use. You can sit down and execute it, yes, with patience, with disgruntlement in Adderall,

Starting point is 00:13:54 sure. Now, the objects being discussed here are infinite, but the reasoning about them is entirely finite, even mechanical. You know exactly what you're doing at exactly every step. When I'm wrestling with a guest's argument about, say, the hard problem of consciousness or quantum foundations, I refuse to let even a scintilla of confusion remain unexamined. Claude is my thinking partner here. Actually, they just released something major, which is Claude Opus 4.6, a story. state-of-the-art model. Claude is the AI for minds that don't stop at good enough. It's the

Starting point is 00:14:32 collaborator that actually understands your entire workflow thinks with you, not for you, whether you're debugging code at midnight or strategizing your next business move. Claude extends your thinking to tackle problems that matter to you. I use Claude, actually live right here during this interview with Eva Miranda. That's actually a feature called artifacts, and none of the other LLM providers have something that even comes close to rivaling it. Claude handles, interalia, technical philosophy, mathematical rigor, and deep research synthesis, all without producing slovenly reasoning. The responses are decorous, precise, well-structured, never sycophantic, unlike some other models, and it doesn't just hand me the answers. The way that I've prompted

Starting point is 00:15:16 it is that it helps me think through problems. Ready to tackle larger problems? Sign up for Claude today and get 50% off Claude Pro when you use my link, clod.aI slash theories of everything, all one word. Now, what I just said, the finitists would disagree. So not everyone accepts the output here. These finitists are mathematicians who reject infinity entirely. To a finitist, alif not is not a real mathematical object. It's at best a useful fiction.

Starting point is 00:15:50 at worse. It's a symbol just pointing at nothing. Hilbert flirted, by the way, with a version of this position. A stronger stance is called ultra-finitism, and it goes even further. They would say that some super-large numbers, like let's say 157 to the power, 157 to the power 157, those are just suspect. There's no physical process that would ever instantiate something like this. It's an interesting melding of math and physics in a sense, these ultra-finites. I spoke, by the way, to Dror Barnettin, who is an ultra-finitist, and I spoke to Norman Wildberger, who's a regular finitist. I'll put links in the description. There's also Nelson and Volpin, who I haven't interviewed, who have similar views like this, to an ultra-finitist asking, what is too to the power of alif not, is like asking,

Starting point is 00:16:39 what's the orange about jumping? It's a question of nonsense. Although I think I heard a Tarantonian on Jarvis Street, having a cogent conversation starting with that question. Most working mathematicians find infinity indispensable, so Hugh Wooden, who by the way is one of the most important living set theorists alive, or living, is someone that I'll be speaking with soon on the Theories of Everything channel. You can subscribe for that. He spent decades saying that the continuum hypothesis does have a definite answer. His position is that ZFC is just simply too weak to detected that you need some stronger axioms, large cardinal axioms. I believe this is called his Ultimate L program. Now, to get even more trippy, if this wasn't enough for you, there's an

Starting point is 00:17:24 interview with Emily Real on Infinity Categories, where she presents infinity categories at the most basic level. It's still super challenging, though. Actually, I'm working on a book with Penguin Press, and I'll be covering Infinity Categories and Hugh Wooden's work in it. By the way, you may wonder, why can't you divide by zero? That's an elementary question and it's a reasonable question and it turns out that in projective geometry which is what Roger Penrose uses

Starting point is 00:17:50 for a twister theory, you can use a division by zero and it makes sense when you take the complex numbers and you union them with infinity or a point, but he calls the point infinity for a reason. Using infinity lets you make

Starting point is 00:18:05 one divided by zero perfectly well-defined statement. That's how, useful infinity is. So let's summarize. The old view is that imagine you're counting and you do one, two, three, four, five, six, blah, blah, blah, blah, you never stop. You don't arrive at infinity. That's called a potential infinity. It's just a process that keeps going. We used to think that that's all infinity was. Then Cantor came along, along with Dedekin, and said, no, actually you can complete that whole set. You could treat it like a collection and do math with it. It's not a process

Starting point is 00:18:38 anymore, or it's not only a process, you can treat it like a thing. That's called actual infinity. You can then ask the question that every guy is afraid a woman will ask them, how big is it? And you can compare it to other infinite sets. For instance, the reels. So you can compare the natural numbers with the reels, and you can show that they're not equal, and the reels has more than the naturals. And then you could also use another method called Hartog's method, so you can use machine one to produce the cardinality of the reels, and you could use machine two to produce a different set of cardinalities of infinities. And the question about how do these two machines relate

Starting point is 00:19:12 at their lowest level, at their first level, actually, is called the continuum hypothesis. Many people stated as, is there an infinity between the natural numbers cardinality and the real number cardinality? And, of course, all of this has to be caveated because of those persnickety finitists and ultra-finitists who just say,

Starting point is 00:19:32 all of what you're speaking about is either a useful fiction. is just somehow useful for you to do some computations but doesn't actually exist. Or, and that's the best case, or it's just nonsense. You're just speaking, Baba-la-la-du-Bid-da, whatever, meaningless, grammatical, seemingly correct strings of symbols,

Starting point is 00:19:53 but they don't refer to anything. And interestingly enough, the finitists and ultra-finitists do have a coherent philosophical position. I like it because it forces you to ask, what the heck do you mean when you say that some mathematical object exists? So, the debate is alive, and depending on your view, the fact that it's a debate at all, that mathematicians can disagree, not about whether a proof is valid, but whether about

Starting point is 00:20:20 the objects that the proof discusses even exist, that tells you something unsettling about the foundations on which the rest of math sits. The Economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments and how the impact markets. They recently published a piece on China's new neutrino detector. They cover extending life via mitochondrial transplants, creating an entirely new field of medicine. But it's also not just science. They analyze culture. They analyze finance, economics, business, international affairs across every region.

Starting point is 00:21:01 I'm particularly liking their new insider feature. It was just launched this month. It gives you, it gives me a front row access to the economist's internal editorial debates, where senior editors argue through the news with world leaders and policy makers in twice-weekly long format shows. Basically, an extremely high-quality podcast. Something else you should know about is that if you go to their app, they not only have daily articles, but they also have long-form podcasts with their editors and writers.

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Theories of Everything with Curt Jaimungal - Curt Jaimungal: What Is Infinity, Actually?

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