Sean Carroll's Mindscape: Science, Society, Philosophy, Culture, Arts, and Ideas - 282 | Joel David Hamkins on Puzzles of Reality and Infinity

Episode Date: July 15, 2024

The philosophy of mathematics would be so much easier if it weren't for infinity. The concept seems natural, but taking it seriously opens the door to counterintuitive results. As mathematician and philosopher Joel David Hamkins says in this conversation, when we say that the natural numbers are "0, 1, 2, 3, and so on," that "and so on" is hopelessly vague. We talk about different ways to think about the puzzles of infinity, how they might be resolved, and implications for mathematical realism. Blog post with transcript: https://www.preposterousuniverse.com/podcast/2024/07/15/282-joel-david-hamkins-on-puzzles-of-reality-and-infinity/ Support Mindscape on Patreon. Joel David Hamkins received his Ph.D. in mathematics from the University of California, Berkeley. He is currently the John Cardinal O'Hara Professor of Logic at the University of Notre Dame. He is a pioneer of the idea of the set theory multiverse. He is the top-rated user by reputation score on MathOverflow. He is currently working on The Book of Infinity, to be published by MIT Press. Web site Notre Dame web page Substack Google Scholar publications PhilPeople profile Wikipedia

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Mathematics has always been the intellectual subject that has a reputation of being the most precise and rigorous and well formulated and completely grounded, right? You have axioms and then you prove things with 100% rigor, ideally anyway. But, you know, starting in the 1800s, there were these events that kind of might be. shake your faith in the ability of mathematics to be perfectly precise and clear and understandable. There was the discovery of non-Euclidean geometry, right? Euclid thought he had figured out geometry, and people generally had the idea that he was putting his finger on something true. This is what we call geometry. And we realized that there were different axioms you could choose that would give you different kinds of geometry, spherical geometry, hyperbolic geometry, etc. But okay, we could still handle that. Then Cantor comes along and shows that not only is infinity an important concept,

but there are different kinds of infinity. There's a bigger infinity that is the number of the real numbers, the continuum, than is the number of the integers, right? So that shook people up. They didn't really like that. A lot of people denied that it was true, but still, you were proving things with theorems and so forth. So you can understand the motivation of David Hilbert, the very famous mathematician who in the early 20th century proposed to really get serious about this program of axiomatizing all of mathematics. Hilbert had this idea you could just write down a reasonable set of axioms that would cover everything, and then not only could you prove all the theorems that were interested in proving, but you could prove that they were consistent, right, that you had a

complete system that would cover everything that you wanted to cover. So you can also, feel very bad for Hilbert when Kurt Gödel came along just a few years later and showed that that ambition could not be realized. In his efficiently powerful formal system, either the system was inconsistent somehow or there would always be true statements that you can't prove. Statements that essentially say, I am unprovable. So if you prove them, they're false and your system is inconsistent. If you can't prove them, they're true. but of course not provable. Hilbert was upset by this.

He didn't like it, but nowadays, mathematicians take this as a centrally important result. And it was also tied up with a bunch of very similar results by people like Tarski and Turing and so forth. So that opens up not only a lot of good research to be done in math, but also deep questions in the philosophy of mathematics. What does it mean to have true statements you can't prove? What does it mean to have different sets of axioms that lead to different conclusions and so forth?

It's actually not completely different from what happens in physics, right? There was always philosophy of physics, but in the Newtonian world, you think you had things figured out mostly, right? And then in the 1800s, you discover statistical mechanics and things are maybe a little bit unsure, like how did probability get in here? And then, of course, in the 20th century, you have quantum mechanics. still struggling with that. So now the philosophy of physics has these deep questions that we're still trying to figure out. So I'm fascinated by these questions in the philosophy of math. I hope you are too. I struggle with them a little bit. So we brought on Joel David Hamkins, who is one of the world's experts. He was trained as a mathematician, later moved to philosophy and is now

technically a philosopher of mathematics and logic at Notre Dame. He's written many books on the subject, and also, you know, original contributions. He's recognized in philosophy of math as a champion of the view called the set theoretical multiverse, the set theoretic multiverse, not the same as the multiverse we have in physics, not the same as any of them. So we'll talk about that a little bit. And Joel is also very active in explaining these ideas. So he's on social media. He has a substack where he talks about infinity and so forth. He gives public lectures. So I think he is the right person to help guide us through this thicket of really intellectually challenging stuff. So let's go.

Joel David Hankins, welcome to the Mindscape podcast. Oh, it's a pleasure to be here. I have to start. This is more for the audience than for you. But this kind of topic is very dangerous because when I talk about physics to people, I know what I'm talking about. When I talk about economics, I have no. idea what I'm talking about. So I have no, you know, strongly held ideas that I can get in trouble with. But this, you know, philosophy of mathematics is just where I know enough to be wildly wrong,

but opinionating about things. So I'm hoping that you can set me straight on some of my existing confusions. Well, I hope so. I mean, there's a lot of people who have quite strong views about the philosophy of mathematics. And to my way of thinking, well, I mean, sometimes discussions in the philosophy of mathematics have this character. of the, you know, we're making a fundamental mistake and we're going to be wrong. It's sort of like this, they're pointing out this danger that we might be falling in if we have the wrong idea about infinity or about classical logic or, you know, something like this. Whereas a kind of competing point of view is that, look, the philosophy

of mathematics is really about helping us to decide what. are the most insightful investigations mathematically, where can we learn the most? And it's not a matter about being wrong mathematically, but rather about where are we going to find the most interesting mathematics. And so in particular, if sometimes people have a view about the philosophy of mathematics that is saturated with skepticism about infinity or something like that, for example. And my attitude in response to those perspectives is often that, well, unless the competing vision is offering me more insight into questions, if it's only a kind of negative skepticism that's saying, oh, we're all wrong about these things, but it doesn't have any positive benefit, then I don't really find much attraction to that point of view. And I would rather to look at many different philosophies of mathematics and take them as a suggestion about, you know, where should mathematics go.

And we should investigate further on the bases of those perspectives. But, you know, people do get attached. You mentioned in one of your books that David Hilbert was quite annoyed when Gertl proved that he couldn't do some of the things that he had laid out as the agenda for mathematics. Yes, that's absolutely right. I mean, it's sort of shocking how that all turned out. But, I mean, I think if anything, the girdle of girdles results in that matter. I mean, you're referring to the incompleteness theorem, of course, and the Hilbert program.

Hilbert had wanted to sort of solve the problem of the Antinemies in the early 20th century in these very worrisome contradictions that seem to be arising in what were otherwise extremely tempting to Hilbert, the foundational theories, like set theory. I mean, he famously said, no one shall cast us from the paradox that Cantor has created for us, you know, referring to the foundations of mathematics as founded in set theory. He didn't want to give that up,

but the contradictions and the Russell paradox and the Raleigh-40 paradox and so on the other. And ten means at that time were very worrisome. And so Hilbert wanted to settle the matter sort of once and for all by saying, well, look, we have this strong theory. we're going to hold it a little bit at arm's length, right? We're a little suspicious of it because of the contradictions, but we expect it's going to answer all of our questions.

