The Continuum Hypothesis - The Problem that BROKE Mathematics

I have a question for you. Which one of these containers has more things in it? This simple question sparked one of mathematics most important open problems, shaking our understanding of logic, infinity and the nature of truth. It forced mathematicians to confront uncomfortable ideas like whether universal truth even exists and whether logic has hidden boundaries we'll never be able to cross. In this video we'll explore the remarkable journey mathematicians took to solve it, a journey that ended in an unexpected resolution, leaving us with even more questions than answers. It's bizarre! We'll explore how this single problem turned people's understanding of mathematical reality upside down. It's really opening the door to this other world which is mathematically enlightening and enriching. It just shows us that math is richer and more interesting and messier than we thought. But first, how could such a simple question cause all of this? To understand, first what does it mean for there to be an equal number of things in each container? Well if we can pair up apples from each container with none left over, there was an equal amount of apples in each one. In math talk this is called a one-to-one mapping. How do we know if one container had more apples? Well if we can't do a one-to-one mapping, if there are some apples left over. That's so easy! How did mathematicians get so tripped up by that? Well now let's try the same thing with numbers. Let's start with the natural numbers one, two, three, four, five, etc. There are infinitely many natural numbers. No matter how high you count, you can always add more. In math, things that contain objects like numbers are called sets. This is the set of all the natural numbers. Over here is the set of all the even natural numbers, two, four, six, eight, etc. Which set has more numbers in it? Are there more natural numbers or more even natural numbers? Your hunch might be the set of natural numbers is bigger. After all, the set of even natural numbers is missing all the odd numbers. Let's test out our hunch with one-to-one mapping. We can pair one with two, two with four, three with six, four with eight, five with ten. Hang on! For any natural number, we can always pair it with an even natural number. So does that mean there are as many even natural numbers as natural numbers? Is the set of natural numbers the same size as the set of even natural numbers? Yes, and a man named Georg Cantor was one of the first to discover this fascinating fact. He also found that the set of all square numbers can be placed in a one-to-one mapping with the set of all natural numbers. He even came up with a clever way to map the set of all negative and positive integers to the set of all natural numbers, as well as the set of all the rationals. All of these infinite sets are the same size. So then are all sets of infinite numbers the same size? Is infinity a one-size-fits-all kind of thing? Well, we're still missing a lot of numbers. Let's add the irrationals, the algebraics, the transcendentals. Let's add every number until there are no more gaps between negative infinity and positive infinity. This is the real number line. Now the funny thing about lines is that they're continuous, uninterrupted. You can point it anywhere on the real number line and there will always be a real number there. This is different to the natural numbers which are discrete. There exist spaces between them. They're better represented as dots. We can also count them. Even if we would never stop counting, we can at least start counting. Same for all of these other sets. Actually, this one-to-one mapping idea can be kind of thought of as counting, as we're saying this is the first element in the list, this is the second one, this is the third one, this is the fourth one, and so on. So Cantor called these sets countably infinite. Cantor tried and tried, but he couldn't come up with a one-to-one mapping between the natural numbers and the real numbers. In fact, through his now famous diagonalization argument, he proved that there is no way to map the natural numbers to the real numbers. The real numbers are uncountably infinite. Now remember, if we can't do a one-to-one mapping between two sets, it means that one set is bigger than the other. So Cantor concluded that the set of real numbers must be bigger than the set of natural numbers. There are different sizes of infinity. He later discovered there are infinitely many sizes of infinity, but for the purposes of this video, we're only concerned with these two. The size of infinity of the natural numbers is often called aleph null and is represented by this fancy-looking n with a zero. It's the smallest size of infinity, and Cantor hypothesized that the set of real numbers was the next biggest size of infinity, aleph one. He believed that there is no size of infinity between that of the natural numbers and that of the reals. Cantor spent decades wrestling with this question, but despite his best efforts, he could never prove it. So it was called the continuum hypothesis. Little did he know it would just about break mathematics. Many of his colleagues rejected his ideas entirely, claiming a theory of infinity wasn't real mathematics. One even accused him of disrupting the youth. But one of the most influential mathematicians at the time, David Hilbert, thought Cantor's ideas were revolutionary. Not just his theory of infinity, but also his ideas about sets. He famously said, no one shall expel us from the paradise that Cantor has created for us. If you don't know this guy, he was a great mathematician, but his main role in math history seems to have been getting his dreams destroyed over and over again. And this story is no exception. Hilbert made a list of what he thought were the 23 most important problems in math and presented them at a conference for mathematicians. The continuum hypothesis was number one on the list. Now you might be wondering, really? Number one on the list of most important math problems? Why is an abstract question about infinities so important? Well, to understand