How Can Some Infinities Be Bigger Than Others?

The idea of infinity is probably about as old as numbers themselves, going back to whenever people first realized that they could keep counting forever. But even though we have a sign for infinity and can refer to the concept in casual conversation, infinity remains profoundly mysterious, even to mathematicians. In this episode, Steven Strogatz chats with his fellow mathematician Justin Moore of Cornell University about how one infinity can be bigger than another (and whether we can be sure that there isn’t an intermediate infinity between them). They also discuss how physicists and mathematicians use infinity differently and the importance of infinity to the very foundation of mathematics.

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Transcript 

Steven Strogatz (00:03): I’m Steve Strogatz, and this is The Joy of Why, a podcast from Quanta Magazine that takes you into some of the biggest unanswered questions in math and science today.

(00:13) In this episode, we’re going to be discussing infinity. No one really knows where the idea of infinity came from, but it must be very ancient — as old as people’s hopes and fears about things that could conceivably go on forever. Some of them are scary, like bottomless pits, and some of them are uplifting, like endless love. Within mathematics, the idea of infinity is probably about as old as numbers themselves. Once people realized that they could just keep on counting forever — 1, 2, 3 and so on. But even though infinity is a very old idea, it remains profoundly mysterious. People have been scratching their heads about infinity for thousands of years now, at least since Zeno and Aristotle in ancient Greece.

(00:57) But how do mathematicians make sense of infinity today? Are there different sizes of infinity? Is infinity useful to mathematicians? And if so, how exactly? And what does all this have to do with the foundations of mathematics itself?

(01:14) Joining me today to discuss infinity is Justin Moore, professor of mathematics at Cornell. His research interests include set theory, mathematical logic and infinite combinatorics and their applications to other fields of math, such as topology, functional analysis and algebra. Welcome, Justin.

Justin Moore (01:33): Hey, Steve. Thanks for having me.

Strogatz (01:35): Yeah, I’m very excited to talk to you. I should say, maybe for full disclosure, Justin is my friend and colleague in the math department at Cornell. OK, so off we go, then, to thinking about infinity as mathematicians think about it. Actually, maybe before we dive into the math part, let’s just talk for a second about the real world, because we won’t be there for long. Now, am I right, that you were once trained in the world of physics?

Moore (02:02): Yeah, it was a physics double major with math, when I was an undergraduate. I kind of got burned out on physics. I started out favoring physics and also being somewhat interested in math more recreationally. And then somehow, through the course of it, I got more interested in the math and physics.

Strogatz (02:18): OK. Well, what about the physics of infinity? Does it even make sense? Is there any infinite stuff in the real world that we know of?

Moore (02:26): You know this video, The Powers of 10, that was created by Charles and Ray Eames? Where basically every — I think it’s every 10 seconds, you are a power of 10 smaller. Well, at first, I think a power of 10 bigger. You zoom out. And then every 10 seconds, you’re a power of 10 smaller, and you go from the largest scale of the universe down to the smallest scale of subatomic particle. You know, this was made back in, I want to say, the late ’70s or early ’80s. And I think our understanding of some things has evolved a little bit since then, but not tremendously. But I mean, the point is, there are about 40 powers of 10 that separate the smallest scale of length from the largest scale of length, and maybe you can be generous and throw in several extra powers of 10, just for good measure. But it’s fair to say that there’s nothing that you can measure in physics that is larger than, you know, 10¹⁰⁰ or 10²⁰⁰ or something like that.

(03:22) And maybe our concept of things being continuous — continuous motion or whatever— maybe this is all just an illusion. Maybe everything is really granular and finite. But what is true is that certainly physicists have discovered a lot about the world we live in, by imagining that things are smooth and continuous, and that that infinity makes sense. When you go into the parts of physics where they haven’t yet really formalized things, a lot of the issues that mathematicians have with this boil down to the physicists are sort of treating infinity in various sort of cavalier ways, and subtracting infinities from infinities, and maybe not being as accountable for it as a mathematician would like them to be. I don’t think that that’s really a controversial statement. I think a physicist would — most physicists would probably — I mean, OK, maybe you would know better. But I believe most physicists would say that that’s a fairly correct statement.

Strogatz (04:20): So, in terms of your own personal story — I promise I won’t go too deep to embarrass you on this — but what was it that drew you to infinity? Was it somehow that physics felt too small for you? Or you just like the rigor of math, or…?

Moore (04:33): I mean, I think I got interested in math as a whole and grew away from physics before I got interested in set theory specifically. Ironically, it was because I — well, if you take a physics class, at some point, you end up being fairly fast and loose with the mathematics. And you’re either OK with that, or you’re not. I was one of the people who wasn’t OK with that.

Strogatz (04:56): Huh. And I was one that was OK, and I’m still doing it. You know, I mean, those things haven’t worried me too much, although I do respect the care that — the intellectual integrity that pure mathematicians have, you know, worrying about these things.

(05:11): OK, so suppose I were just, I don’t know, like a curious teenager, and I don’t even know what infinity is. What would you say it is? Should I think of it as a very big number? Is it some symbol? Is it a property? What’s a good way to think about what infinity is?