All of the questions will be settled on the basis of this strong theory. But meanwhile, what we really want to do to sort of give us greater confidence in the situation is to prove that it's consistent but purely finitary means. So in the weak finitary theory, Hilbert wanted to prove this sort of safety of the strong infantry theory. And I think it's a very sound thing to want to do. I mean, it makes a lot of sense,

the historical situation that he was in. And so, of course, in order to make sense of how one would prove that a theory is consistent, you're led inevitably to the philosophy of formalism, where you look at, well, what's really going on in the infinitary theory when people use the infinitary theory, they don't have to actually be committed to the infinite sets and the posits that the theory makes, but rather they're just writing symbols on paper

and reasoning in that theory, pushing these symbols around, right? And that that's a kind of finitary combinatorial process, and we might hope to prove in the finitary theory that it was safe. That's what his strategy was. And then, of course, you're a completely, this whole picture, right, by showing not only can the finitary theory not prove the consistency of the strong infinitary theory, but it can't even prove the consistency of itself, of the

finitary theory itself. So it must have been quite disappointing for Hilbert at that time. Well, good. We're going to get to all of these things, which is great. I mean, there's still the looming. These are the crowd-pleasing questions in philosophy at math. But one crowd-pleasing question is the one that I am just struggling with, which is realism or blatantism or anti-realism or whatever. I've had the occasional nominalist on the show, the occasional realist. I lean toward nominalism myself, but I think that puts me in a pretty tiny minority of people who think carefully about these issues. So what is the right answer? The right answer.

I see. So, you know, there's been a really interesting development, particularly in the philosophy of set theory, which is very much connected with these issues, and that is it, it used to be in mathematics and in set theory in particular, if you said that you were a Platonist in mathematics, then this was taken to imply a kind of singularity. I mean, not a singularity in the physics sense. I like that. I mean that there would be only one universe. So platonism is the view right, that there's a real existence to the mathematical objects exist in a perhaps idealized real realm. So it's connected with realism. But also there was this kind of connotation

of uniqueness that the mathematical universe would be unique. And I think in more recent discussion of the philosophy of mathematics, this connection between Platonism and uniqueness of the platonic realm has been severed. And I think this is a positive development because one can be a realist and a pluralist just because you think that there are, say, multiple concepts of set, each of which serves adequately as a foundation of mathematics, but, you know, with incompatible truths. And so we can still be a realist about these perspectives, even if we're real, even if we're pluralist. And so now it's no longer true, I think, if you say I'm a Platonist, it's not necessarily implying that you think there's a unique answer to mathematical questions. You can still be

pluralist and Platonist. I mean, it's sort of like asking, I mean, to put it in the plutonical. Is there only one platonic realm? Why should there be only one? I mean, maybe there's multiple ones, and some of them have certain kinds of mathematical abstract objects and not others and other platonic realms have a different kind of combination. I mean, that's basically the situation of what's going on in the plural foundations of set theory today. If you think that there's, that there are multiple coherent and fully real concepts of set, each of which is giving rise to its own set theoretic universe, with different truths. Some of them have the continuing hypothesis, perhaps some of them don't, you know, the negation and some of them have lots of large

cardinals and others don't. Then you don't think that there's going to be a, unique, determine, and answer to every mathematical question. It's going to depend on which mathematical universe you're in, but you can still think that they're fully real in the platonic sense. So it's sort of like having multiple platonic realms. Well, and just so do we tease the audience who cares about physics a little bit. I mean, they've heard about a lot about the cosmological multiverse and the many worlds of quantum mechanics. So the line you're pushing is a sort of foundations of mathematics multiverse. Exactly. And, and So my papers and which I first started talking about this was called the Sathetic Multigrose.

And I didn't realize the danger I was putting myself in Jeff. I kind of warned you. I started getting lots of emails from physicists who are from people who make the kind of conclation that you just refer to. Really, I think there's not much connection between this kind of mathematical pluralism, multiverse view, and the multiverse. views in physics, I think they're just not connected very much well. But I mean, there is a kind of family resemblance, though, in the sense that in each case, they're positing the view that,

you know, at bottom reality has this pluralistic nature with, you know, multiple coherent universe instead of standing side by side or independently or, you know, on top of each other superimposed or whatever the metaphor is that you're using to refer to that. I think that much is similar between them, but I don't think there's a deeper methodical connection between these views. Agents who are realtors do more than open doors. They analyze market trends, interest rates, comps. They can tell you about flood zones, mixed use zones, and decode acronyms like

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massive sci-fi adventure about survival and science. And what happens when you wake up alone very far from Earth? I really had to make a decision because I caught myself getting that frog in my throat and starting to get teary as I'm narrating some of these sections. And it's like, okay, yo, yeah, yeah, yo, is this indulgent? And I really thought about it. I was like, no, at this point, it would kind of be betraying the trust the author and the listener have in telling this story if I don't go through it. But there's places in this book that deeply emotionally affected me. And I left it on the mic.

That's great. Because it served the story. People will say like, oh, my God, I cried at the end. It's like, yeah, dude, me too. Listen to Eursay, the Audible and IHeart Audio Club on the IHeart Radio app or wherever you get your podcasts. So my own sort of naive physicist's view on mathematical truth, I've learned has been labeled if-then-ism. that is to say that we can postulate a bunch of different axioms, and then we can prove theorems on the base of those axioms,

and the truth of the theorem is relative to whatever axioms we picked, but we shouldn't attach absolute truth to the mathematical results. And that sounds almost like what you're saying, but I think you're saying is what you're saying is much more sophisticated than that. Well, I see. So the view of if-thenism is often... It's often used in a very disparaging way. It is, yeah.

It's meant to be kind of dismissive of the view. And I guess one way of discussing it is this dispute that I've tried to call attention to between not enough philosophical attention is being given to the dispute between strong foundations and weak foundations. Okay. Let me explain what I mean by that. So mathematicians instinctively that seem to prefer. weakening the hypotheses of the theorem. Of course, if you have a theorem, you know, and it's from some assumptions, you're making a conclusion, then it's a better theorem if you weaken the assumptions. It's a stronger theorem. It applies to more cases and so on. So you can either weaken

the assumptions or strengthen the theorem and you're going to have a better result by doing that. And so I think there's this kind of instinctive reaction to just always prefer the mathematical result with the weaker foundations. And I think for very sound reasons, that's absolutely right. But when you're talking about the foundations of mathematics where you're trying to identify what are the core principles that sort of underlie all of our mathematical truths that we care about, then if you apply that sort of instinctual preference for weak foundations, you're going to end up with a weak foundational theory. That's what it would mean.

Okay. But the reason we liked the theorem with the weak hypothesis was that it applied in more cases. And so if you have a preference for weak foundations, then you're sort of implicitly suggesting that there would be multiple alternative mathematical realities. It's connected with pluralism. A preference for weak foundations is sort of admitting, well, you know, maybe there's lots of different kind of mathematical realms in which those axioms are true, but maybe not stronger ones, right?

