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Where Does It End? — Adrian Moore on The Infinite
0 Playsin 9 hours
Infinity may seem simple, just the absence of limits. But the closer we examine it, the more it unravels into paradox and mystery. Can some infinities be larger than others? How can an infinite hotel be fully booked yet still have room for more guests?
In this episode of Multiverses, I’m joined by Adrian Moore, professor of philosophy at Oxford, to explore these questions. We dive into Hilbert’s Hotel, Cantor’s revolutionary work on transfinite numbers, and the philosophical and even theological implications of the absolute infinite—the place where maths itself seems to break down.
Along the way, we ask: Is infinity something we can ever truly grasp? Or does it forever retreat beyond our understanding?
If you like these topics, where science, maths give way to the unstable ground of philosophy ... subscribe!
Adrian's book: The Infinite
Recommended
Transcript
Complexity of Infinity
00:00:00
Speaker
Infinity is a concept that might seem simple, just the absence of limits. Yet the closer we look, the more it seems to retreat from our understanding.
00:00:12
Speaker
Does infinity come in different sizes? Could an infinite hotel be fully booked and yet still have room for more guests? And what happens when even mathematics can't keep up?
The Absolute Infinite and Theology
00:00:24
Speaker
In this place, which Cantor calls the absolute infinite, Would we find God if only we could reach it? Who better to pick through these topics than Adrian Moore, Professor of Philosophy at the University of Oxford.
00:00:40
Speaker
He's thought deeply about how the infinite connects with mathematics, metaphysics and even theology. He's written many excellent books.
Cantor's Hierarchy and Hilbert's Hotel
00:00:49
Speaker
Most relevant to today's discussion is The Infinite, where he chronicles the thousands of years of history where humans have grappled with this concept.
00:00:58
Speaker
Today, we discuss Cantor's hierarchy of infinities, Hilbert's paradoxical hotel, and whether the infinite is something that we can ever truly comprehend.
00:01:10
Speaker
Beyond the maths and the philosophy, this is a story of human curiosity. What is it that draws us this finite beings to this this idea so vast, so unknowable, yet so irresistible?
00:01:26
Speaker
I'm James Robinson, and this is Multiverses.
00:01:39
Speaker
Hi, Adrian Muir. Thank you so much for for joining me on Multiverses. um ah Thank you very much for the invitation, James. It's a pleasure to join you.
Children and Philosophers on Infinity
00:01:51
Speaker
So I think the infinite is something that children start to think about fairly early on. and In my experience of having a seven-year-old boy, ah you know he's he's already noticed that there's bigger and bigger numbers. And he asks, you know well, is there a final number? And there is none.
00:02:08
Speaker
So that's kind of maybe one route which um I think in childhood, we we first encounter the infinite. Another perhaps quite different route is we is we might look out at some unbounded ocean or the stars above and and start to think, well, you know here is some great majestic thing that's beyond my comprehension. um and And we might think of that as the infinite as well.
00:02:33
Speaker
ah But these are two very different routes. Are they really pointing towards the same thing? Is is there something even there that they point to?
00:02:44
Speaker
So I think that's a ah really ah interesting question to to start with. And um if I may, ah before I go any further, if I may preface my remarks with a reference back to your seven-year-old son, um and that there's no doubt that um infinity is a concept that even young children latch onto at quite an early stage and are exercised by and are fascinated by.
00:03:14
Speaker
And it's also a topic which has always been of interest to philosophers. I mean, if you go right back um to the ancient Greeks, it was one of the concepts that they wrestled with.
00:03:28
Speaker
And and and I'm flagging this because it reminds me of something that I ah very often say to people when they ask me what philosophy is or what a philosopher is.
00:03:42
Speaker
um And one of my favorite answers to that question is a philosopher is somebody who's never really grown up. I mean, I think what we do a lot of the time is people,
00:03:53
Speaker
ask the questions that children ask. I mean mean, the fact that children notoriously are asking why all the time, which is something, of course, that philosophers do as well.
Interpreting Infinity: Numbers, Spaces, and God
00:04:05
Speaker
And the fact that children do ah think about the infinite and are soon exercised by the fact that there's no biggest number or wonder whether there's anything outside space.
00:04:20
Speaker
um And it is striking that you know this this is ah concept that that um grabs our attention at such an early stage.