why the continuum hypothesis was such a big deal, we need to know why Hilbert called this conference in the first place. Math was going through a pretty big problem at the time, and it was in desperate need of some fixing. There were all these new theories that were vastly outside of our physical realm. Math was becoming more and more confusing and counterintuitive. Paradoxes and inconsistencies were starting to pop up all over the place. The most famous being Russell's paradox, which stated simply is, does the set of all sets that do not contain themselves contain itself? Either answer leads to a contradiction. Mathematics is constrained only by logical inconsistency, like you can do whatever you want, just don't write down a contradiction. Hilbert's dream was to repair mathematics. He wanted a system powerful enough to prove or disprove every mathematical statement clearly and decisively. No more mysteries, no more paradoxes. Everything would be either provably true or provably false. This doesn't seem like a terribly unreasonable ask. After all, most of us think of math as this objective thing. A lot of the time when I ask people why they like math, they say because there's a definite yes or no answer. Hilbert believed we just needed the right The answer to his dream, or so he thought, came in the form of Zermelo-Frenkel set theory with the axiom of choice, or ZFC for short. That was a lot of words. Let's break it down. ZFC is basically the official rulebook of mathematics. It consists of a small number of carefully chosen axioms that define what sets are, how you can combine them, and how you can build new mathematical objects safely by rules of deduction. Think of these axioms as statements that are so obvious they don't need a proof. Like if two sets have exactly the same elements, they're identical. Or you can always collect together elements to form new sets. The brilliance of ZFC is that it provides a framework where mathematical statements can be rigorously proven. Each proof becomes a careful, logical progression. It seemed like the perfect fix for all these inconsistencies and paradoxes that were going on. By starting from basic axioms that we all agree are true, and following precise logical rules, you'd end up with theorems that are simply undeniable truths. To prove something true, you start with some axioms and, through a logical progression, prove that the statement follows logically from the theory. To prove something false, you need to show that the negation of the statement follows logically from the theory. ZFC remains the standard foundation of math today, powerful enough for essentially all mathematics we do. And because ZFC is fundamentally about sets, the continuum hypothesis became one of its first big questions. The continuum hypothesis wasn't just a weird question about infinity. It had deep implications for the power of axiomatic systems. Some of the greatest minds were in the room, and many tackled it. But years turned into decades, and no progress was made on solving Cantor's continuum hypothesis... until this guy came along. Kurt Gödel. You may know him from Incompleteness fame, or you may not, but either way, he was a super smart dude who had a knack for thinking outside the box. You could say he was eccentric. He actually later died because he was so worried about his food being poisoned that he refused to eat and starved to death. Anyway, Gödel was the first person to make any progress on the continuum hypothesis in decades. But there was a slight problem with his solution, and that was it wasn't really a solution at all. What Gödel found was more surprising and profound than anyone could have guessed. He proved that you cannot disprove the continuum hypothesis. Now when I first heard this, my response was, what? How can you prove that you can't disprove something? And why isn't that just a proof that it's true? So to answer these questions, take this analogy. Imagine you go to a new country, and the only law is all buildings must be rectangular prisms. So you travel around, and you find a city, and all buildings are rectangular prisms. Cool. A mathematician would say You also notice that all the buildings have this property that the number of corners or vertices minus the number of edges plus the number of faces always equals two. That's interesting, you think. But then you're like, well, actually, it's not that interesting. This fact follows logically from the law that all buildings must be rectangular prisms, as all rectangular prisms have It would be impossible to have a city where buildings were both rectangular prisms, and this number was anything other than two. You keep doing tourism, and you notice that all the buildings in the city are red. So you wonder, does the fact that all buildings are red also follow logically from the law that all buildings must be rectangular prisms? You're not sure. But one thing you are sure of is buildings being red is logically consistent with the law that buildings must be rectangular prisms. To clarify about the word consistent, because it's an important word that we're going to keep using. Mathematically, it just means that if two properties are consistent, they can exist together without causing a contradiction. There is nothing logical that says they can't go together. If two properties are inconsistent, they cannot exist together without causing a contradiction. They just do not make sense together. So again, you haven't shown that buildings being red follows logically from buildings being rectangular prisms. But you definitely know the two properties can exist together without causing any problems. You know, you can have a city where both things are true. I mean, you're in it. Gödel did something equivalent with math and the continuum hypothesis. ZFC theory are the laws of mathematics. Gödel built a mathematical universe that satisfied the laws of ZFC and where the continuum hypothesis was true. He showed that having the set of reals being the next biggest size of infinity after the set of naturals was consistent with ZFC theory. However, he didn't prove that this follows logically from the ZFC axioms. He