But if you have a slightly different view about what's going on when we're searching for mathematical foundations, what we really want to do is find the fundamental truths, the fundamental mathematical truths that are going to help us to prove, you know, and answer the mathematical questions that we care about, then maybe we want to have very strong but still true mathematical axioms. We want them to be strong because the strong axioms are going to be the ones that paint the co-heum. picture of mathematical reality. And this is a very strong reason. If you're, say, developing a physics theory, it's sort of the same situation. You want your physical theory to be, to have a lot of answers to the physical phenomenon

that you find troubling, right? But you want it to still be true. So, you know, or not refuted or whatever, not falsified you've had it or whatever. And so it seems like in physics maybe the preference would be towards the strong foundations. You want a strong theory that has all the answers, right? And from the mathematical point of view, I think that still that also makes sense to have a strong foundation. So there's this tension between the weak foundations and strong foundations. And it tends to be the weak foundations have a kind of pluralist aspect or affinity to them.

And the strong foundations have a kind of monist perspective because if you, you thought there was only one true mathematical reality in which every mathematical question has a definite truth value, then the true foundational theory is the one that's true in that unique realm. And so we want to find out what those truths are. And so this is pushing you towards strong condition. But then the thing that I'm slowly coming to understand is the important distinction between a set of axioms, at least in first order logic, and maybe you could explain to us what that is, versus models of the axioms, right? So in arithmetic, we can prove things about addition whatever and the natural numbers, but then we can get different models of the natural numbers that

are compatible with the same axioms. I think that probably most people listening don't know what that means or how important it might be, because one argument for realism is that, sure, you can get lots of models for the same axioms, but one of them is the right one. And it's not pinned down by the axioms. So to express this view that one of them is the right one is exactly the monist. This is the universe perspective. There's one true mathematical reality, and we're trying to figure out what that one is, what the truths of that model are.

But of course, because of the incompleteness, we can never write down, you know, we can never describe fully a computable list of axioms that are capturing that truth fully. That's exactly what Gertl's theorem is telling us we can't do. But we can get closer and closer, right? We can write down more and more axioms and try to figure out stronger and stronger theories that are true in that one true universe. Okay, but if you're a pluralist, maybe you say, well, I don't know if there is this one true universe because precisely,

because we already understand so well how different truths can be instantiated in multiple different models. We have a very rich understanding of how that could happen, and we have actually very little reason to think that there's a unique standard interpretation of our ideas. So this is especially true in the case of set theory where, I mean, maybe it's helpful to think about geometry. Sure. If you go back to geometry, you know, for thousands of years, as geometry was viewed as the study of the mathematical properties of space,

There was the one true geometry. That's what everyone was trying to figure out. What's true in plain geometry, say, or higher dimensions, whatever. But then with the discovery of non-Nuclidean geometry and hyperbolic space and spherical geometry, elliptical geometry, and so forth, the concept splintered and it became pluralistic. Right. But still real, right? So it relates. We're talking about rapport because we don't think that somehow geometry is only now a formal activity because we have multiple incompatible models. Just Euclidean geometry is fully real. I have points in logic. But hyperbolic space is also fully real. I mean, that's an abstract thing. You can be a realist about geometry, even though we're basically all pluralists in geometry. Okay, but a similar thing happens. in set theory because when set theory first started, you know, we, Zermello wrote down the fundamental principles that he thought would govern the reasoning about sets, and that was, it came to

be known as Zermelo set theory, and then, you know, additional axioms were added when they were realized that certain things we wanted to do with sets were missing from Zermelos theory, so we added replacement axiom to get Zermelo-Frankel set theory, and the axiom of choice was present from the start in Smellos theory. And then this picture emerged of looking at the satiritic universe as this sort of cumulative hierarchy growing up from nothing. Or maybe you start with this sort of irrelevant objects, the atomic objects that are not sets, but out of which you will form the sets. And then on the next layer, you can make sets of those er elements and then sets up sets of the er elements and so forth. And you can keep going transfinitely.

And so the universe sort of grows in this cumulative manner. And so there was this kind of picture of the set theoretic universe, the one true universe, is something like that. And then this amazing independence phenomenon, this pervasive, ubiquitous independence phenomenon, basically all of the fundamental open questions in set theory have turned out to be independent of the CFCS. essentially every question about infinite commentatorics that you might ask is either completely

trivial and you can prove it immediately or achieve it or else it's independent of the CFC accents. And so this was maybe first started to be realized with Gertil's theorem in 1938 that the constructible universe satisfies the axiom of choice and the continuum hypothesis. And then in 1963, Paul Cohen proved that the continuum hypothesis could fail in a certain model constructed using the forcing method. And since that time, forcing has been used in thousands and thousands of independence periods to show this pervasive independence phenomenon. Basically, every question that we are interested in is not settled by the axioms. Okay, there's a couple of ways to react to that situation. If you know and you have a proof that your theory can't settle the axioms,

then a monist, a universe view person is going to say, well, it's because the theory is too weak. We need to make it stronger to settle those questions, right? And it's pushing you again to a stronger theory. But the pluralist is going to say, well, the independence phenomenon itself is evidence for pluralism because we can already see how it could be, and in why this is true or is false, you know, if you use the right conception of set, you're going to get a satiritic universe in which, you know, the truth value is one way or the other,

and we can sort of make it what we want. And the special thing about this set theoretic case is that these models are all standard in a sense. They're all well-founded with respect to each other. They have the same ordinals. So they're not completely weird models. If you want to make CH true, you can go to Girdl's constructive universe, and it has the same ordnals as the universe that you started in. And if you want to make it false, you can make a forcing extension,

which has the same ordinance as the one you started it. And so they're not these totally weird, non-standard conceptions. They're kind of nice, actually. I mean, they seem set theoretically perfectly acceptable. And this is sort of this fact is part of the basis. what I call the dream solution argument that. Some people hope to settle the continuum hypothesis, they describe it as a still open question. Whereas I say, no, no, the continuum hypothesis is settled, even though it's independent. It's settled by our understanding of how it behaves in the multibrears.

We can make it true. We can force it to be true or force it to be false. We can make it true in inner models and so forth. Then we understand all of that quite deeply. And that's the answer to CH, not one. You should explain to us what the continuum hypothesis is. Oh, I see. Okay. Right. So, okay. The continuum hypothesis is a question that was formulated by Cantor, who at the end of the 19th century proved that the reels form an uncountable set.

So he proved that there are different sizes of infinity. There's the infinity of the natural numbers, you know, 0, 1, 2, 3, 4, and so on, and the infinity of the real numbers, which is the continuum of the number line, and Cantor prove that those two infinite sets cannot be put into one-to-one correspondence with each other. We cannot make a list that contains all the real numbers on it. And it's the famous diagonal argument, right? If you had such a list of numbers, then you could write down another number,

which was different from the first number in the first digit after the decimal point. and different from the second number in the second digit and so forth all the way down. And that number, it's a perfectly good number. We determined it's decibel, and it's different from all the numbers on the list. So it shows you can't have a list that contains all the numbers, all the real numbers. So therefore the reels are uncountable. It's a bigger infinity than the natural numbers. It's an uncountable infinity.

And the continuum of all this is is the question, well, the question, the continuum of hypothesis is whether or not there is any infinity in between them. So we have the natural numbers, that's the countable infinity, and we have the real numbers, the continuum, that's an uncountable infinity. And is there anything strictly in between? The continuum I bother this, it says no, there's nothing in between. But we know now that it's independent of the zermelo axiom. So Cantor struggled with the question his entire life,

and he never had an answer, and it was open for many, many years, decades. It wasn't until 1938 that Gerdl had first proved that you can't refute it from Zermelho's theory because he produced this, what's called the Constructible Universe, in which the ZFC axioms together with the Condinian Rwara Walthusisis are true. So therefore, it's safe to assume that there's no infinity in between. And he proved actually a vast generalization of the congeneralysis, the generalized continual hypothesis, which says that for every set, there's no infinity between that set and the power set, which is the number of subsets of the sets.