Moore (05:26): Yeah, I mean, I guess it’s — it can be an idealized point at the end of the line, all right? It can be a formal symbol. You know, you can think of it kind of the way … a formal symbol in the same sense as say, we introduce -1, right? And I remember when I was a little kid, that teachers wouldn’t be willing to make it clear whether it was safe to talk about negative numbers. And, right, that sounds silly in hindsight, but at some level, right, does -1 exist in the real world? But you can formally manipulate it and you can formally manipulate infinity at some level, but you have to maybe exhibit a little bit more care. You can also use infinity as a means of quantifying how many there are of something. And that opens more doors there, because you can talk about there being infinite sets, some of which are larger than others.

Strogatz (06:15): OK. All right. So you’ve mentioned this word “sets,” and we’re going to certainly be talking a lot about sets today. I did say that your interests include set theory. Do you want to say any more about what you mean by a set?

Moore (06:26): I guess I… The answer is both yes and no. So I think it’s OK to fly by the seat of one’s pants and just view it as just, you know, an undefined notion and use it kind of intuitively. But it also was sort of used as a mechanism to provide the foundations for mathematics, when people realized that we needed to have some, make some careful foundation of what mathematics is.

Strogatz (06:49): Uh huh. That’s interesting. Because I — so like, as little kids, we learn to count on our fingers, or our parents probably start saying words, and then they might point at things and say, “1, 2, 3…” And we learned sounds — kids like that when they’re very little, I know, right? I mean, if you have little kids yourself, or relatives. So there’s that side of things. And I think most people would imagine that numbers are the foundation of math. But you’re saying, and I think most mathematicians would agree, that there’s something even deeper than numbers, which is this concept of sets, right?

Moore (07:22): I think the concept of “set” came about as a foundational concept because it’s so basic and so primitive. And if you’re, if you’re wanting to have something to use as a fabric for mathematics, you want to start out with something where its basic properties seem very primitive, and then start from there. And then the idea is that you then use sets to encode things like the counting numbers, and things like the rational numbers, and the real numbers, and so on. And then from there, all sorts of other more complicated mathematical constructions, like manifolds, or, or whatever.

Strogatz (07:57): So I can remember, in a Sesame Street episode that I used to watch with my kids. It was in a movie; I think it was. That there’s a character who was ordering fish for a room full of hungry penguins. And he asked the penguins to call out and, and they say, “Fish, fish, fish, fish, fish, fish.” And so then the waiter calls down to the kitchen, “Fish, fish, fish, fish, fish.” And then somebody else says, “No, you got that wrong.” And someone else says, “Well, why didn’t you just say they ordered six fish?” But it makes the point that this idea of a number kind of comes after this collection of objects of fish. And then another character is surprised and says, “Does it work for spark plugs? And cinnamon rolls?”

Moore (08:42): I mean, I think also, it’s just if you’re interested in trying to understand, can you prove this? Or can you prove that? And you’re trying to set up the rules for how you would prove things or whatever, you would like to have the basic principles be as simple as possible. And so rather than try to write down rules for how arithmetic works, you start by writing down simpler rules for simpler things, and then build arithmetic out of these more basic building blocks.

Strogatz (09:08): OK. So then, and this reminds me of “New Math,” too, when as a kid in the ’60s, we used to be learning about intersections and Venn diagrams and unions, right? That was the beginning of set theory. They were teaching it to us in — I don’t remember — it was second or third grade; my parents didn’t know why. But it was, I guess, mathematicians of your type, or others who thought kids should learn sets, either before or at the same time that they’re learning about arithmetic.

Moore (09:33): Yeah, most of what people study in set theory, I mean, these days is really how infinite sets work. Because our intuition about infinite sets isn’t as good as our intuition about finite sets. And I think that’s a lot of why the drive for foundations was there. It was in part because we would like to write down, OK, what are we fairly sure should be the properties of infinite sets and sets in general, and then try to develop what is true about infinite sets from there?

Strogatz (10:03): OK, so why don’t we have a few examples? Can you tell me some examples of things that are infinite sets?

Moore (10:08): Well, like the natural numbers. Like you were saying — like 1, 2, 3, 4, 5, 6, 7, 8 and so on — but also things like the rational numbers. You know, fractions like two natural numbers over each other, or maybe a negative fraction. But then there are also things like the real numbers, where — you know, anything that you can express with a decimal, including things like pi and e.

Strogatz (10:28): Mm-hmm. So they could have infinitely many digits after the decimal point.

Moore (10:32): Yeah, yeah, infinitely many digits. They don’t have to repeat.

Strogatz (10:35): Uh huh. And what about things like shapes or points or geometric things, not just numerical things?

Moore (10:41): Yeah, you can talk about collections of geometric shapes too.

Strogatz (10:45): OK, so this is a nice feature of sets: that we can, with sets, unify or at least have a common language for talking about arithmetic, geometry, … .

Moore (10:54): Right.