didn't show it was a theorem. He just showed that the continuum hypothesis is consistent with ZFC. In the same way, red is consistent with being a rectangular prism. Even though this isn't a proof of the continuum hypothesis, it shows that we can't disprove it. Remember that to disprove something from a theory, you need to show that its negation follows logically from the axioms. But how could that be when here was a place where ZFC and the continuum hypothesis lived together in harmony? It'd be like trying to prove that being red and being a rectangular prism was impossible while you're looking at the red rectangular prism city. He showed that the negation is not provable because there's a place where the CH is true. So how exactly did he do this? He took the laws of ZFC and figured out what does that construct? And he started constructing. And he took everything he constructed, called out the constructible universe, and started figuring out what its properties are. The constructible universe is the bare necessities universe. At every stage, everything in L has to exist. Gödel basically just kept putting in everything he was absolutely contractually obligated to build. And then he counted how much stuff that was. And what do you know, you get exactly all of one real numbers by necessity. This was an entirely new kind of proof, and it left us in an awkward position. Gödel proved that you can't disprove the continuum hypothesis, but he didn't prove that the continuum hypothesis was true. It might be true. And the obvious next question for mathematicians was, well, can we prove that the continuum hypothesis is true? Nobody really had any ideas on how to do this, and progress stalled for over 20 years. Until in 1963, a guy named Paul Cohen came along and turned everything on its head again. He proved that the continuum hypothesis cannot be proven from ZFC theory. Cohen was trained as an analyst, not a set theorist, but he wanted to get famous and for some reason thought logic was the best way. He asked a mathematician at Berkeley, what's the biggest open problem in mathematical logic? The mathematician replied, the continuum hypothesis, but you shouldn't work on that because there are no leads on how to solve it. Cohen didn't listen and went on to win the Fields Medal for his results. The Fields Medal is the achievement in mathematics, kind of like the Nobel prize of math. So I guess his plan worked, he did become famous. In a lot of ways, Cohen did the opposite of what Gödel did. He built a mathematical universe that was consistent with ZFC and where the continuum hypothesis was false. If we go back to our city analogy, it's like he forced in a bunch of non-red is also consistent with the law that all buildings must be rectangular prisms. You'd be forced to conclude that the property of being red must not follow logically from the property of being a rectangular prism. However, once again, he didn't show that the property of not red follows logically from the law. Cohen started with Gödel's constructible universe and forced in a bunch of stuff until the continuum hypothesis was no longer true. How did he do this? He invented an ingenious technique called forcing. Forcing is pretty complicated. So before we go over what it is directly, let's start with a helpful analogy. An analogy that I've often used in talking to other mathematicians is a lot of times we like to think about fields and a field is a place where you can add and subtract, multiply and divide, and it all works out really nicely. So the rational numbers or the real numbers or the complex numbers, these are all examples of fields. A thing that we do a whole lot in math is build field extensions. We start with some field and we make it bigger. This idea of extending an object, we do it all the time in math. So you could start with the rational numbers. You cannot solve the polynomial x squared minus two inside of the rational numbers, right? The root of two is not a rational number. But you could take the rational numbers and you can build a bigger field that has root two. And then there are a couple other things that just have to come along for the ride. If you add root two, then you have to add root two plus root two because you have to be able to add things. And then you have to add like one over root two. So that's what I mean by there are some things that have to come along for the ride, just logically must. So that's kind of how forcing works. You have your model of ZFC, you add your new thing. That's not going to change ZFC, but it can change other stuff. So I guess the next question is, what did he add? Cohen added more real numbers to Godel's constructible universe. But how? The constructible universe was already meant to encapsulate all of mathematics. With the fields, the rationals and the reals are a small part of the universe of mathematics. It's easy to grab things from outside of them. But this was meant to be the entire universe of mathematics. Where was he going to pull more numbers from? How do you expand an entire universe of mathematics? So it turned out the only way we could begin to tackle this problem was to imagine what a random number would look like. We can start bookkeeping our expectations about what a larger universe would look like. The way we do this is to imagine we choose a real number at random between zero and one. And we one at a time assign probabilities to every question we could ask about it. 50% chance is between 0.2 and 0.7. 