Cantor prove the number of subsets of the set is always a strictly larger infinity. So there's no largest infinity. You can just keep making bigger ones. And therefore, by the replacement axiom, the number of infinities is larger than any one of that. I mean, there are more infinites than anyone is a way of saying. It's provable in ZFC, but it's not provable if you don't have this replacement axiom, the thing was added by Frankl. So then Cohen proved in the 60s that the negation of CH is also consistent with the axioms,

and that's this forcing method. And so basically the situation is that if you have a set theoretic universe, you can manufacture from it using the, you know, the ontological resources that are available in that kind of platonic realm. You can make another one in which the continuum hypothesis is true and another one in which it's false. And so it's this kind of, it's a very pluralist point of view to think, this is an answer to CH to the continuum problem. It's independent and it's, you know, realized sort of densely in the multisies. Hey, everyone. It's Cal Penn.

I'm the host of Earsay, the Audible and I Heart Audio Book Club. This week on the podcast, I am sitting down with Ray Porter, the narrator of Andy Weir's audiobook Project Hail Mary, massive sci-fi adventure about survival and science. And what happens when you wake up alone? very far from Earth. I really had to make a decision because I caught myself getting that frog in my throat and starting to get teary as I'm narrating some of these sections. And it's like, okay, yo, yeah, yo, is this indulgent?

And I really thought about it. I was like, no, at this point, it would kind of be betraying the trust the author and the listener have in telling this story if I don't go through it. But there's places in this book that deeply emotionally affected me. And I left it on the mic. That's great. because it served the story. People will say like, oh my God, I cried at the end.

It's like, yeah, dude, me too. Listen to Eursay, the Audible and IHeart Audio Book Club on the IHart Radio app or wherever you get your podcasts. So there's not a fact of the matter full stop about whether the continuum hypothesis is true. There's a fact of the matter about relative to where you are in the set theoretic multiverse. Right, exactly. This is a way of understanding the CH. the multiverse answer to the continuum problem. It's exactly how you describe.

And so I want to go ahead. No, you go ahead. So I had mentioned this dream solution because it's related to the question that you've been asked slightly before, namely some people want to settle the continuum of wealth system by finding the missing axiom, the sort of the obviously true principle, the set theoretically natural principle, which happens to. settle CH1 way or the other. This is what I call the dream solution. You find the missing axiom, which has the character of an axiom in terms of its self-evident affinity with the concept

of sense. So we would want to add it to the theory and it would settle the question. And I argue this is impossible. It's never going to happen. No. And the reason is we already know what it's like in a world where CH is true or where CH is false. And it's perfectly fine set theoretically, though this is the important character of the forcing arguments is that they're set theoretically robust. And they're not weird models, weird non-standard models. They're just alternative concepts have said

in which the answer to this specific question comes out differently. And therefore, any proposed axiom that settles the continuum hypothesis is going to therefore fail in one of these universes that we have just said are set theoretically acceptable. And therefore, we can never find such a principle to be obviously true or manifestly true for sets in the way that is required. So if you have a principle and you prove that it settles CH, then that fact itself is undermining for the self-evident nature of your axiom. Is there a principle difference between this idea of adding more axioms or changing existing axioms? I mean, you started with the example of geometry, which is the one that I know the best. That makes sense. Versus the idea of sticking with some axioms but considering different models of them.

I mean, I know about in number theory, there are these non-standard integers. So in other words, you can model the axioms of number theory with the good old integers that we know and love, but there's also extras. There's other models with extra integers that don't sort of connect to them. But could we just sort of try to capture which model we're in by throwing in new axioms? So it's the same issue or are these two separate issues? Right. So you're referring to the piano axioms of arithmetic.

I think is the most common theory. And there's a whole subject called models of PAA, which is all about exploring those non-standard model. And I've done a lot of work in that area. And so the problem of non-centered models is inherent. There's no, you can't ever solve the non-centered model issue by adding axioms. Because every theory will have non-centered models. Even if you think that there's a one true model of arithmetic,

that you're interested in, then the set of statements that are true in that model, that's called true arithmetic. And it also has non-centered models. It's a consequence of the compatibility. So you can make always non-centered models that have all the same truths as the standard model. And that's quite an interesting situation. But actually, the question of whether we have a right to say that there is a unique intent model of arithmetic is actually not so clear as one might think. I mean, it's quite common

amongst mathematicians and everyone to think, well, look, when I talk about the natural numbers, you know, zero one, two, three, and so on, this is describing a unique mathematical structure. Yes. But that is it, really? Because this, and so on, of course, is hopelessly vague. And if you try to capture what you mean exactly, then to prove that, you know, that there's a unique structure that you're talking about. Right. Now, Datican proved a categoricity theorem about the natural numbers, namely he identified the fundamental principles that are true about the successor relation, for example.

If you think about the natural numbers with the operation of adding one successor operation, then he said, well, zero is not the successor of any number. Okay, because, I mean, we're not talking about negatives. We're just talking about the natural numbers, so starting from zero, zero, one, two, three, and saw as the structure we're trying to capture. Zero is at the successor of anything. The successor function is one-to-one, which means that if X and Y have the same successor,

then X and Y are the same number to begin with. So it's a one-to-one function. And finally, the third axiom says that every number is generated from zero by successor. So we have to say it in the right way, because otherwise it's circular, because we're trying to say what we mean by finite numbers. So we can't say, oh, you can get to the number from zero by applying the successor finite be many steps. That's a kind of circular account. We want to say what we mean

without being circular. And so we can say it with the second order induction next thing. We can say for any set of numbers, if it contains zero and it's closed under successor, then it contains everything. So it's just the induction. but stated not in a first order manner, because we're not just talking about definable sets of numbers or something, but for any collection of numbers whatsoever, for any set of numbers with this property, then it would contain all of them. Okay, and then Dedek improved that if you have two models of this theory, Dedek and arithmetic, then they're isomorph.

So there's only one model of the theory up to isomorphism, and therefore that's what we mean by the standard model of arithmetic. That's them all. But, okay, so it's quite common to point to this categoricity theorem as the answer to the question, do we know what we mean by the standard model of arithmetic? And the answer

is, well, yeah, because Thetekin gave us the theory and there's only one model of that theory, and that's the model we need. Okay, but how successful is that answer? Because you know, we were interested in picking out exactly the

structure of the natural numbers because we wanted to have a kind of absolute understanding of what it means to be finite, what it means to be a finite number that's part of the goal. So if we were worried about whether it's sort of the concept of finiteness was indefinite, then the Datican theory is supposed to be saying, well, the concept of finiteness is definite because it's provided by the Detican theory. But if you look at the Detican theory, this third axiom is making this statement about arbitrary sad. And so it's grounding the concept of finiteness and saying it's absolute because of some claim about arbitrary sets.