Strogatz (10:55): I suppose we could talk about a set of functions, if we were taking a precalculus course. You know, like the set of the set of continuous functions, if we were in a calculus course.

Moore (11:04): Sure. Yeah.

Strogatz (11:05): Or whatever. So yeah, so this gives us a common language for all different parts of math.

Moore (11:09): Right.

Strogatz (11:10): And — but it’s a relatively new idea as a foundation of math in terms of the overall history of math, wouldn’t you say?

Moore (11:16): Yeah, I mean, I… Well, modern mathematics as we know it, it’s about somewhere between 100 and 150 years old. But I usually associate it around — the first part of the last century was when, really, we started to see all of the major parts of mathematics as we know them today start to develop and really become distinct subjects of their own. And that was also around the same time as [Bertrand]Russell discovered his paradox, which spurred the need for some sort of rigorous foundations for mathematics.

Strogatz (11:49): Uh, huh. We should mention — yeah. So Bertrand Russell, we’re talking about now, is often better known as a philosopher or a pacifist, and yet he was quite a strong mathematician and logician, someone interested in logic as a part of math.

Moore: Yeah, yeah.

Strogatz (12:04): So as you say, he was one of the people who helped get set theory really rolling. And even before him, there was this gentleman, Georg Cantor, who we’ll be talking about quite a bit, in Germany in the late 1800s.

(12:17): OK, so how within math, let’s say, do mathematicians use infinity? You mentioned how helpful it can be. Where does it get used?

Moore (12:27): Yeah, so, in a calculus class, it’s a useful symbol for doing certain calculations. Talking about how a function behaves as the input becomes very large. You can talk about the limit at infinity, or ratios of quantities as a number goes to zero or infinity or something like that. That’s a notion of infinity that’s kind of in the first sense that I mentioned, where you view infinity as an idealized point at the end of the line.

(12:53) But you can also talk about it as — you know, you can, you can talk about counting the number of elements of some collection or some set, and keeping track of either how finitely many elements it has or maybe, if it has infinitely many elements, trying to distinguish between different sizes of infinity. I mean, everyone understands — or pretends to understand — the distinction between being finite and being infinite. And I think Cantor’s remarkable discovery was that you can, for an infinite set, you can make further distinctions. You can distinguish between it being what’s called countable and then what’s called being uncountable. Or even just in general, higher uncountable cardinals than distinctions between different uncountable cardinals.

Strogatz (13:34): So OK, let’s go there. Because this is, this really takes us into the heart of our subject. I think the average person hearing the word “countable” for the first time might think it means literally countable, like something that has 10. You know, if there’s 10 spark plugs on the table, I could count them — 1, 2, 3, up to 10. But you and other mathematicians use countable to mean something a little different than that.

Moore (13:56): It just means that you can assign a natural number to each element of the set so that no natural number gets used twice.

Strogatz (13:56): So something can be countable and infinite.

Moore (13:57): And infinite. So the natural numbers are obviously countable because they count themselves. But maybe a little bit less obvious is that the integers including the negatives of the natural numbers, that those are countable.

Strogatz (14:18): So let’s talk about that for a second. So if a person hasn’t thought about that before, it’s interesting. Because like — so you said, you’re going to consider all the numbers, all the positive integers, all the negative integers and zero.

Moore (14:29): Yeah.

Strogatz (14:30): And you could do it wrong. Like if you started at zero and started counting to the right, and you go 0, 1, 2, 3, you’d never get back to the negative numbers. And so then you would have failed to count all the integers.

Moore (14:41): Yeah.

Strogatz: But what should you do instead?

Moore: What you can do is, you can count, you know, 0, 1, -1, and then 2, -2, 3, -3, 4, -4, 5, -5. And if you list them in this way, then you eventually list everything.

Strogatz (14:55): Beautiful. So this zigzagging argument where you’re hopping back and forth between the positives and the negatives is a nice, organized, systematic way to show that if you think of any integer, eventually it will be on the list.

Moore: Yeah. Yeah.

Strogatz(15:07): So that’s great. So OK, so the integers are countable. Cantor also discovered some other things were countable that were — I don’t know if he was surprised, but a lot of us are surprised when we first learn about it. Like, like what?

Moore (15:21): Yeah, I think two good examples that are surprising are the — first, the rationals. So the collection of all fractions of two integers are countable. That’s actually pretty easy to see when you, when you think about it, because you can just list all fractions with denominator 1 — or numerator and denominator absolute value at most 1. And then, at most 2, at most 3, at most 4. And at each stage, there are only finitely many fractions where the numerator and denominator are at least in magnitude at most n. And then you can exhaust all of the rationals that way.

Strogatz (15:55): So like, if I were picking the number n to be 3, you’re saying I could have a number like 1/2 or 2/1, or 0/3, because the numerator plus denominator add up to 3?

Moore (16:06): Yeah. Another one, which is, again, kind of surprising, is if you take the number of words that you can write down in the Latin alphabet, or any alphabet that you’d like. There are at most countably many finite words, or finite strings of symbols coming from this alphabet. If you think about all words or all sentences, all pieces of literature, if you like —

Strogatz: Ooh.

