▶︎ 0:00 I'd like to tell you about a certain confounding mystery at the center of set theory, and that is concerned with Cantor's continuum hypothesis. We saw how Cantor proved that the set of real numbers is an uncountable infinity, and therefore, it's a strictly larger infinity than the set of natural numbers. So we have the countable infinity of the natural numbers, and we have the uncountable infinity of the real numbers. When you prove that one infinity is bigger than another, it must be the most natural question in the world to ask, is there any infinity in between? That question is precisely what the continuum hypothesis is concerned with.
▶︎ 0:46 So Cantor wondered. He knew the set of real numbers was a bigger infinity than the natural numbers, but he didn't know whether there would be anything in between them. So the continuum hypothesis he formulated is the assertion that there is nothing in between. But it was an open question for Cantor. He became obsessed with this question. He spent his entire life trying to prove it, trying to refute it, and failing at both. So he never knew the answer and died before he came to an answer, and the answer is quite an interesting story that unfolded over the decades that followed.
▶︎ 1:22 So Cantor had laid out a strategic program for attempting to prove that the continuum hypothesis is true, namely that there is no infinity between the natural numbers and the real numbers, and what he wanted to do was to prove that it's true for certain very simple kinds of sets, and he succeeded in that. And he wanted to work his way up to more and more complicated sets. Say he proved it for closed sets of reals. It's also true for open sets of reals and so on, and then he wanted to move to the next level of complexity of sets and prove that those sets are never going to arise as an intermediate cardinality between the natural numbers and the reals. Regrettably, though, he died before learning an answer to this. In certain respects, his program was continued to a certain extent, and I'll talk about that in a bit.
▶︎ 2:12 The eminent mathematician David Hilbert had placed the continuum hypothesis as the very first of his famous list of questions at the dawn of the 20th century that ended up guiding so much of mathematical research during the 20th century, and the continuum problem was the number one question on that list. Nobody knew whether it was true or false. No one could find a set of intermediate cardinality. We didn't even have candidate sets.
▶︎ 2:41 So what is your view? Do you think the continuum hypothesis is true or do you think it is false? Can you identify a set of reals that's intermediate in cardinality between the natural numbers and the reals? So what you need is a set that's not countable because it's bigger than the natural numbers, but it's not equinumerous with the real numbers either, a set of real numbers that is intermediate in size between the natural numbers and the real numbers. Is it possible? Or maybe there's no such thing. That's exactly what's at stake in the continuum hypothesis question.
▶︎ 3:15 Of course, one can set the stage by mentioning that there's ubiquitous equinumerosity. So we have many, many different sets that we can prove are equinumerous. For example, in the context of the discussion with Galileo, we mentioned, Galileo even proved that different line segments of different lengths are equinumerous because of this one-to-one correspondence that's provided by this fanning, and Galileo also showed that a given line segment is equinumerous with the whole real line, and maybe a different way of seeing it than his argument is to just look at the arctangent function, which is a function from the real line into a bounded interval, say, from minus pi over 2 to pi over 2. It gives a one-to-one map from the whole real line to an open interval. So that shows that those two sets have the same size.
▶︎ 4:12 So we have many more instances of equinumerosity. For example, we can see that the power set of the natural numbers is equinumerous with Cantor space, which is the set of binary sequences, and that's equinumerous with the real line. So these sets are all equinumerous with each other. We have many, many instances of such equinumerosity. Let me give another very surprising result due to Cantor, namely...
▶︎ 4:52 Let me sit down here. Cantor considered the question whether the real line might be equinumerous with the real plane, R squared, the set of pairs of real numbers X comma Y. On the one hand it might look, well, the plane seems to have way more points on it than the line, because it's two-dimensional and so on. There are lines sitting inside the plane, but they take up the tiniest fraction of the plane. They have zero area and so on. And so it might seem reasonable to suppose that there should be more points in the plane than there are points on the line.
▶︎ 5:34 But amazingly, Cantor was able to observe that, in fact, the real line and the real plane are equinumerous. They have the same size. And when he discovered this, he wrote immediately a letter to Richard Dedekind, in which he said, this is 1877, "I see it, but I don't believe it." So he gave the argument to Dedekind. It's such an incredible and counterintuitive result. There are exactly the same number of points on the line as in the plane, and the same is true in any higher dimensions. In space, where four dimensions, five dimensions, any finite dimension, even countable dimension, if you think about the infinite dimensional spaces.