So how can that possibly be satisfactory to say, well, you know, finite truth. Truth about finite numbers is definite and absolute because, well, it would seem to beg the question about the definiteness of the meaning of arbitrary set of numbers. And we can see this how this happens already in the fact that different models of set theory, different models of Surmella set theory, well, they each have a unique, dedicated model of arithmetic. I mean, I'll tell you some morphism. They each have their standard model. But it's not the same model in all of that, when we know this already, that this can happen. So different models of set theory can have the standard model of arithmetic,

satisfying different truths. For example, we can make a model of set theory that thinks, say, some large cardinals are inconsistent or other models think that they're consistent. But consistency is an arithmetic statement. It's visible. You can formalize it as a statement about arithmetic. And so these standard models in these models of set theory can have different arithmetic truths in them, even though they both think that that standard model satisfies the

in theory, but they have a different interpretation of that induction axi because they have a different concepts of set. I take this just to show that the attempt to show that our understanding of the standard model of arithmetic has failed because it's based on set theory, but if we're, if we think that there's indefiniteness or pluralism in set theory, then it can't possibly work to establish definiteness or monism in arithmetic. I mentioned Gertl a couple of times. Obviously, his incompleteness theorems are central to this sort of whole grenade that got thrown into Hilbert's program. But I've heard a lot of claims on the internet that people say Gertl's theorem incorrectly.

So could you just tell us what the theorems are actually telling us? Oh, I see. So there's a variety. I mean, there's a whole sort of collection of theorems that are connected, all of which are on this theme. In fact, I'm writing a book right now called 10 Proops of Gerl Incompletence. Oh, very good. Of course, it's not just Gerl's theorem, but there's Rosser's variation and Taraski's theorem and Smolian has a version. And there's a way of thinking about the Russell paradox also is connected and so on.

So there's a whole kind of spectrum. But the basic theorem, one of the ways of stating the basic theorem, well, I mean, let me give you a sort of simplified version that I often, when I prove Gerl's theorem in my classes, I often prove a kind of simpler version. You cannot describe a computable list of true axioms of arithmetic that prove all and only the truth of arithmetic. So you cannot write down a theory of true axioms of arithmetic that prove all of the truths of arithmetic. And that's one way of stating the theory.

So there's no computably axiomitizable theory of arithmetic, which is true and proves all of the true statements that we. So if you know that the halting problem is undecidable, which is a result due to Alan Turing in 1936. Yeah, you should tell us what the halting problem is. Sorry, I know what it is, but. Yeah, no problem. Okay.

It's often the case that students who are taking philosophy of mathematics or logic class and so on that they often already know a little about the holding problem and computability theory, and so this is a kind of quick route into the inconvenience theorem on the basis. So Alan Turing introduced a sort of abstract notion of concept of computability, sort of these Turing machines that run programs,

and you can take a Turing machine. It runs on a paper tape, and it's writing, zeros and ones on the tape and moving according to the dictates of the program. So it's very rigid instructions, the program is saying, well, if you're in this state and you see the symbol zero, then you should change it to a one and move right and change to this other new state. And so the program is a sort of finite list of instructions like that.

And when you have a computational process, you know, you set it up with the input written on the tape. It's in the start state and you let it run. And it just is moving back and forth and changing stuff on the tape and ultimately maybe it halts. and then you can read off the output of the computation about what's on the T. So the halting problem is the question, given a program and an input, does it halt? Thanks. You know, either it runs forever, maybe, or maybe it halt at some state, right, at some stage of computation, maybe after a billion steps or after a Googleplex number of steps or, you know, however many

steps it is. Maybe it halts, maybe it doesn't. Okay, the halting problem is the decision problem, given a program and a program, input and so the question whether it halts or not. And what Turing proved is that this problem has no computable procedure that will answer, that will answer correctly in all instances. So in other words, there's no computable procedure that will answer correctly whether or not any given program and input halts. So now suppose towards contradiction that we had a theory that Gerdl's theorem is about, a theory that answered all the true questions, that proved all the

true questions, then we could solve the halting problem because given an instant, consider the following algorithm, okay? I have my fixed theory. I'm supposing towards contradiction that there's a theory, a true theory that proves all and only the true statements of arithmetic. Now, given any Turing machine program and input, I can formulate the arithmetic assertion that asserts that that program halts. I'm not going to run the program. I'm just formulating the statement that it halts. And that is possible to formalize in arithmetic. Because you're really just saying, well, there is a number that codes a sequence of numbers. And those numbers in that sequence are each coding sort of snapshots of computations. And they're obeying the program from one step to the

next and so on. And the first one is like the starting configuration and the last one it's showing a halting configuration. So the existence of such a number is exactly equivalent to the halting of the program. And so it's an arithmetic assertion. So we can form this assertion that says this program halts on this input. And then we can go over to our theory. And we can just systematically start searching for proofs, either a proof that it's true that it does halt or a proof that it doesn't.

And if the theory is, as we supposed, it knows all the answers, then we're either going to find a proof that it halts, or we're going to find a proof that it doesn't halt at some stage. And when we find that proof, we can answer the question. And so if there were a theory that knew all the answers, then we could solve the halting problem. But we can't solve the halting problem because that's what Turing proved in 1936, and therefore there can't be such a thing.

So this is, yeah. Yeah, go ahead. So Girdle proves good. That's a very helpful way of thinking about it. I think that the difference between someone who is good at this and not is whether they've sort of internalized the halting problem, the undecidability of it. I have not quite yet. I'm getting there. I'm trying to get better.

But so if Girdle is telling us that, you know, in the set of well-formed statements we can make in some system, some of them are going to be true but unprovable if the system is consistent. Right. Do we know how many of them there are? Are most true statements unprovable versus the ones that are provable? Is that even a sensible question to ask? Yes, it is a sensible question. Actually, this question has come up multiple times on math overflow. I should think that's exactly that.

What's the density of independent statements, right? And you can answer, it's strictly in between zero and one, I mean, the density. So there's certainly infinitely many statements because if something is independent, then I can just say if fee is true but not provable, then fee and fee and fee and fee and fee and fee and fee and fee and fee and fee. I mean, it's sort of trivial state, but it's infinitely many. So there's at least infinitely many, okay. But also you can make other things like you could say, for example,

you know, let's see, so how does it go? If you have an independent statement, then you can take, if you take any other, let's see, I'm trying to remember how the argument goes, it's just this trivial syntactic thing where you combine fee with other statements. And of course if fee is fixed, then there's a certain proportion of all statements

that are like fee or psi for any psi. And so, and fee is fixed. So it's, you know, like a certain, it's pretty small, but you can make a certain proportion of all the statements that involve Fee in this trivial way. And you can use that to show that the density can't be zero because there's this tiny percentage of state and can't be one for similar kind of

reasons. So do you think there is a number, the fraction of statements that are well-formed and provable? Oh, I see. Well, the question is would be, does the density converge or now? Right. I think it probably oscillates. I don't know.

That's my number. That kind of question is it's a little off-putting because it depends highly on the formalism, on the formal language. Like if you have different syntagic rules and so on, you're going to get a different percent for sure. Yeah, okay. And maybe it's going to, you know, not convert. Maybe you can make it not convert just by changing the syntactic rules of the formal language. But therefore, it shows that the question itself is not the kind of factor that we want. Yeah.

It's related to this, for example, there's a theory with mine that's said. the following. I proved it with Alexei Miasnikov. The halting realm is undecidable, yes, except that the title of my paper with Alexi is that the halting problem is decidable on a set of probability one. So you can decide almost every instance of the halting.