Moore (16:30): — anything which not only exists now but could potentially exist at some time in the future. You know, you put those infinitely many monkeys at the typewriter and look at what the outputs are that they could generate in a finite amount of time. That’s all just a countable set.

Strogatz (16:44): Wow. So all possible books in all, let’s say, in Latin, in all possible languages that we know?

Moore (16:50): In all possible languages. Yeah. I mean, if you even like, you can have a countable alphabet if you like. That doesn’t make anything any larger.

Strogatz (16:56): So countable would seem like a very big infinity. And yet —

Moore (16:59): Yeah. The first surprising thing is that those sets that seem to be larger than the natural numbers actually are the same size as the natural numbers. They’re countable. But then there’s the other surprise, which is that the real numbers, the set of decimal numbers, are uncountable.

Strogatz (17:13): So there’s this remarkable point that you’ve been mentioning that there can be sets that are not countable. And I guess, maybe the simplest example would be: Think of a line that goes off to infinity in both directions. So like an infinitely long, straight line. The real line as we would call it. That is uncountable.

Moore (17:32): Right. If you, if you hand to me a list, a purported list of all the elements on that line, there’s a procedure called the diagonal argument, which allows you to produce a new point that’s on the line, but not on your list. That was Cantor’s famous discovery.

Strogatz (17:49): So that was a really totally astonishing discovery, I guess at the time, right? That now you could suddenly talk about two infinite sets and compare them.

Moore (17:58): Yeah, yeah. And the distinction between countable and uncountable is a really useful one in math. Basically, countable sets, you can still talk about sums which are of countably infinite length. That’s something which gets taught at the end of a standard — end of a second semester calculus course. Whereas sums over uncountable sets are less meaningful, or at least you have to define them in a more delicate way. That said, something more along the lines of an integral or something like that.

Strogatz (18:30): OK, so now that we have this distinction of countable, like the whole numbers — 1, 2, 3, 4, 5 — and uncountable, like the points on a line. There’s another question which I think would be good if we could spend some time on that. Called the continuum hypothesis. Could you, could you tell us what that is?

Moore (18:50): Yeah. So Cantor wondered: Is there, is there something in between? You can — you know, the natural numbers sit inside the real numbers, and the natural numbers are countable. The real numbers are uncountable and larger than the natural numbers. Is there a set of real numbers which is larger than the natural numbers, but smaller than the —

Strogatz (19:10): Smaller in this sense of counting.

Moore (19:12): — smaller than the line? Is there a set of points on that line, on the number line, which is larger than the natural numbers, larger than the rationals, but smaller than the whole line itself? The assertion that there is no such intermediate set is called the continuum hypothesis. And that was Hilbert’s first problem, whether the continuum hypothesis is a true or false statement.

Strogatz (19:35): Uh huh, so Hilbert was a great mathematician of this — maybe a little bit later generation but not much later. And in the year — what was it, 1900 or so, I think — he announced or gave a list of what he thought were some of the greatest problems for the future, at point 20th-century mathematicians to work on. And I think this was the number one question on his list?

Moore (19:58): Yeah, this was the number one question.

Strogatz (20:00): Wow. So it was big to think about this. Cantor, you say, called it a hypothesis. He thought was going to turn out to be true.

Moore: Yeah.

Strogatz (20:07): That there was no infinity sandwichable between those two that he already knew about

Moore (20:11): Yeah. And the thing is, it survives the test of looking for counterexamples. I mean, if you start looking at all of the sets of reals, subsets of the line that you can write down a description of or that you can construct by some means. He tried this. And he proved, I mean, well, he showed that there aren’t counterexamples. There are even theorems early on that say that sets of this or that type can’t be counterexamples.

Strogatz (20:40): That’s amazing. Let me make sure I get this. I’ve never heard this statement: Just the mere fact that some of them are describable makes them, in a sense, not good enough.

Moore (20:49): For instance, a set which is closed has all of its limit points. Cantor proved that this can’t be a counter example. It’s either countable or it has the same size as the reals.

Strogatz (21:00): So if it if there is a counter example, it has to be indescribable.

Moore (21:04): Yeah, it has to be complicated.

Strogatz (21:06): Wow. But of course, it’s possible there is one, just that it’d be some really bizarre thing.

Moore (21:12): Yeah. So that kind of brings us to something that’s getting back to this foundational question. You know, around that time they were starting to try to formalize what the axioms for mathematics were. And sometime later, around the — in the 1930s, [Kurt] Gödel proved that actually any sort of intelligible axiom system that you might have that attains the modest goal of formalizing arithmetic on the natural numbers, is necessarily incomplete. There are statements that you can’t prove from this axiom system, and you can’t refute them from the axioms, using standard finite proofs.

(21:52) And this was, I think, pretty shocking. Because it tells you that the goal of somehow algorithmically trying to settle all your problems in mathematics and produce some sort of algorithmic foundation, some complete foundation of mathematics is, in some sense, doomed. Or at least has to be governed by some higher intuition beyond just — I don’t know — what was available at the time.