▶︎ 6:18 Let's give the argument. I want to find, if I'm claiming the number of points on the line is the same as the number of points in the plane, well, that can be proved by establishing two facts. On the one hand, there are at least as many points in the plane as there are in the line, because I can view the line as sitting inside the plane. The x-axis is sitting inside the plane R squared. So, to show that they have the same number, I just need to find a one-to-one correspondence between points in the plane and a set of real numbers.
▶︎ 6:50 So that's exactly what Cantor did, and what I'm going to do now. Given two real numbers, say maybe I have 3.14159265 and so on, that's the X coordinate, and the Y coordinate, maybe it's 2.718281828 and so on. So pi and e, for example, or given any two reals, I'm going to associate this pair of real numbers with a single real number, and the way I'm going to get that real number is simply by interleaving the digits. So I take this three, and then this two, and then this one, and this seven, and this four, and this one, and this one, and this eight, and so on. So I'm just taking first a digit of X, and then a digit of Y, and then a digit of X, and then a digit of Y. And so on. And I'm assembling them into one real number.
▶︎ 7:50 So what have I done? I've mapped pairs of real numbers to real numbers, and I claim that this is a one-to-one mapping of R squared into R. I guess, to be a little bit definite, because sometimes real numbers have more than one representation, but when I'm interleaving the digits, I need to be working with a specific representation of the real number, so let's never use the all nines case. So if a number has two representations, it's because one of them ends in all nines and the other one ends in all zeros. Let's just, by convention, always use the eventually all zeros case, for the numbers for which this issue arises. So the resulting number I get will never be all nines. Because the only way I would get all nines is if each of them were ending in all nines, but we said we're never using that case.
▶︎ 8:47 Given a real number, I can decode it into X and Y to see which pair of numbers it came from, because I can just do the unraveling. I can take the first digit and the next digit, and then the next one, and the next one, and so on. So given the combined real number here, I can recover X and Y from it, and that shows that that mapping is a one-to-one mapping of pairs of real numbers into the real numbers. It's not an onto mapping, because, for example, I'm never going to get the number that has a nine in every other place here. By this association, I'm never going to get 5.90919091 or whatever. If it has nines in every other place, then that would have had to have come from a number that was eventually nines, but we said we're never using that one.
▶︎ 9:35 So this number, which does not end in all nines, but it has nines in every other place, is a particular real number. It's a perfectly good real number, but it will not arise in the image of this association, and therefore that interleaving of digits is not a one-to-one correspondence between R squared and R, but rather it's a one-to-one correspondence between pairs of real numbers and some of the real numbers. But that's good enough to show that there are at least as many real numbers as there are pairs of real numbers.
▶︎ 10:09 So this is the argument that shows that the real line is, okay, so ultimately, what we've done is we've shown, let me just explain it a little better here. We've shown that the real plane, the set of pairs of real numbers is at least as big as the real line, because the line is in the plane, but we've also shown that pairs of real numbers can be associated in a one-one way with a set of real numbers, so the reals are at least as big as pairs. And therefore, because of these two things, and because of the Cantor-Schroeder-Bernstein theorem, I get to deduce that the real line is equinumerous with the pairs of real numbers. So the line is the same size as the plane.
▶︎ 10:58 And now, of course, it follows that I can also do it more times. One can argue directly that we could, given triples of real numbers, X, Y, Z, I can interleave their digits just as easily as I did with pairs, but also I can just deduce it, because once I know that this one is equinumerous with this, I can just do times R on both sides. So I get R cross R is equinumerous with space, and therefore all three of them, the line, the plane, space, they're all equinumerous. They all have exactly the same number of points. And similarly in any finite dimension. So let's now ask another
▶︎ 11:47 A question about equinumerosity. If you wanted a big set, would you rather have the set of functions from the reals to the integers, or the set of functions from the integers to the reals?
▶︎ 12:03 The question is, which is a bigger set? Let's write it down, like this. We're comparing the set of functions from the integers to the reals versus the set of functions from the reals to the integers. This exponential notation, when I have X to the Y, it's the set of functions from Y to X, from the exponent to X. Which one is bigger?
▶︎ 12:36 Maybe you want to pause the video and think about it. Are there more functions from the integers to the reals, or from the reals to the integers? That's the question, and let's think about this situation.