00:04:35
Speaker
But the question that you've raised is whether we should even really be talking about a concept or whether there's perhaps more than one concept. um you You mentioned the fact that your seven-year-old son has already noticed that um numbers go on indefinitely, that there's there's no biggest number.
00:04:55
Speaker
um But you also talked about the kind of experience that we have when we look out at the horizon or look up at the night sky and see all those stars shining and wonder whether space goes on forever.
00:05:10
Speaker
And there's perhaps a third route in that you might've mentioned as well, which is something else that small children think about, which is, you know, is there a God? um And if there is, is God in some way infinite? and and And does that have anything to do with the fact that numbers go on forever?
00:05:34
Speaker
ah My own view is that they are clusters of concepts here. I think perhaps we shouldn't be thinking of the infinite as as a single concept. um There's a lot of interrelated ideas that are going on here and sometimes interrelated in quite unexpected ways.
00:05:58
Speaker
um And one of the issues is...
Infinity in Mathematics and Mysticism
00:06:01
Speaker
um how how how they're related, what the character of the interrelations are, how these different concepts ah hook up with one another, and why it is that we use the same word in connection with all of them.
00:06:19
Speaker
um And unfortunately, I don't have anything very pithy to say in response to that. um I mean, it's not because, if I may blame my own trumpet, it's not because I have nothing to say in response to that. it's It's almost the opposite problem. there's there's There's too much to be said. I mean, there's there's a lot going on there, obviously.
00:06:42
Speaker
but But one thing that I think is very striking, and maybe we'll get on to a discussion of this in due course, is that some of the very, very technical mathematical work that's done on the infinite um in its own interesting way ah turns out to have connections with some of these more mystical aspects of the concept.
00:07:13
Speaker
um And as I say, maybe we'll get on to a discussion of that in due course. But I'm flagging it at an early stage.
Hilbert's Hotel Paradox
00:07:21
Speaker
um And in particular, the other thing that I'd like to flag at an early stage is that we find this, especially in the work of the great German mathematician Gael Cantor,
00:07:35
Speaker
um Cantor was um ah a 19th century mathematician, brilliant mathematician who who did outstanding work um and is generally reckoned to be the father of what's often called transfinite mathematics, formal mathematical work on the infinite.
00:08:04
Speaker
um And he was also somebody who was deeply religious. um He had quite a profound belief in God as as well.
00:08:17
Speaker
um And he himself thought that there were interesting and deep connections between the mathematical work that he was doing and some of his own religious convictions.
00:08:32
Speaker
um And we can explore that if if if you if you want to. But as I say, I thought it was perhaps going to be helpful to flag at an early stage how even the most technical
00:08:46
Speaker
ah on the mathematical infinite has a bearing on these more mystical or theological or metaphysical conceptions of the infinite.
00:08:59
Speaker
yeah Yeah, I definitely want to get to that. um I think another, it just struck me as you were speaking that perhaps another way
00:09:09
Speaker
these quite disparate ideas ah maybe intersect or or maybe it's better, but they have something in common, which is that with all of them, there are paradoxes that that you can find with the idea of God, you know, can God make something so big he couldn't move it or or with but space and time, of course, there's the paradoxes that Zeno um
00:09:33
Speaker
proposed or wrote about in ancient Greece, you know, Achilles and the tortoise, perhaps we could go into that. And then, of course, Cantor's formal mathematical work, um it's I mean, I would say that it creates paradoxes to me, perhaps to a mathematician, that I know this is all very straightforward, but it it does have very counterintuitive results, which I would say ah quite paradoxical to my understanding.
00:10:03
Speaker
Yeah, well, that's i mean that's certainly true. um and um
00:10:11
Speaker
Whether you think that these paradoxes ultimately troubling or not, whether you think that they point to some profound contradiction that we just can't get our minds around, or whether you think that they're nothing more than results that are counterintuitive,
00:10:39
Speaker
um whatever your stance, they're at least paradoxical to that extent. i mean, some of the formal results are um at at the very least counterintuitive.
00:10:53
Speaker
And some of the puzzling features of divine infinitude, if you are somebody that believes in god again, at the very least, a counterintuitive.
00:11:08
Speaker
Some people think it's worse than that. Some people think that there's something fundamentally contradictory involved in the very notion of the infinite. um And that either this goes to show that we really just can't make sense of the infinite,
00:11:28
Speaker
um or it goes to show that talk of the infinite is a way of gesturing at something that can't properly be expressed or, you know, something along along those lines.