Okay. For certain Turing machine models, if we just used the sort of standard model, that Turing had a one-way infinite Tink with zeros and ones, on the, and then what we proved is that there's a set of programs which, as the number of states grows larger, as the size of the program grows larger than the proportion of programs that are in our set, the ones that we're going to decide the question for, gets close and closer to

100%. So as the size of the program grows, the proportion of programs that fall into our process is increases towards 100%. So that's the sense in which almost every we're answering. But, okay, the theorem is totally unsatisfactory. It sounds pretty great when you say it, because we're solving almost every instance of the halving problem. But when I tell you the proof, you'll say, and the proof is just the following.

We look at the program and we just run it until it repeats a state. Okay, so, and, okay, it's the one-way infinite tape model. half of them fall off on the first step. Yep. They move the wrong way, and that sort of causes it to crash. Or if you count that it's halting, that's fine too. So half, 50% fall off.

Yeah. And okay, and the others go the other way. I see where this is going. But then as long as they're not repeating a state, it's basically a random walk. Because if it's in a new state, from that state, in that situation, half of them are going to go left and half are going to go right. So until you repeat the state, it's a random wall, but then there's this theorem of polio, the recurrence theorem,

that says with probability one, you come back and have fall off at that point. And so with probability one, the probability one behavior of a turning machine calculation is that the head falls off, that the probability one behavior. And we can solve the holding problem for those. Okay. That's the critical of theorem, basically. It's like that.

Okay, so it's totally... That counts. I don't care. Yeah. Okay, but the problem with it, and the reason I brought it up is that that theorem is dependent on the formalism, because if you use a two-way infinite tape, it doesn't work at all. Right. But there's other models of computability that don't have, and it's an open question.

So the current bound on the two-way model is we can decide the halting problem 13.5% of the time. It's one over d squared. is what we can do, which is about 13.5%. And those are the programs that never, that don't have any instruction to halt. You look at the program and if it never says halt, then you know it's not going to halt ever. And so you can solve the holding realm for those programs and that's proportion, if you calculate it, that's proportion 1 over e squared, which is about 13.5% of the time.

So you can solve the, you know, you can solve the holding room. Originally, it was quite interesting because Alexi came to me and he said, well, he had this concept of black hole problem in decision theory. It has nothing to do with black hole. I get it. In perfect. Okay, but the black hole problem is it's a difficult problem, which is undecidable, or maybe it's NP complete or something in complexity theory.

It's a difficult problem, but the difficulty is concentrated in a very tiny region outside of which it's easy. Okay. So such a problem is not suitable for, say, encryption, because if you could rob the bank 95% of the time, you know, rather than... They're happy, yeah. So, you know, even any non-trivial time. Okay, he came to me. He had already found black holes in various problems, famous problems.

And he said, well, does the halting problem have a black hole? And the first observation was this 13.5% thing, you know, that I made. And so then we said, okay, now we've got a plan because all we need to do is find more and more stupid reasons why the program is going to halt for sure or is not going to halt. And we just wanted to add up to more than 50 percent because then we can say most. That was the strategy, right? Okay. But then we were thinking about it more. And I realized that this head falling off was like an incredibly super deep.

The best one, yeah. But it had 100% of the probability already, just that one reason. And so that's how we got the result. But, okay, but one can definitely criticize the theorem because it's dependent on the computational model, right? And that would be similar to your question about this density. If it's dependent on the formalism, then maybe it's not the right question. Agents who are realtors do more than open doors. They analyze market trends, interest rates, comps.

They can tell you about flood zones, mixed use zones, and decode acronyms like HOA, APR, MLS. They connect you to lawyers, contractors, even Phil, the Sewardcope guy. They negotiate, coordinate, advocate for you, close the deal with you, and hand the keys to you. They bring you home. Realtors are members of the National Association of Realtors, right by you. Hey, everyone, it's Cal Penn. I'm the host of Earsay, the Audible and... I heart audiobook club. This week on the podcast, I am sitting down with Ray Porter, the narrator of

Andy Weir's audiobook Project Hail Mary, massive sci-fi adventure about survival and science. And what happens when you wake up alone very far from Earth? I really had to make a decision because I caught myself getting that frog in my throat and starting to get teary as I'm narrating some of these sections. And it's like, okay, yo, yeah, yeah, yo, is this indulgent? And I really thought about it. I was like, no, at this point, it would kind of be betraying the trust the author and the listener have in telling this story if I don't go through it. But there's places in this book that deeply emotionally affected me, and I left it on the mic. That's great.

Because it served the story. People will say like, oh my God, I cried at the end. It's like, yeah, dude, me too. Listen to EIRSA, the Audible and IHeart Audio Book Club on the IHeart Radio app or wherever you get your podcasts. Perfectly fair. We did have the little thing in there about Girdle's theorem that the system of axiom has to be consistent, right? And I know that some people, maybe Roger Benrose counts here, think they can just look at a system of axioms and tell you whether it is consistent or not. But it's more subtle than that, I take it.

I mean, you gave the example of the continuum hypothesis. You can either add the continuum hypothesis or not. to set theory. And I've been very amazed to learn. It was a while ago, but I was still amazed that in number theory, you can add the consistency, an extra axiom that says it's consistent, or maybe you said number theory, am I thinking correctly? Or you can also add another axiom and says it's not consistent, and you get equally consistent systems either way. Yes, that's absolutely right. So if you start, say, with piano arithmetic, it's usually denoted PA, then the second Second incompleteness theorem says that no theory can prove its own consistency.

So therefore, it follows, it's just immediate from that, that you can add the negation of the consistency assertion as an axiom and it will still be a consistent theory. So PA plus not con-PA is how people say it. So it's the theory of piano arithmetic plus the single assertion that says piano arithmetic is inconsistent. Now, if you think about it philosophically, it's a lot of it. It's a kind of incoherent theory because it's asserting that a certain theory is true and simultaneously asserting that that theory is inconsistent,

which requires, it seems, a certain kind of cognitive dissonance in order to... Yeah. Except that it's a perfectly consistent theory, we know. I mean, if PA itself is consistent, then that theory is consistent, and that's a consequence of Girtle's second incompletence theory. So wait, sorry, I want to just repeat that very slowly so that we get it. Sure. If PA itself is consistent, then the theory defined by the axioms of PA plus an axiom that says the PA is not consistent is consistent.

Yes, exactly. This is why this is very hard for me. My poor physicist's brain is not up for the task. You know, it's, but it's not just you. It's everybody because like, you know, when I'm teaching and it's, so on and giving a proof of, you know, and it's like there's a fixed point and you have these negations and so it's and this self-referential aspect to it. It's so easy to get tied in knots.

It's definitely confusing for everyone. But what you said is exactly right, the way that you said it is exactly. And so this makes it for me, apparently not for anyone else, it makes it hard for me to be a mathematical realist because there's no fact about the matter about whether those axioms are consistent or not. Oh, I see. No, but if you're a realist, I mean, if you're a platonist, then you would say, well, that theory is merely consistent, but it's not true. It's not a true. Yeah, okay. It's not true in the intended model, right? And that, you know, brings, it brings you to the whole question. Well, what is this intended model? And do we actually have a good reason to think that there is one? And we have the categoricity theorem, as I mentioned, but that seems kind of circular because it's based in set theory. But if you're a set theoretic universeist, then you would. wouldn't have any problem with, of course, the uniqueness of the standard law of arithmetic either. And so you would think that that's the wrong theory. You want to add PA plus con-PA, not the negation.