▶︎ 12:47 If I think about this function space, functions from the integers to the reals, of course it's equinumerous with functions from the natural numbers to the reals, because the integers and the natural numbers are the same. So basically, how many infinite tuples of real numbers are there? A function from the natural numbers to the reals is basically just a list of real numbers. So I have a list of real numbers. How many countable lists of real numbers are there?
▶︎ 13:20 Well, if I have this interleaving idea, then you can see that actually given any countable list of real numbers, I can imagine interleaving their digits in a certain way. For example, I put the first digits of the first real here, there's X1, I mean X0, and then the next digit's here, X1, X2, and so on. And now I can imagine interleaving the digits in this windy path, similar to how we showed that the set of pairs of natural numbers is equinumerous with the natural numbers. Here, what I'm doing is interleaving all infinitely many real numbers, interleaving their digits in such a way that I get one real number out. And so this argument turns into a proof that the number of infinite sequences of real numbers is the same as the number of real numbers. So this set has the same size as the real numbers, because of this kind of argument.
▶︎ 14:14 But this set, the set of functions from the real numbers into the integers, it's at least as big as, in fact, we can see it's exactly as big as the functions, as the two-valued functions from the real numbers, the functions from the real numbers into the set zero and one only, if the functions always have value only zero and one. But this set is equinumerous with the power set of the real numbers. And so therefore, this set here, Z to the R, the functions from the reals to the integers, it's at least as big as the power set of the real numbers, which is strictly bigger than the real numbers. And therefore, the answer to the question of which of these sets is bigger is that there are way more functions from the reals to the integers than there are functions from the integers to the real numbers.
▶︎ 15:07 So this kind of analysis of the sizes of the various sets that arise is maybe the natural preliminary kind of steps that one takes when you're thinking about the continuum hypothesis. You want to look at all the sets that you know about and see if there are any candidate sets that might be in between the reals and the natural numbers in size. So let's think about another sort of instance, calculate. So let's think about.
▶︎ 15:38 Another sort of instance, calculate how many elements of a certain other natural set there are. And that is how many continuous functions are there on the real numbers. We know how many functions there are on the real numbers. There's a lot, because there's at least this many, and that's more, there are more functions from the reals to the reals than there are reals, precisely because the functions from the reals to the reals include all the functions from the reals to the two element set, and that's equinumerous already with the power set. So there's a lot of functions from the reals to the reals, but a lot of those are not continuous functions.
▶︎ 16:17 In fact, the only continuous functions with two values have to be constant. So let's think about how many continuous functions from the reals to the reals are there. Suppose I have a continuous function F from the real numbers to the real numbers. It has some graph, maybe like this. There's F. I can think about this graph. Maybe I have a good understanding of the graph.
▶︎ 16:49 And now let's think of the graph as sitting inside the plane, the real plane. And I could look at all the points below the graph. The set of points below the graph somehow determines the function, because the function will be the envelope of those set of points below. But I don't actually need all the points below. What I really want to do is just take the rational points below.
▶︎ 17:19 So I'm going to look at pairs of rationals, P, Q, such that they're both rational and it's below the graph of the function. So I'm looking at the part of the plane, the part of the rational plane that sits below the function. I claim that information is enough to recover the function, because it's enough to recover the function at any rational coordinate, because I'm going to know all the numbers that are below the function, and the limit of those numbers will be the value F. So I'll get the value of the function at all the rational points, but that's enough to determine the value of the function at all the points, because there's rational points converging to any other real number. So if I know the rational points below the function, then I will be able to recover the function.
▶︎ 18:12 So what I'm saying is, what do I call this? If I have a function F, then I'm associating with F a certain set of rational numbers, let's call it A sub F, and it's a subset of Q cross Q. So therefore, the number of continuous functions is less or equal the number of sets of rational pairs. But Q cross Q, well, Q is a countable set, and so that's equinumerous with N cross N, the natural numbers, and that's equinumerous with N itself.
▶︎ 18:48 So therefore, what we've shown is that the number of continuous functions is less or equal the power set of Q, which is equinumerous with the power set of the natural numbers, which is equinumerous with the real numbers. So what we've shown is that the number of continuous functions is bounded by the number of real numbers, but of course there's also at least as many continuous functions as real numbers, because for every real number I have the constant function. And so therefore, the number of continuous functions is exactly the same as the number of real numbers. So most functions, therefore, are not continuous, most in this sort of equinumerosity sense.