00:11:41
Speaker
um And one of the things the that anybody has to do who's trying to come to terms with the infinite and trying to make sense of the infinite is they have to decide...
00:11:54
Speaker
um what what what to do about these paradoxes, what what the appropriate reaction is. um ah Again, I'm very much in your hands whether whether you think it's worth discussing particular paradoxes to see what we might make of them.
00:12:16
Speaker
I'd be happy to do that. And as you indicated earlier, there are there are plenty of them to choose from. ah Yeah, I wonder if you have, do you have any favorite ones that you think really, don't know, cut to the heart of the matter? Well, maybe we could talk about Hilbert's Hotel. Are you familiar with Hilbert's Hotel?
00:12:36
Speaker
Yeah, I'd love to visit. ah Okay, I mean, but some of your ah viewers may not be familiar with Hilbert's Hotel. So let me see if I can um introduce this paradox, family of paradoxes, actually.
00:12:53
Speaker
Okay. So before we before we go any further, we need to ah talk about something that lies at the very heart of contemporary formal mathematical work on the infinite.
00:13:09
Speaker
And that's the very idea of what it is for two sets or two collections to be the same size as each other.
00:13:21
Speaker
Yeah. Sometimes you can tell that ah two sets are the same size as each other just by counting.
00:13:33
Speaker
So let's suppose that you're ah in a ah room at a meeting and for one reason or another, your mind is wandering and you're just wondering whether the number of men in the room is the same as the number of women.
00:13:54
Speaker
um And you look around you and um you can quite quickly settle that question simply by counting.
00:14:06
Speaker
And it may be that you count the number of men and you see that there are 12 men in the room. And then you count the number of women and you see that there are 12 women in the room.
00:14:17
Speaker
And you conclude that these two sets are the same size as each other, the set of men in the room and the set of women in the room. are the same size because there are 12 of each. And that's very straightforward and very familiar.
00:14:29
Speaker
And again, this is an operation that very small children are accomplished at performing. But sometimes the question arises whether two sets are the same size as each other, and you can't resolve it by counting.
00:14:49
Speaker
ah But nevertheless, you can resolve it. um And ah a simple example of what I have in mind is, um suppose the question arises, um is the number of firstborn twins ah that are that are alive, let's let's suppose we can find our attention to twins where both twins are still living.
00:15:17
Speaker
Is the number of firstborn twins that are still alive? the same as the number of second born twins that are still alive.
00:15:29
Speaker
And I think neither you nor I know exactly what either of those numbers are. ah um ah i mean, I'm sure nobody knows what either of those numbers are. We might be able to make an estimate, but it would only be an estimate.
00:15:45
Speaker
um Nevertheless, we can answer the question whether those two sets are the same size. We we can see that they are. There must be just as many firstborn twins as secondborn twins.
00:15:59
Speaker
ah And the reason that we can answer the question, even though we're not in a position to count the two sets, is that we can see that the members of the sets can be paired off with each other.
00:16:13
Speaker
um To each firstborn twin, there corresponds a secondborn twin, a unique secondborn twin. and vice versa to each second born twin they correspond as a unique firstborn twin and vice versa they can be paired off with each other and so we're in a position to conclude that those two sets are the same size is um irrespective of how many they are there are we don't need to go out and count now so far so good um
00:16:51
Speaker
ah But the sets that we've been thinking about so far, the number of men in the room, the number of firstborn twins, are finite sets.
00:17:02
Speaker
But paradoxes begin to arise if we turn our attention to infinite sets. So you know we've claimed that if you can pair off all the members of one set with all the members of another, then we have to conclude that they're the same size.
00:17:22
Speaker
Well, now think about the set of natural numbers, 0, 1, 2, 3, 4, etc.
00:17:35
Speaker
And think about those that are even, 0, 2, 4, 6, 8, etc.
00:17:46
Speaker
Well, obviously, these can be paired off ah to each. Natural number, there corresponds a unique even number. It's double.
00:17:59
Speaker
And to each even number, there corresponds a unique natural number. It's half. And so it looks as if we're obliged to say that there are just as many even numbers as there are natural numbers altogether.
00:18:16
Speaker
And yet there's a very strong intuition tugging in the opposite direction that no, surely there are more natural numbers than even numbers because the natural numbers include all the even numbers, but it also they also include all the odd numbers as well.