And that's a strictly stronger theory because con-PA, the consistency of PA is not provable in PA. So it's an extra axiom. And that theory doesn't prove its consistency. So we can do it again. So we can have PA plus KANPA plus KANP. So we're climbing what's called the consistency strength hierarchy. So there's this tower of theories, and it's getting strictly stronger each time where we add the consistency of the earlier theory. But even after you do it, you know, omega many times, infinitely many times, you're not done because then you could assert the union of all.

of those statements and it's a computable list of axioms because we just described how to generate the axioms. So the consistency of that theory at Omega is also formalizable in arithmetic so we can say that it's consistent and so on. We can keep going transfinitely. It's a transfinite hierarchy. So one way of thinking about Gerdo's theorem is that it has identified towering above any theory that you have is this hierarchy of consistency strength. And that hierarchy is, it provably exists because of the incompleteness theorem. And it's getting stronger and stronger consistency assertions. And then the remarkable thing that happened is in set theory, we found these strong axioms

of infinity, the large cardinal axi. And they have exactly this feature that the stronger axioms, say, the exactus, say, the existence of a measurable cardinal is implying the consistency of the smaller ones of an inaccessible cardinal, say, or a molo cardinal is in between those and so on. So these, the large cardinal axioms are axioms that SEPTher has discovered that express, you know, profound infinite combinatorial infinities, the existence of these infinities realizing different combinatorial patterns. They're not these weird logical self-referential things, and they're not talking about consistency at all. They're just infinite combinatorics, but expressing sort of natural principles

of infinity. And they have the character that Gertl predicted because of the consistency hierarchy. They are growing in consistency strength. And so one can take the attitude. So the large cardinal hierarchy is lends a lot of support to this monist picture because Gerdl said, look, any given theory that you might have is inadequate, but there's this tower of things that you should probably be committed to, namely the consistency of that theory and the consistency of that and so on this, this consistency strength hierarchy. And then the set theorists find this incredibly tall tower of axioms, the large cardinal axioms, that instantiate exactly that predicted feature in a very robust way. And so it seems like we're on the path for the one true

theory of sets. So this is, I think, a lot about how the universe view set theorists look at the large cardinals. It's identifying this path upward, the one path upward. But you don't believe it. You don't buy it. No. Well, I mean, I find quite a lot of attraction in the pluralism. perspective. And one way of responding to this one road upward is that there's another aspect, which is that, sure, the large-carnal hierarchy is increasing in consistency strength, and that's quite remarkable, and it makes it sort of an automatically very attractive theory. But yet, there are also many set theoretic statements that are independent of all of the known large cardinals, including the continuum hypothesis. So we know for a fact. Gertil had hoped that the

large cardinals would settle the CH question. There's this famous quote that appears, there must be axioms so abundant in their consequences and verifiable consequences and so on, that we would be compelled to adopt these axioms. And he was referring basically to these large parnell axioms which have all these remarkable consequences, and he hoped that they would settle the continuum of this question.

But it was proved by Levy and Solibay that this is not true, that given any of the standard large cardinal axioms, it's been proved in almost every case, that it's consistent with, if those large cardinals are consistent, then we can manufacture forcing extensions in which the continuum hypothesis is true or in which the continuum hypothesis is false, in which the large cardinal retains its large cardinal nature. It seems to the quasi- outsider that infinity, the notion of infinity, is to blame for a lot of these weirdnesses in the philosophy of math. I mean, and there is a minority view that is tempted to say, like, maybe that's because infinity isn't real. Or we should just stick with mathematics that doesn't rely on that.

Like, even as you said right at the beginning, we talk about infinity, but all of our talk about infinity is used with a finite number of symbols. So is there any bear there? Is that like a reasonable source of skepticism for the whole thing? Sure. There's a whole sort of new research area called, which is looking at sort of potentialism, which is a way of understanding what you just remarked on. I mean, of course, the concept of potential infinity

versus actual infinity goes all the way back to Aristotle. So the question is whether, can we, can we ever have a completed infinity? So maybe the natural numbers, they're infinite. For the potentialist says they're infinite, yes, you can have as many as you like, and you can always have more, but you will never have all of them. So they are potentially infinite. Whereas an actualist would say, yeah, you can have all of them

and then continue to use that actual infinity to go on and do further instructions. So it's a historical, classic debate between these two prospective. And I mean, the funny thing is that throughout most of history, I mean, for millennia, almost all mathematicians were potentialists quite explicitly, including GALS and so on. I mean, until quite recently, whereas nowadays, almost all mathematicians are actualists. And it sort of made a change around the late 19th century. It's not universally true. For example, Galileo was critical of the potentialist point of view.

And one of the things Galileo said was, well, look, if you're a potentialist, then you think that all of these other things are possible. And so you are committed to an infinity, an actual infinity of possibilities. And so he was trying to undermine the potentialist perspective by saying that the potentialists were committed to an actual infinity of possibility, which is a way of being non-potentialist. So that's very interesting. and actually there's a whole, in the dialogues of two new sciences, it's really quite interesting. Really?

I recommend it. And so, okay, I mean, instead of what I call the Aristotelian version of potentialism is, you can think about it in terms of possible worlds. And the sort of current approach to it is to introduce explicitly this modal language of possibility and necessity with crypti models and so on. So the sort of possible worlds of numbers are these finite initial segments, the natural number. That's one concept of potentialism. What you can have at any moment are all the numbers up to and including some of them. And you can have more if you want. Those are the possible ones. And then you can define a kind of modal language for this situation. Like, for example, every number possibly has a successor, meaning that whatever number you have, you know,

maybe if it's currently your largest number, then it doesn't yet have a successor, but it possibly has a successor because you could make it true in a bigger world that that number did have a successor. But then that one is going to have one. Okay, so necessarily every number possibly has a successor is one of the truths of this kind of perspective. Now there's a competing perspective of potentialism for the natural numbers where if you have a number, does that mean you're committed to all the smaller numbers already? Do they come in order

or not? And let me try to convince you that it's really quite reasonable to GINA-Hyda, because maybe I think probably most of your listeners know what a Google is, the number of Google. It's 10 to the 100, or you could write it a 1 with 100 zeros after it in decimal. And maybe some of your listeners also know what a Googleplex is, right, which is 10 to the Google, so that would be 10 to the 10 to the 100, or as you wrote it in decimal, it would be a one with a Google number of zeros after it. That's a Googleplex. Okay.

It's an enormous number. And, but I just described it pretty easily. And we know a lot about this number. For example, it's even. I mean, it's, the prime factors are two and five only because it's 10. It's a power of 10. And so, okay, we can say quite a lot about this number. Yeah.

But could we recite this number? Well, I mean, like if I wanted to recite the digits of it, right, there's a Google number of zeros there. And suppose I was really good at saying digits, you know, maybe I could say a million digits every second. Then even if I did that since the beginning of time, since the Big Bang, I would not even be close to like the tiniest fraction, less than 1%. I mean, the tiniest amount, way less than 1%. So I couldn't possibly recite this number in decimal that way. But now let's think about the sort of typical number less than a Googleplex. A Googleplex was a one with a Google number of zeros.