▶︎ 19:37 One can get quite good at calculating these kind of equinumerosity classes, and judging which sets are equinumerous with each other sets, and when are they less, and so on. And so gradually, we build up this stock of examples of sets, many of which are countable, others are equinumerous with the real numbers, and so on. And it's never the case that we find sets that are strictly intermediate. No one has ever defined a certain set of reals that's intermediate between the natural numbers and the real numbers, which is evidence for the continuum hypothesis. So let's turn now to Cantor's strategy.
▶︎ 20:16 For solving the continuum hypothesis, he wanted to start with the simple sets, and so one of the classes of simple sets that he started with was the closed sets. Cantor proves that every closed set of real numbers is either countable or equinumerous with the whole of the real line. In other words, no closed set is ever going to be a counterexample to the continuum hypothesis. Let's explain how that argument goes.
▶︎ 20:56 If you think about a closed set, then it might have some intervals in it, or it might be very complicated and so on, it might have some isolated points in it though. In the general case, maybe we have some isolated points, but if I have a lot of isolated points, say, in a bounded region, then they would have to have a limit point that occurs in it. Maybe I have a lot of sequences like that, and so if I have a closed set, let's call it C sub-zero, we're going to start with this set, and we undertake a certain process called the Cantor-Bendixson process, the Cantor-Bendixson derivative. We form the next set, C1, by casting out the isolated points.
▶︎ 21:47 For example, with this convergent sequence that I had here, this point is isolated, which means that it's sitting inside a little open neighborhood of which it's the only member, and this one also is sitting inside a little open neighborhood in which it's the only member, and so on. All of these points on the sequence are isolated, so we cast them out. But this limit point, every neighborhood of this point includes some of these other points converging to it, because it was a convergent sequence, it was a closed set, so we included this limit point. Therefore, the limit point itself is not isolated, and it won't be cast out. In this case, of a convergent sequence, C1 will have only this limit point remaining. And then in that set, it will be isolated.
▶︎ 22:31 Of course, we want a C2, we're going to iteratively cast out the isolated points. In the case of a convergent sequence, then initially, we have the convergent sequence with the limit point. When we cast out the isolated points, we get just a limit point, and now it's isolated in that set, and so it's cast out at the next step. But one can imagine more complicated kinds of sets.
▶︎ 22:53 For example, let me try to draw one. Maybe, say, I have three points here, and for each of these points, I have a convergent sequence converging to it, and here is a convergent sequence, and here is a convergent sequence converging to this point. Let me use some color here. I've got these three green points, and then the white points are converging to the green points, but then I've got now sort of epicycles. For each of the white points, I'm going to add a blue sequence converging to it. I've got now these blue points converging, and they're not interfering with each other, so every white point is the limit of a sequence of blue points. It's sort of hard to draw, but I hope you get the idea.
▶︎ 23:57 The set altogether is a closed set. It consists of these three green points, and each green point is a limit of the white points, and each white point is a limit of the blue points. The blue points are all isolated, because I said they're not interfering with each other, so every blue point, I can put a little open ball around it, and it's the only point from this set that's in that open ball. So they're all isolated. When I perform the Cantor-Bendixson process, at the first step, the blue points get thrown out, but the white points stay because they're not isolated, because at the previous level, they're limits of the blue points, they're not isolated.
▶︎ 24:33 At the first step, I throw the blue points out, and then I have only the green and the white points. But now, in that set, the white points have become isolated. And so they will be thrown out at the next step, but the green points will not. The second round, I'm throwing out the white points, but not the green points. And then finally, I have only the three green points, which are now isolated, and so they get thrown out.
▶︎ 24:58 One can make more and more elaborate such sets. I can imagine now adding red sequences converging to the blue ones, and so on, and it would make the whole process last one step longer before I get the empty set. And you can do that for every finite number N, you can produce a closed set which will last for N steps in this Cantor-Bendixson process. And in fact, by taking a set that lasts for one step, and then next to it you make a set that lasts for two steps, and next to that you make a set that lasts for three steps, and so on, and you take all those sets together, now you've got a set that will last for omega many steps before it becomes finished. And if you take that kind of picture in a convergent way and add a limit point to it, now you've got a set that will last for omega plus one many steps, and so on.