00:18:33
Speaker
So that's already paradoxical in in as much as at the very least it's counterintuitive. I mean, there there are now questions that remain to be addressed about what we should say, what's the best way to describe what's going on here?
00:18:47
Speaker
um And and um you know do do we have to say there are just as many even numbers as natural numbers? Are there reasons for resisting saying that, et cetera, et cetera?
00:19:00
Speaker
But now let me turn my attention to what's called Hilbert's Hotel, which is a variation on the same theme. um where the paradoxical nature of of this result really kicks in.
Sizes of Infinity and Transfinite Arithmetic
00:19:16
Speaker
So Hilbert's Hotel is a hotel named after the um ah 19th, 20th century mathematician David Hilbert, who used to discuss it in lectures that he gave on these topics.
00:19:33
Speaker
And he asked you to imagine... um a hotel that has infinitely many rooms. and And he also asked his audience to imagine that at a particular point in time, every single room is occupied, therefore. ah And a stranger turns up late at night, and it's a cold, wet, miserable night. It's raining heavily.
00:20:05
Speaker
um and he would love shelter for the night, and there's no other hotel nearby. And he asks whether he can be accommodated in the hotel for the night.
00:20:18
Speaker
And the receptionist says, well, I'm sorry, we're absolutely full. And the guy says, but am I not right in thinking that this is um a hotel with infinitely many rooms?
00:20:32
Speaker
And the reception says, well, no, you are you're right. You are right in thinking that. And the stranger says, well, then you can accommodate me. ah All you need to do, as long as the other
00:20:48
Speaker
occupants are amenable to this, all you need to do is to ask the person in room one to move into room two. the person in room two to move into room three, the person in room three to move into room four, et cetera.
00:21:07
Speaker
ah They'll all be accommodated. Nobody will miss out because there are infinitely many rooms available. And that will free up room one, says the stranger. um ah ah and I can move into room one.
00:21:23
Speaker
And the receptionist says, well, what a brilliant idea. Let's let's do that. ah But then the next thing they know, infinitely many people turn up wanting to be accommodated.
00:21:38
Speaker
um And again, they're met with the same response. Well, I'm sorry, we're now fully occupied. And again, they tried this ingenious move, reminiscent of the discussion that we were just having.
00:21:55
Speaker
And they say, well, no, you can accommodate all of us. um Ask the person who's currently in room one to move into room two. Ask the person who's currently in room two to move into room four.
00:22:11
Speaker
Ask the person who's currently in room three to move into room six. etc, etc They'll all have a room to move into.
00:22:23
Speaker
That's no problem. And by the time you've done all of that, the infinitely many odd numbered rooms will be available. And these ah infinitely many people that have shown up can all be accommodated.
00:22:41
Speaker
ah And as I say, ah you know, that's paradoxical, at least to the extent that it's counterintuitive.
00:22:52
Speaker
Whether it's any worse than counterintuitive is is a further question. But at the very least, there's something a little puzzling and rather shocking about but the the situation as I've just described yeah it's very hard.
00:23:11
Speaker
So going from a physics background, kind of hard to test these things, right? I mean, it's a concept. So, i mean, it must be true. i mean, it's just based on the mathematics. There doesn't seem to be anything wrong.
00:23:24
Speaker
um But it is one of those things that we kind of like to see it happen. um Yeah. Well, and I mean, that's an absolutely fair point that you're making. I mean, this is a purely conceptual exercise. and And when we're imagining an infinite hotel like this, know,
00:23:43
Speaker
a It very much is an exercise of conceptual imagination. ah There's no way that an actual physical hotel could could be infinite. um i mean, you might try to to envisage even a and ah physical hotel with infinitely many rooms by envisaging the rooms getting smaller and smaller.
00:24:08
Speaker
um So, you know, each room is sort of half the size of the one next to it. And and you can sort of squeeze them all in that way. But it's, I mean, you know better than I do, you're a physicist and I'm not, but you know, I'm i i'm sure that even that isn't going to make ultimate physical sense.
00:24:27
Speaker
um there are going to be problems about what's going on at the very small level. um and And physicists, as as you know better than I do, ah ah tend to recoil the minute they encounter the infinite.