(22:16) And what Gödel proved — one of the things that he proved later was that one of the statements that you can’t prove or refute is the statement that your axiom system is consistent in the first place. That it doesn’t lead to any contradictions. That statement can be coded as some sort of statement about number theory, about arithmetic on the natural numbers, but not in a particularly natural way. If you go and talk to one of the number theorists in the department, they wouldn’t regard that as a problem or a statement of number theory, even though technically it is. And so it was — a question that was left from Gödel’s time was whether the continuum hypothesis — or whether there’s some other natural mathematical statement, which is undecidable based on the axiom system that we were working within.

Strogatz (23:02): So there’s this concept of axioms. We should probably try to remember what those look like. Because if we’re doing very careful math, we have to lay down some definitions, but also some things that we take — I don’t know why I don’t want to say “we take for granted,” but that we accept as bedrock.

Moore (23:19): Yeah, yeah. So this is, I mean, this is something that the Greeks did that, that was, you know — one of the achievements in formalizing geometry — was to, rather than try to define what geometry is, sort of view it as: You are going to write down a few undefined terms, and then write down the rules or axioms that govern how these undefined terms behave. For them, it was things like a point and a line. And when a point is on a line, those are the undefined concepts. And when a point is between two other points on a line, those are undefined concepts. And then you write down a set of axioms that govern how these concepts work. And if you’ve done it right, then everyone agrees that these properties are obviously true of these, these things. And so therefore, these axioms are things which are sort of self-evidently true.

(23:19) So for geometry, you know, there’s this famous parallel postulate, which — you couldn’t derive it from the other ones. And it was somewhat revolutionary, when it was discovered that you can actually construct models of geometry which satisfy all of the axioms but not the parallel postulate. And therefore, the parallel postulate’s not provable from the other axioms. So in some sense, what Gödel had done is develop a method for doing that, but at the level of models of mathematics, or at least models of this axiom system that we have for mathematics.

Strogatz (24:45): Aha, that’s an interesting way to say it. So, like, where we have Euclidean geometry and then we also have these more newfangled non-Euclidean geometries which, famously, Einstein used in general relativity, but they get used in other places too. And they’re logically as good as Euclidean geometry. But now instead of just talking about geometry, you’re saying it’s sort of like we could have the traditional — well, I’m not sure what the words are. What’s the analogue of Euclidean geometry? Is there traditional mathematics?

Moore (25:16): That’s an open question. I mean that, I mean — I think it’s partly a philosophical question. Maybe it’s a sociological question, because it’s a matter of what is mathematics, right? It comes back to that basic question. And I think that the axioms that we have the ZFC axioms which were developed a bit over 100 years ago, are ones which we generally agree that these are true, or these are, these are properties that “set” should have, but they’re not complete.

Strogatz (25:44): Well, wait, let’s unpack all of that. That sounds good. So ZFC, why don’t we start with that? Those are the names of some people and a thing.

Moore (25:51): Yeah, yeah. “Zermelo-Fraenkel set theory” with something called the “axiom of choice.” Yeah.

Strogatz (25:55): OK. And so those are rules of the game that are widely accepted.

Moore (25:59): Yeah, it’s a list of axioms that are — it’s rather lengthy, but not that lengthy. Things like, if you have two sets, there’s a set which has both of them as their, their elements. The pairing axiom, that you can take the union of a collection of sets, and that’s a set. And so on.

Strogatz (26:15): OK. So there’s the ZFC way of doing set theory, and that’s, you say, proposed at a certain time and people like it, but then you said it’s not complete?

Moore (26:26): Yeah. So it is something that you can write. A computer algorithm to list the axioms. It’s an infinite set of axioms. But with the exception of two sort of clusters of axioms, it’s finite. If you are not paying attention, you would actually think that these, each of these other clusters of axioms are single axioms. But they’re actually an infinite family of axioms. You can generate a computer program that will spit out all of the axioms. We tend to believe that ZFC is consistent because we haven’t discovered any contradictions. If you believe that, then by Gödel’s incompleteness theorem, ZFC is not going to be able to prove that it’s consistent.

(27:03) And so there are statements, such as the consistency of ZFC, that ZFC can’t prove. That’s an interesting point. Because again, we believe that ZFC is consistent. And that’s, I mean, one of the reasons that, I mean… Most mathematicians, they’re going to work is based on the faith that ZFC is consistent. Right? But that’s something that we regard as a true statement. But it’s not something that ZFC itself is sufficient to prove.

Strogatz (27:27): I’m just thinking. Along the way here, we’ve been mentioning Gödel. I don’t know that we’ve said who he is. Do you want to tell us briefly?

Moore (27:34) Yeah, he was. I mean, he was kind of a revolutionary logician. This, the Incompleteness Theorem was one of his major achievements. And his other major achievement was to show that the continuum hypothesis cannot be disproved using the ZFC axioms.