▶︎ 25:50 And so you see, in fact, in the Cantor-Bendixson process, this was the birth of the ordinals for Cantor. Cantor invented the ordinals precisely to make sense of this Cantor-Bendixson process. And now what Cantor observed is that in the Cantor-Bendixson process, whenever you're casting out isolated points, you can only be throwing out countably many. There can only be countably many isolated points in a given closed set in the real plane, say, or in the reals.
▶︎ 26:26 And the reason for that is that a point is isolated because there's an open ball around it. But we can take such a ball with a rational center and a rational radius, and that ball will never be used for another purpose because that point was the only point in that ball. So every time a point is thrown out, it's thrown out for this sort of countable reason, and therefore, we've only thrown out countably many points each time. And so then using that, Cantor proved that in countably many steps, at a countable transfinite ordinal stage, the process must end.
▶︎ 26:58 And therefore, every closed set of real numbers is equal to the union of a countable set together with a perfect set, which is a closed set with no isolated points. And Cantor knew that every perfect set, every non-empty perfect set, was equinumerous with the whole real line. So therefore, this shows that every closed set is the union of a perfect set and a countable set. Maybe that perfect set is empty.
▶︎ 27:29 So therefore, every closed set is either countable, because when you finish the Cantor-Bendixson process you get down to the empty set, and so the set just consisted of the countably many isolated points that you threw out. Or else, you got down to a set which was nonempty, in which case it's a nonempty perfect set, in which case it's equinumerous with the whole real line. So therefore, every closed set is either countable or equinumerous with the whole real line. In other words, the continuum hypothesis holds for closed sets.
▶︎ 28:02 This was generalized by Suslin to the Borel sets, which is a continuation of Cantor's program of working his way up in complexity, and I think analytic sets also by Suslin. But the project stalled at that point in the early 20th century. No one could show that more complicated sets had this property of always being either countable or equinumerous with the real line. One wants to maybe make the move to the projective sets, which are the sets of reals that you can define by a property that quantifies over integers and also real numbers. So there's a rich hierarchy of complexity there. And the Cantor program for proving the continuum hypothesis stalled at a relatively low level of the projective hierarchy.
▶︎ 28:58 Subsequent work showed in set theory in a deep way that actually if there are certain kinds of large cardinals, if certain kinds of extremely strong infinities exist, then in fact, Cantor's program continues throughout the projective hierarchy. Every projectively definable set of reals will be either countable or equinumerous with the whole real line. And that work can be seen as a kind of fulfillment of, or eventual continuation, I mean, it happened only decades later in the '60s and '70s and '80s, I guess. So the full result fulfills Cantor's idea that we're going to show the continuum hypothesis is true by working our way up in complexity. But even with that result on the projective sets, it's not finished. It doesn't show that every set of real numbers is either countable or equinumerous with the whole real line.
▶︎ 29:50 So how are we left? Well, the situation is that on the one hand we have many instances of sets that are countable. The integers, the natural numbers, the rationals, the finite binary sequences, the integer polynomials Z of X, the set of algebraic real numbers, the computable real numbers, those sets are all countable. And on the other hand, we also have many instances of sets of reals of size continuum, the intervals, the non-trivial intervals, the open sets, the uncountable closed sets, so also the power set of the natural numbers, the space of continuous functions, and so on, those sets are all size continuum, equinumerous with the reals. And we don't have any natural example of a set of real numbers that we can define that we think might be intermediate in size. And so this is how matters stood for a long time after Cantor.
▶︎ 30:49 The question just seemed hopelessly difficult to answer. Then finally, in 1938, the great logician Kurt Gödel made a remarkable observation. Kurt Gödel proved that it is consistent with the axioms of set theory that the continuum hypothesis is true. What he did was he gave us a model construction method. So he said, well, if we have a model in which the axioms of set theory are true, the Zermelo-Fraenkel axioms of set theory, then Gödel described a way of describing a kind of submodel of that model, and then he proved that, in that submodel, it's known now as the constructible universe, or L, Gödel's L, the constructible universe has the property that the continuum hypothesis is true there.