But like, you know, the numbers less than it also have about a Google many zeros also, right? Most of them do. And the sort of typical one, you know, it's going to be sort of random digits. A Google many of those digits. And most of them are not going to have very simple descriptions at all, because there's just way too many of them. And for a truly random one, the sort of easiest way to say what the number is is going to be to recite the digits. But as we just said, that is going to, even if you recited a million digits every second since the beginning of time, you wouldn't be able to even describe a single one of those numbers. And so there's really no sense in which you could ever hold such a number as an object of thought in your mind.

It's just not possible. You couldn't really say anything about it because you couldn't even describe what the number is very easily. So, okay, so this is a kind of potentialism where you might think, look, I can hold a Googleplex in my mind and analyze it and discuss its features and sort of numbers that are easy to describe relative to Googleplex, like a Googleplex. plus 10 or, you know, sort of in the neighborhood of a Googleplex, I can sort of jump to other places. But most of the numbers less than, and I just can't have any thoughts about it all. And so maybe a bigger mind in a bigger universe with more time or something would be able to describe that. So we could be a kind of potentialist where some numbers come into existence sort of earlier, but not all the smaller numbers do. And that's a fundamentally different

perspective on the nature of this potentialist. And it's going to have different modal truths and so on. The most recent work on potential infinity, I mean, potentialism in sepheryri is sort of severing the idea of potentialism with infinity, with which it was connected for millennia. And the idea rather is that, look, the core idea of potentialism is that the universe of mathematical objects might be unfinished. This is the essence of potentialism.

Even if fragments of that universe have actual infinities inside them. For example, in set theory, we can imagine, okay, the cumulative universe grows forever transfinitely, but I can sort of chop it at an ordinal. It's a kind of potentialist conception of set theory, but if I chop it high enough, then I'm going to have infinite actually infinite sets already existing at that level. And so you realize, well, potentialism isn't really about infinity. It's about whether the universe is completed. And so there's quite a bit of work about potential set theory

and sort of looking at different conceptions of potentialism. And I really found it quite fascinating because, of course, this is an idea that goes back to Aristotle. It's a philosophical question about the nature of our mathematical reality. But it gave rise to this whole mathematical theory of looking at, you know, what's possible, what are the modal truths. You know, we get S4.2 in certain accounts. That's a certain modal theory and S4.3 and other ones.

And only S4, there's quite a lot of technical work going on. And as a result of that mathematical analysis now, it sort of feeds back into the philosophical question by pointing out that there are these different conceptions of potentialism, which I was able to hint at a little bit. And so I just love that situation where a philosophical question, gives rise to this mathematical analysis, which then feeds back in and ultimately refines the philosophical understanding. I think that's just fantastic. Well, and it was really interesting

that you at least used the physical universe in which we live to make a point, right? Which I think is fine. I mean, the best theories we have to describe physics all use the continuum in some very deep way. But no one has experienced most of the, the numbers in the continuum, right? So there is some back and forth to be had, I guess, about whether or not we are helping ourselves to too much that we need to describe the world when we have all of these infinities and their puzzles. Right. I think that's absolutely right. I heard once that Feynman had been asked at some stage, which assumption is, you know, that's sort of basic to physics is most likely eventually to be overturned. And I think this is the

story I heard maybe you know better than me was that the continuity of space and time was his answer. And it's sort of like what you're talking about because, okay, I don't know much physics, but it seems, you know, I wouldn't find it completely outrageous if suddenly we discovered that space is discreet on a very tiny scale or something. I mean, I would think that that's not necessarily incompatible with what we know about on mechanics and so on. If the scale is small enough or something, then maybe the effects are sort of smoothed out at the scale.

We're looking at it and so on. So I could imagine maybe it's discrete on a much tinier scale, in which case all of these infinities are somehow irrelevant to physics, ultimately. If it really is just discrete and finite, then maybe the whole universe is finite in that way. And so what would be the effects? I remember sitting in my office when I was a grad student with you, would and my advisor, and he said, well, you know, a lot of people have this understanding of

infinity because they see this sequence of telephone poles on the highway in the Arizona desert receding into the distance, you know, they imagine it's infinite. But if the universe turns out to be fine, then that picture is not actually going to hear it. And so how could it possibly be the basis of our understanding of this mathematical idealization. And so maybe we don't have good reason to think that P.A. is consistent. Right. I mean, famously, physicists don't frequently interact with philosophers of physics. I get the impression that in mathematics, there's a closer relationship to the philosophy of math. Is that just an outsider's mistaken view, or is that right?

Well, you know, I mean, my work, of course, is directly in the middle of between the two, so I'm interacting with people on both sides, and so it seems quite rich and robust from my experience, the interaction. You know, but, okay, there's a lot of mathematicians who don't really know any philosophy and either sometimes dismissive. I'm sure there's physicists like that, too. In fact, I'm certain. There are.

But mathematics by its nature is, you know, know, a bit more abstract, theoretical, et cetera, you know, it kind of, and logic is right there in philosophy as much as in math. So it seems like there is a little bit more room for productive given take to the average mathematician in a way that the average physicist who is sort of building some detector could not possibly care less about philosophy. But aren't the statistics questions that are fundamentally, some of them, fundamentally philosophical? Oh, there are.

And I live exactly the middle there, but therefore I need to bump into colleagues in the physics department who really, you know, needs some persuading to think that philosophy is worth paying attention to. But so, I mean, the last question then is I think you've given us a pretty good impression that not only did Gertl and Tarski, et cetera, proved some really fascinating things, but that the excitement is still ongoing, that this, you know, really a lot of progress being meet. What do you see as the next big thing to be thinking about in math slash philosophy? Oh, I see. Well, that's great. I mean, there's so many. Sarah. That's fine.

I mean, this work on potentialism is really, I kind of, I view it as a subpart of the larger dispute on pluralism, which is informing quite a lot of work that's happening now in the flood. philosophy of set theory in particular. But also, I guess, more generally, the philosophy of mathematics generally is becoming more connected with mathematics, I think, in a way. Instead of, I mean, maybe one could see it as a possible problem with some previous work in the philosophy of mathematics is that it wasn't enough connected with what mathematicians found interesting about the philosophy of mathematics. And I think that issue is becoming much less,

that the philosophers are becoming more sophisticated mathematically and more mathematicians are getting interested in the philosophical questions, and there's quite a lot of collaboration. It's a difficult subject, and I imagine it's very similar in physics, because to do philosophy of mathematics really well and to understand some of the, you know, to master the, you know, to master the, important questions and phenomena, you really have to have a high level of mathematical skill, I think. But you have to, of course, also have a philosophical outlook. And so it's this combination, which isn't so common, I think. And you need someone with the technical and mathematical skill, but also the philosophical way of thinking. And so it's difficult to make advance. And I think

probably philosophy of physics must be similar in that way. Yeah, but that makes me happy. That means I have less competition for what I want to do. It's more work to convince people that's interesting, but at least I can move at my own pokey pace and still make some progress. All right, Joel David Hankins. Thanks so much for the Mindscape podcast. A pleasure, yeah, really.

Thank you so much for inviting me.