Strogatz (27:49): Some people think of him as the greatest logician since Aristotle. And Einstein, who was his friend and colleague at the Institute for Advanced Study, said he loved having the privilege of walking to work with Kurt Gödel. I mean, he was in the same intellectual league with Einstein. If you haven’t heard of him, I recommend you look at a book about him called Journey to the Edge of Reason. A terrific book about Gödel’s life. But OK, so he’s, right, so he’s a mid 20th-century, early 20th-century logician. And you say he proved that — well, say it again about the continuum hypothesis?

Moore (28:23): Within any model of set theory, he constructed a smaller model of set theory which satisfies the continuum hypothesis. And so what that shows is that you can’t disprove the continuum hypothesis within the axioms of set theory. From one model of set theory, if you have one, then I can produce a new one, which satisfies the continuum hypothesis.

Strogatz (28:43): I see. So there could be versions of set theory, sort of smaller versions, that are still adequate to do arithmetic, I take it.

Moore: Yeah.

Strogatz (28:51): But in which, OK, the continuum hypothesis is true, just like Cantor guessed.

Moore: Yeah.

Strogatz (28:56): And then. But then — there’s a big “but” to this story.

Moore (28:59): Yeah. So many, many years later, [Paul] Cohen developed a technique called forcing that allowed him to enlarge models of set theory. And using this, he proved that you can’t prove the continuum hypothesis. Except his technique can also be used to prove that you can’t refute it. This, yeah, this technique called forcing is really, it’s very powerful. Forcing and the technique of building a smaller model within your model of set theory. These are the sort of two tools that we have for building new models of set theory from old models of set theory.

Moore (29:32): Going back to the geometry analogy. I mean, even these models of the hyperbolic plane, which were the non-Euclidean models of geometry — those themselves start by taking the Euclidean plane or a subset of it and building the model of geometry like the points and lines there. The points are just ordinary points on this disk. And the lines there are circles in, certain circles in the original geometry. The point that I’m trying to make is that this is a kind of a fruitful thing you do in mathematics. You oftentimes start with some structure that satisfies your axiom system, like a geometry that’s satisfying your axioms of geometry, and you manipulate it somehow and produce a new thing, which maybe satisfies a different set of axioms. That’s what Cohen and Gödel were doing, was that they were taking a model of the axioms of set theory — and therefore, in some sense, a model of mathematics — and manipulating it using various techniques to produce new models, which satisfied either that the continuum hypothesis is true, or that the continuum hypothesis is false.

Strogatz (30:36): So this is really amazing to me, and I’m sure to many people that, you know… Like, Plato has this philosophy that, that there are certain ideal forms out there and truths that — maybe we can’t see them here on Earth, but in some Platonic realm, their truth exists.

Moore: Yeah, yeah.

Strogatz (30:57): And you would feel like the real numbers exist, whether human beings think about them or not, and that the continuum hypothesis is either true of the real numbers, or it’s not. But you’re telling me?

Moore (31:09): Well, I mean, yeah, there are different schools of thought on this. I mean, you couldn’t — you can view it as, there’s this thing that I think goes under the name, that generic multiverse view, that there is nothing more that you can say. There are just all of these models of set theory. And the best that we can do is try to understand what’s true in each of them and move around between them. And that’s a very non-Platonic view of things, a kind of formalist view of things. You might also take the viewpoint that there is some maybe preferred model of set theory. That is, you know, the reality that we live in, and all of these other models, they’re models of the axioms, but they’re not really what we’re trying to describe with the axioms. I think the analogy with geometry is somewhat illustrative there, right? I mean, you can produce many different models of geometry. But we still live in a physical world that has a geometry and maybe that’s the, the geometry that we most care about.

Strogatz (32:03): I see. So in the same way that we could give Euclidean geometry some preferred status because it’s the one we’re used to. It’s the one which has been around long, because it’s sort of the easiest and most obvious, but we still think these others are good, and they have their domains where they’re useful and interesting.

Moore (32:20): But maybe the thing that’s worth pointing out there, too, is that even our understanding of — Well, first, I’m not sure that we live in a Euclidean geometry. But there’s, there’s a question about that. But even our understanding of the physical world is greatly enriched by understanding all of these other geometries, this free exploration of other models of geometry. And the same is true with set theory. I think, even if in the future, we settled on some consensus as to what is a new axiom for set theory, arriving at that destination is something that surely will not have been possible without all of this exploration that occurs beforehand.

Strogatz (33:00): What would proving or disproving the continuum hypothesis mean? For each of these camps? What’s at stake?

Moore (33:08): Yeah, that’s — OK, so I think the camp that takes this sort of “all worlds” viewpoint just would say that this is a meaningless question. That Cohen and Gödel and their techniques for building lots of models of set theory is kind of the end of the discussion. And you know, we’re going to produce lots of new models of set theory, maybe, but we’re never going to have a final answer for saying that the continuum hypothesis is true or false. The people that take the viewpoint that there is some sort of truth or falsity to that statement, would presumably try to come up with some new axiom and presumably some heuristic justification for why this axiom should be true — either a heuristic or maybe a pragmatic justification for why it’s true. And then once you argue that this axiom should be accepted, that it somehow encapsulates some intuition we have about mathematics or sets, then if this axiom also proves or disproves the continuum hypothesis in a sort of formal sense of the word, then you would view that CH is true or false.