▶︎ 31:45 Also, the axiom of choice, he proved is true there. So even without assuming the axiom of choice in the ambient set theory, the axiom of choice is true in the constructible universe. And so this result shows simultaneously that if the axioms of set theory are consistent without the axiom of choice or the continuum hypothesis, then they're also consistent with both the axiom of choice and the continuum hypothesis being true. It's quite a remarkable achievement.
▶︎ 32:12 And so at bottom, this model construction method is very similar to the logic underlying that which occurs in geometry. When, for example, you consider the parallel postulate in geometry and the question of whether it's possible to prove it from the other axioms of set theory, this was eventually observed not to be the case, by doing a similar kind of model construction method. If you say, take the Poincaré disk model of hyperbolic space, of hyperbolic geometry, then that is a model of geometry that you can define relative to Euclidean geometry, and you can prove that all of the other axioms of geometry are true in the Poincaré disk, except for the parallel postulate. It's not true there.
▶︎ 33:01 So it's very similar to what Gödel did. Gödel said, if the axioms of set theory are consistent, if we have a model of this theory, then we can build another model inside it by interpreting our notions slightly differently. And in that model, the continuum hypothesis is true. In the Poincaré disk, we provide another alternative conception of geometry in which the parallel postulate is false. Of course, that's not the only model that you can build in geometry, but that analogy is quite robust.
▶︎ 33:31 So what does it mean? Have we answered? Does Gödel answer the continuum hypothesis question? He proves that it's consistent, and how is that different from proving that it's true? Well, of course, to know that it's consistent or even relatively consistent, is just showing us that we can't refute, we shouldn't expect to refute the continuum hypothesis. There can't be any proof of the negation of the continuum hypothesis from the other axioms of set theory unless those axioms are inconsistent themselves, because Gödel showed, if the axioms of set theory are consistent, then they are consistent with the continuum hypothesis being true.
▶︎ 34:13 So there can't be a refutation. So what Gödel showed, in other words, is that you cannot prove the continuum hypothesis is false unless you can already prove a contradiction from the axioms of set theory without any extra assumption. So it doesn't quite prove that the continuum hypothesis is true. It's rather just showing that you can't prove that it's false, and that's not the same thing. So there matters stood for another 25 years.
▶︎ 34:37 So we had Gödel's result in 1938 showing that the continuum hypothesis is relatively consistent with the other axioms of set theory, but it wasn't known whether it was a theorem or not. Cohen showed that it is not. He showed the analogous result for the negation of the continuum hypothesis. Namely, if the axioms of set theory are consistent, then they are consistent with the continuum hypothesis being false.
▶︎ 35:01 So the situation is just like the parallel postulate in geometry. We can't prove it from the other axioms of geometry, and we also can't refute it from the other axioms of geometry. Similarly, in set theory, we cannot prove, we cannot expect to prove that the continuum hypothesis is true, and we also cannot expect to prove that it is false, because if the axioms of set theory are consistent at all, then it's consistent with them that CH is true, and it's also consistent with them that CH is false.
▶︎ 35:27 So this latter result was proved by Cohen with the method of forcing, which is also a model construction method. Whereas Gödel had started with a model of set theory and built an inner model in which the axioms of set theory were true with the axiom of choice and also continuum hypothesis, Cohen's argument went in the other direction. Given a model of set theory, he built an outer model, a larger model in which the continuum hypothesis failed. And that method has been subsequently developed far beyond Cohen's applications of it.
▶︎ 36:03 We now have hundreds, maybe thousands of forcing arguments in set theory, and so the situation is that not only is the continuum hypothesis independent. To be independent means you can neither prove it nor refute it from the other axioms if those axioms are consistent. The situation is that almost every statement, almost every non-trivial statement in infinite combinatorics in set theory has been recognized to be independent of the axioms of set theory. So, we have a pervasive, ubiquitous independence phenomenon in set theory. Hundreds, thousands of statements being independent, having this exact same status as the continuum hypothesis, with respect to the other axioms of set theory.
▶︎ 36:48 Gödel had hoped to settle the continuum hypothesis, this sort of independence problem, by means of adopting extremely strong axioms of infinity, so he had hoped that the large cardinal axioms would settle the continuum problem. We have different concepts of infinity going way beyond the continuum, so the concept of inaccessible cardinals, Mahlo cardinals, weakly compact cardinals, Ramsey cardinals, measurable cardinals, supercompact cardinals. There's this enormous hierarchy of strength, these strong axioms of infinity, they're widely recognized now as amongst the strongest known axioms in mathematics.