100% chance of being irrational. And 100% chance of being different from every wheel we already have. Forcing is the language that Cohen developed to keep track of all these questions and assign them an abstract likelihood of being true. In doing this, and doing this all of too many times, to randomly add that many more new numbers to our universe, he developed a consistent language for describing a universe that with probability one satisfies every individual axiom of ZFC, but not the continuum hypothesis. One expert I talked to literally said that it's not so easy to explain forcing even to a professional mathematician. So if you didn't get all that, don't worry. What you need to know is that when Cohen was done, there were Aleph null many natural numbers, Aleph one many real numbers from Gödel's constructible universe, and Aleph two many real numbers from Cohen's universe, which were considered to be all the real numbers. Remember that the continuum hypothesis was the assertion that there is no size of infinity between that of all the natural numbers and that of all the real numbers. But now we have the size of the constructible real numbers in between the two, thereby showing that the continuum hypothesis is false. So here's where we're at. Gödel constructed a mathematical universe that satisfies ZFC, where the continuum hypothesis is true, and Cohen constructed a mathematical universe of ZFC where the continuum hypothesis is false. So where does that leave us? A statement is independent of a theory just means that you can't prove the statement from the theory, and you also can't refute the statement. In other words, you can't prove the negation of the statement. Let's just say Hilbert's not happy. Let's take a moment to really appreciate why, because it's a bit subtle, and the nuances definitely confused me for a while. So something that's been implicit in this video is the idea of mathematical models, like Gödel built a mathematical universe and then Cohen extended his mathematical universe. Another word that mathematicians use for mathematical universe is mathematical model. But what does built a mathematical model mean? Isn't there just one math? Like, the math? Well that's actually not how set theorists think about things, and it's largely due to results like the continuum hypothesis. Set theorists think in terms of mathematical models. We have a theory, like ZFC, and rather than there just being one math it describes, it can describe multiple models that we fit to the theory, like Gödel's model and Cohen's model both fit to ZFC theory. This is a pretty different way of looking at math than what we're used to, and I bring all of this up because it has a very important consequence. For a statement to be provable from the theory, it needs to be true in all models. For example, remember that relation we had between vertices, edges, and faces earlier on? We agreed that that followed logically from the law that all buildings were rectangular prisms, right? Notice how this relation still holds in both cities. We can see that in any city that's made up of rectangular prism buildings, this relation will be true. However, as we've known the entire video, the property of red or not red does not follow logically from the law that all buildings must be rectangular prisms. It can change from model to model. The color of the buildings is independent of the law that all buildings must be rectangular prisms. The law simply says nothing about the color of the buildings. Drawing a parallel with ZFC and the continuum hypothesis, here we have two models of math, one where the continuum hypothesis is true and one where it's false. Therefore, it must not follow logically from the axioms of ZFC. We cannot arrive to it by starting with the axioms and logically following the rules of deduction. The theory simply doesn't say anything about the continuum hypothesis. If you produce by some brute force algorithm every single possible theorem from ZFC, the proof of the continuum hypothesis and the proof of the negation of the continuum hypothesis will never show up. Considering the continuum hypothesis is one of the first questions you would ask when building a theory of sets, it raises a lot of questions about the state of mathematics. Is the continuum hypothesis considered solved? Is its independence a problem that we need to fix? Is it okay that there are some questions our current math just can't answer? I mean it gets into contentious philosophical stuff. There are generally two main philosophical camps when it comes to answering these questions. The first is Platonism. Platonism is the idea that there is some kind of objective realm in which all mathematical structures exist. There is one true mathematical universe and there is a true answer to all of our questions. We all agree from these mathematical facts that there is more than one universe of set theory. The position of a set theory Platonist is usually that amongst these universes one is the biggest among them all and that everything else is just a sub-universe of that and under this position when we are asking questions of like the continuum hypothesis what we are trying to figure out is is it true or false in this biggest of this universes to one who believes that there is this ultimate universe then there is still hope for us to resolve the hypothesis. Some set theorists are currently working on building new axioms of ZSD in hopes of being able to solve the continuum hypothesis. At the opposite end of the spectrum to Platonism is Formalism. Formalism is the mathematical philosophy that there's nothing more to math than shuffling symbols and rules on a piece of paper. Math is the consequence of our starting axioms and we get to choose whatever axioms we want as long as they're consistent. Different starting rules lead to different mathematics. To the formalist this is the end of the story. We have our full answer. The continuum hypothesis is true or false depending on what model you're in. There's a mathematical multiverse and all universes are equally valid. I don't know if I think there is a real mathematical universe. To me it's more like we're trying to understand what happens if you assume this rather than what's the real truth of the matter. I think it just tells us ZFC doesn't settle all of our questions and that means we get to ask more questions. This contrast in how mathematicians view mathematics raises an even deeper question. Are we discovering mathematical truths that have always existed or are we inventing mathematics as we go along? Are mathematical axioms like those of ZFC something humanity created or something we uncovered? This question of whether math is invented or discovered comes up again and again when you study mathematics. 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