Strogatz (34:12): So that’s sort of where we are now. That there really are these two camps at the moment.

Moore (34:16): Yeah, to a degree. It’s been so long since the continuum hypothesis was shown to be undecidable based on the axioms, that I think most mathematicians have kind of gotten used to the fact that maybe that’s the most that you can say. And I think it would be amazing at this point if mathematicians as a whole could rally around some new heuristic that, you know, everyone could agree ought to be true. And maybe that will never happen. Maybe, maybe the community has too many different viewpoints in it. To be fair, I think it — I think it’s somewhat of a consensus view, but not a universal view, that ZFC is the set of true axioms for mathematics. There are certainly people that take the view that anything infinite just doesn’t exist. And it doesn’t make any sense to talk about and we shouldn’t be talking about it.

Strogatz (35:05): Well, that’s a time-honored tradition. I mean, that’s — Aristotle was telling us to watch out about infinity. And throughout the history of math, people even as great as [Carl Friedrich] Gauss were very careful about this concept of completed infinity, which is what Cantor opened up this can of worms for us. But I don’t know that it’s worms. It seems like it’s — you know, what’s the harm? It’s that we’re letting our imaginations go and discovering a lot of interesting things.

(35:30) But I do have a question. As someone who’s not a set theorist, I don’t want to ask it in an impolite way. But it might come out sounding a little impolite, which — you know where I’m going, right? Like, how does this affect me? Does the rest of math feel the vibrations that are happening within set theory? Or are we sort of insulated from what you guys are doing?

Moore (35:49): That’s a good question. I think most mathematicians never encounter a statement which is neither provable nor refutable within the usual axiom system for mathematics within ZFC. And the set theorists have to a degree discovered an explanation for that. There’s a model of set theory which is larger than Gödel’s original model but smaller than the universe of all sets called Solovay’s model, that [Robert] Solovay discovered around the time of Cohen’s work. And the remarkable discovery is that this model — what’s true in it can’t be influenced by forcing. And therefore, essentially, if you can phrase something about what’s true in that model or false in that model, it’s something which is largely immune to independence phenomena.

(36:35) The catch is that this model of set theory is not — does not satisfy the axiom of choice. So the axiom of choice is — this is another can of worms here. But one of the reasons why the axiom of choice is different from the other axioms is that it’s not constructive. All of the other axioms tell you that some set that you have a description of is, in fact, a set. That’s just how the axioms work. But the axiom of choice tells you that given a collection of sets that are non-empty, you can select something from each one of them — hence choice — but it doesn’t tell you how you’re going to make the selection. This was an axiom that, on the one hand, allowed us to construct all kinds of weird, paradoxical things. You know, I guess, in the ballpark of 100 years ago or so, like non-measurable sets, whatever that is. There’s this famous decomposition of the sphere, that Banach-Tarski paradox, that —

Strogatz (37:29): Oh, this is interesting.

Moore (37:32): — you could cut the sphere into finitely many pieces, and then reassemble them into two spheres that are the same dimensions of the original sphere. And now the reason why that’s absurd is that you ought to be able to assign a mass to each of the — you know, to the original sphere, and then assign a mass to all these pieces that you can cut it up into, and those ought to add up to the original mass. And then when you rearrange them, that process shouldn’t change the mass. But somehow, when you reassemble them, you have twice the mass that you started out with. Now, the point in that argument — where things go wrong is this cutting up of the sphere that the axiom of choice allows you to do is so bad that you can’t assign masses to these pieces that you have.

(38:11) Now, that paradoxical behavior led people to think that the axiom of choice is somehow perhaps problematic. Maybe it’s, it’s gonna lead to some sort of paradox within mathematics itself. And therefore, the axiom of choice shouldn’t be accepted. One of the things that Gödel proved at the same time as he proved that you can’t disprove the continuum hypothesis, is that it’s also safe to assume the axiom of choice. That is, if the axioms of ZF without the axiom of choice are consistent, then so too is the set of axioms of ZF with the axiom of choice. It gives you a lot of weird, exotic things, maybe, but from a foundational point of view, it doesn’t pollute the water.

(38:51) Sometime later, there was the discovery of this thing called Zorn’s lemma, which turned out to be equivalent to the axiom of choice. And it’s really very fruitful for developing a lot of different branches of mathematics. It’s something that — you learn about it if you’re an advanced undergraduate, or if you’re a graduate student in math. It’s somehow part of just the required learning for a graduate degree in math. And because of this extreme utility, it’s something that we just accept these days. I think most mathematicians are not comfortable working without the axiom of choice, just because in many cases they might be using it without even knowing it.

(39:31) So I think this is also an instance of how we might settle the continuum hypothesis. It’s that we discover some axiom in the future which is so useful in developing mathematics further, that we just regard this axiom as being true. To a degree, that’s what happened with Zorn’s lemma. And with the axiom of choice, it wasn’t something that was initially viewed as true. In fact, it was sort of initially viewed with some skepticism.