▶︎ 37:30 Regarding Gödel's hope to settle the continuum hypothesis on the basis of the large cardinal axioms, these hopes were basically dashed by the Lévy-Solovay theorem, which shows that none of the large cardinal axioms that we know about currently can settle the continuum hypothesis. The argument uses Cohen's method of forcing, because we can show that all of the large cardinal axioms are preserved by what's called small forcing extensions, forcing using a partial order that is of size less than the large cardinal in question. And now, the relevance of that result is that you can turn the continuum hypothesis on and off by going to a forcing extension and that forcing is a small forcing extension.
▶︎ 38:17 And so therefore, whatever strong axioms of infinity are true, you can make CH true or false, and true, false, true, false, in an iterated succession by forcing arguments that will preserve the large cardinals. It can't be the case that those large cardinals imply CH is true or that they imply CH is false precisely because of this. In other words, the continuum hypothesis is not only independent of ZF, and ZFC, but it's independent of the extensions of ZFC that arise by adding whatever kind of large cardinal axioms you want for any of the known large cardinal axioms.
▶︎ 38:57 How are we to settle the CH question? Is it true or not? Or maybe, actually, there's a certain philosophical perspective that's embedded in that question, the way of asking it. Do we think that there's a fact of the matter about whether the continuum hypothesis is true or not? And so this brings me to the debate on pluralism in the foundations of mathematics, and particularly the foundations of set theory.
▶︎ 39:23 One philosophical attitude that you might have about the foundations of set theory is that there is a unique set theoretic reality for mathematics that we are trying to axiomatize with our axioms of set theory. There is a unique cumulative hierarchy of sets that one achieves by starting with nothing and then constructing in a sequence of levels by adding all possible subsets of earlier elements. So, we build up the cumulative hierarchy of all sets in this manner, and in that way, we arrive at the unique intended model of set theory, the set theoretic universe. And in that universe, either the continuum hypothesis is true there or it is not true. And the fact that ZFC, that the Zermelo-Fraenkel axioms don't settle it, or even with the large cardinal axioms, is just telling us about the weakness of those theories rather than about what's really the case.
▶︎ 40:22 So, on this view, the universe view, one holds that every set theoretic question and maybe indeed every mathematical question, if we view those mathematical assertions as being interpreted in set theory, every mathematical statement will have a final definitive truth value in this one intended set theoretic universe. That's the universe view. So if you hold the universe view, then you think there is an answer to the question about whether CH is true or false, and we're trying to discover what it is, and maybe the further question to ask is, well, by what means would we come to such an answer?
▶︎ 41:04 An alternative perspective is known as the multiverse view, or partaking in set theoretic pluralism, and this is the idea that there are multiple distinct concepts of set. There isn't just one set theoretic reality, but rather multiple independent set theoretic worlds, and some of them have the continuum hypothesis being true, and some of them have the continuum hypothesis being false. And the multiverse view proceeds from the idea that the central discoveries of set theory in the past half century, in the latter part of the 20th century and since that time, are all about constructing these different models of set theory so as to exhibit different combinations of truths. We want CH to be true, or we want it to be false, but also we want Martin's axiom true, or we want Suslin trees to exist or to not exist, and so on. So, we take all of these questions that arise in set theory, and what set theorists do nowadays is they build a model of set theory exhibiting exactly that pattern.
▶︎ 42:06 And one can think about that situation as leading us to the view that actually there are multiple distinct concepts of set, there are different kinds of ways of thinking about what sets are, and those different models of set theory are providing for coherent, possible pictures of the way that set theory might be. And the point is that they're incompatible with one another, and so from this point of view, maybe there isn't a unique answer to the question about whether CH is true or false.
▶︎ 42:40 Sometimes it's true with some conceptions and false in other conceptions. There's a certain kind of strategy for solving the continuum problem, the dream solution. This is what many set theorists hope for. What they want to find is a certain natural principle that is obviously true in the intended sense of set theory, that's manifestly the case. They want to identify the missing axiom. We want to find an axiom which we all agree is fulfilling our main conception of the concept of set, but which also has the property that it implies CH or that it implies not-CH.