Strogatz (39:56): But let me see if I can, since it does… We’ve been talking now a lot about the axiom of choice: Its relation to the continuum hypothesis. Is there a pithy way to say what that is?

Moore (40:06): You know, the axiom of choice and the continuum hypothesis have kind of a curious relationship because they… OK, the continuum hypothesis, from a set theorist’s point of view, it allows you to construct a lot of exotic things. It allows you to do an infinitely long, even uncountably long construction, where you’re doing everything in a very controlled way, an algorithmic way. And building some weird object where you’ve maintained a lot of control along the way. In the absence of the axiom of choice, the continuum hypothesis, as I stated it originally, that there is no set of reals which is intermediate, that’s something which doesn’t have the same bite as if the axiom of choice is true. And the reason for that is that, for instance, in the absence of the axiom of choice, you can talk about even stronger versions of the continuum hypothesis. Like, every subset of this number line, the real number line, is either countable, or there’s a copy of the Cantor set that lives inside of it. Like, there’s kind of a tree of points, a binary tree of points that sits inside of your set. And this is a very concrete way of saying it has the same size as the real numbers.

Strogatz (41:14): So for the rest of us in math outside of set theory, should we be losing any sleep over the — what seems to be — kind of indeterminate status at the moment of the continuum hypothesis? We’re told it’s undecidable in the standard model of set theory. You know, does it matter? Does it affect the rest of math?

Moore (41:35): The answer mostly is no. But it’s not entirely no. The continuum hypothesis. It’s true in the Solovay model, for instance: Every set of reals is either countable or there is a closed set of reals inside it which is uncountable and has no isolated points. But there are statements that show up in mathematics, questions that show up naturally, kind of organically in other fields, where it turns out that they are dependent on either the continuum hypothesis or something else, which is independent of the axioms of ZFC. One example of this is something called a medial limit, which is a device that is useful in probability and some parts of probability for taking limits of things and still maintaining that things are measurable. Medial limits are something that you can construct using the continuum hypothesis, but they’re not something that you can build in ZFC.

Strogatz (42:27): This makes me happy, I have to say. I mean, I want to believe that math is one big web. And that, like there’s an old saying, “No man is an island,” from whoever, I don’t know. But anyway, I don’t want any part of math to be an island. So I would hate to think that set theory is somehow some — I mean, no one would say it is, but even the part that contains the continuum hypothesis, I don’t want that to be divorced from the great continent. And it sounds like it’s not.

Moore (42:52): Right. If you take a Hilbert space, and you look at the bounded operators, and the compact operators, these are well-studied algebras of objects that are studied in mathematics. You can take a quotient of them. Studying what’s called the automorphism group of that is something that a mathematician might ask about. And indeed, Brown, Douglas and Fillmore asked about that in the 1970s. And it’s known that whether the continuum hypothesis is true or false is related to whether there are very complicated automorphisms of that algebra or not. That’s something that is, you know, a standard object in a functional analysis course that you would teach at the graduate level. And these are sort of very, very basic properties of this object.

(43:34) But the point is, this is something that’s, on the face of it — this is not a problem in set theory. Different set theorists have different takes on why the subject is important. But to me, this is why the subject is — what it’s important for. It’s that it plays this unique role of being able to let you know when you’re asking the question that might not be decidable, based on the axioms. Because you don’t want to be studying this problem that you can’t decide without any success for years and years and years. And if someone can tell you that, “Well, you’re never going to actually come up with a solution to that problem, because you can neither prove nor refute that,” right? That’s a good thing to know.

Strogatz (44:13): All right. Well, to me this a very uplifting message you’re giving, Justin, that — John Donne! That’s the name I was looking for, John Donne. And let’s say this in the modern way: No person is an island. And the same with no part of mathematics. There is — even the most esoteric seeming things on the outer reaches of set theory are still linked into very down-to-earth parts of math, in probability, in the functional analysis that underlies quantum theory. So, this is news to me, and I just want to thank you for enlightening us. This was fun. Thanks.

Moore (44:46): Thanks for having me.

Announcer (44:46): Explore more math mysteries in the Quanta book The Prime Number Conspiracy, published by The MIT Press, available now at Amazon.com, Barnesandnoble.com, or your local bookstore. Also, make sure to tell your friends about this podcast and give us a positive review or follow where you listen. It helps people find The Joy of Why.

Strogatz (45:12): The Joy of Why is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Funding decisions by the Simons Foundation have no influence on the selection of topics, guests or other editorial decisions in this podcast or in Quanta Magazine. The Joy of Why is produced by Susan Valot and Polly Stryker. Our editors are John Rennie and Thomas Lin, with supported by Matt Carlstrom, Annie Melcher and Zach Savitsky. Our theme music was composed by Richie Johnson, Julian Lin came up with the podcast name. The episode art is by Peter Greenwood and our logo is by Jaki King. Special thanks to Burt Odom-Reed at the Cornell Broadcast Studios. I’m your host Steve Strogatz. If you have any questions or comments for us, please email us at [email protected]. Thanks for listening.