▶︎ 43:26 In other words, we want to find the missing axiom that settles CH. This is the dream solution. This is how the universes maybe hope to settle the continuum problem. They want to find the missing principle, everyone agrees that, "Yeah, that should be an axiom," and then we prove from that missing principle that the continuum hypothesis goes this way or that way. It settles the question.
▶︎ 43:52 I've argued, however, that this is a mirage. It will never happen. And the reason why the dream solution is impossible is because in the past half century, we have come to have this enormous familiarity with all of these different models of set theory that were constructed by the forcing method and by the inner model constructions, and they all seem completely set theoretic. They are perfectly reasonable set theoretically. Some of them have CH being true, and some of them have CH being false, and we have a deep understanding of what it's like to live in those worlds where the continuum hypothesis is true and what it's like to live in those worlds where the continuum hypothesis is false.
▶︎ 44:37 So the situation isn't just that, look, CH is independent and we don't know which way it's going to go and we don't know anything more. Rather, we have an extremely deep experience of living in these mathematical worlds which have the different outcomes for the CH question. And because of that, if someone were to propose a principle that settled CH one way or the other, it would directly contradict our experience in those contrary worlds. We could never accept that principle as manifestly true for sets precisely because we have the experience now of the contrary situations, and that experience shows that those conceptions of set are perfectly reasonable and fully set theoretic. There's nothing wrong with those conceptions. So this is why I've argued that the dream solution is impossible.
▶︎ 45:35 So let me just read a quote from my book on this matter. What I wrote was, "Our situation with CH is not merely that it is formally independent and we have no additional knowledge about whether it is true or not. Rather, we have an informed, deep understanding of how it could be that CH is true and how it could be that CH fails. We know how to build the CH worlds and the not-CH worlds from one another. Set theorists today grew up in these worlds, comparing them and moving from one to another while controlling other subtle features about them. Consequently, if someone were to present a new set theoretic principle, phi, and prove that it implies not-CH, say, then we could no longer look upon phi as manifestly true for sets. To do so would negate our experience in the CH worlds, which we found to be perfectly set theoretic. It would be like someone proposing a principle implying that only Brooklyn really exists, whereas we already know about Manhattan and the other boroughs."
▶︎ 46:35 On a final note, I want to draw this analogy with geometry, again, a little more fully. Are we monist or pluralist about geometry? I think almost everyone today is a pluralist about geometry. We recognize that there are different kinds of geometry. There's Euclidean geometry and there's various kinds of non-Euclidean geometry, spherical geometry, hyperbolic space, and so on, and they're all fully real. Euclidean geometry is real.
▶︎ 47:07 One can be a Platonist about Euclidean geometry and also a Platonist about non-Euclidean geometry, and the geometers come to have a deep understanding of the nature of, say, hyperbolic space and moving around. What is it like to walk around in hyperbolic space? You can find videos on YouTube that show you actually the sort of bizarre visual experiences that you would have in those kinds of geometries, and the top geometers have extremely deep insight into the nature of those alternative geometries that help them to realize certain facts are true and to prove theorems about them. So these different geometric conceptions give rise to fully real mathematical realities that instantiate those different perspectives on geometry, and what I'm arguing in this pluralist view and the set theoretic multiverse is that it's exactly the same situation in set theory.
▶︎ 48:01 In geometry, for thousands of years, there was the view that what geometry was about was the one true geometry of space. That's what the subject of geometry was about. But with the discovery of non-Euclidean geometry, that perspective splintered into a spectrum of different possible conceptions of geometry, which are now taken as fully real, and that's the nature, the pluralist nature, of geometry. Similarly, in set theory, maybe previously there was the idea that what set theory was about was the one true set theoretic universe, but now we have come to recognize that actually that concept splinters and what we have and what we're faced with is this diversity of different set theoretic conceptions, alternative conceptions that come to different answers as to the CH question and many, many other questions, and they're all fully real concepts of set.
▶︎ 48:56 And in a sense, this multiverse picture provides an answer to the continuum hypothesis question. The answer to the CH question is that it depends on whether you're in a set theoretic universe, where it holds or where it fails. It holds in some and it fails in others, and we have a deep understanding of how to transform those universes from one to another with the method of forcing or by going to inner models in just the same way that we have a deep understanding of how the parallel postulate works in geometry.
▶︎ 49:28 So I hope you've enjoyed this discussion of the continuum hypothesis and the issues involved with set theoretic pluralism. Thank you very much.