00:24:41
Speaker
um But mathematicians and philosophers are more likely to feel at home. ah in this more conceptual territory, I think. um And and what a lot of what a lot of mathematicians would say, and for that matter, what a lot of philosophers would say, um is that um you know even if it's physically absurd or even if it doesn't make any physical sense, Hilbert's Hotel makes perfectly good conceptual sense.
00:25:18
Speaker
um And all it is is counterintuitive. You know, the idea that a hotel could be full at a given time and still have the resources to accommodate a newcomer without anybody having to move having to move on ah Well, puzzling, yes. Logically contradictory, no.
00:25:47
Speaker
Makes perfectly good sense. um and And that is, as I say, that's the reaction that that um ah many philosophers, many mathematicians have to to this kind of paradox.
00:26:04
Speaker
I do want to get back to the the points about physics, but I think at first now might be a good time to talk about one of Cantor's great contributions, because I think you've lined it up very well in terms of um saying, okay, well, look, actually, um the maths shows us that the the size of the set of even numbers is the size same as the size of the set of all the real numbers. And and that seems odd because one seems to fit within the other.
00:26:34
Speaker
And it might be natural for the people to be thinking, okay, well, It just turns out that all infinities are the same size and you know once we get to that number, it's just one thing and it's weird.
00:26:45
Speaker
But it turns out it gets even weirder than that, at least and again in my mind. um And I think it's just a wonderful, yeah, a brilliant example of Cantor's genius, how he he proved that. so yeah Yeah, well, let's use this as ah as a cue to start talking about Kantor then. and I mean, as I've already mentioned, um he was a brilliant 19th century mathematician who was the first person who did really serious, rigorous, formal work on the infinite.
00:27:18
Speaker
um And precisely what he was doing was capitalising on this idea that we've been toying with of what's involved in two sets being the same size.
00:27:32
Speaker
um And as you quite rightly point out, um if all we had was the example of the even number, is um then um you know youve you've got these puzzling results like Hilbert's Hotel. But um nevertheless, there would be scope for the view that it's actually ultimately not that counterintuitive.
00:27:58
Speaker
um because you might think, okay, so what this goes to show is that we can distinguish between various different finite sizes.
00:28:11
Speaker
um You know, we might be in a position to say that the number of men in the room is greater than the number of women because there are 13 and only 12 women.
00:28:24
Speaker
um And so there's an example of one set that's bigger than another. So you've got these various different finite sizes that sets can be, and then you've got the infinite.
00:28:36
Speaker
um And um if it turned out that all infinite sets were the same size, you might take the view that actually that's not particularly counterintuitive, that in fact that's intuitive. You know, that's in a way what you might have expected.
00:28:55
Speaker
that once you get into the realms of the infinite, it's no longer possible or appropriate to start trying to to distinguish between different sizes.
00:29:08
Speaker
That's what you might think. And that's exactly what Cantor showed to be incorrect. um And this is and you know one of the most striking and most remarkable features of his work.
00:29:24
Speaker
Cantor showed that there are cases and where um you've got two sets and it's not possible to pair off all the members of one of them with all the members of the other.
00:29:41
Speaker
ah And what you have to conclude is that one of these infinite sets is bigger than the other. They're both infinite, but one of them bigger.
00:29:52
Speaker
is infinite, as it were, to a greater degree than the other. And Kantor showed that that that can happen. um Now, I don't think, unfortunately, that it's possible for us to go over the proof in this context. um Apart from anything else, um it's it's best done visually.
00:30:16
Speaker
um And some of your viewers will already be familiar with this result, and I'm hoping that the viewers that are not familiar with it well will take our word for it, at least for the time being. And then there are places where they can look it up if they are interested.
00:30:34
Speaker
But this is what Kantor showed, that some infinite sets are bigger than others. But actually, that's not the end of it, because he also showed...
00:30:45
Speaker
um given any infinite set, there always will be another that's bigger than it. um
00:30:57
Speaker
ah So ah in in particular, I mean, we were ah talking earlier about the set of natural numbers, um and he showed that the set of sets of natural numbers has more members than the set of natural numbers.
00:31:18
Speaker
So if you start off with a set like the set of natural numbers and then consider the set of all its subsets, that will always give you a bigger set than you started off with.
00:31:31
Speaker
But then you can consider the set of sets of sets of natural numbers. You can consider all the subsets of that new set, and that will be even bigger.
00:31:44
Speaker
And then you can consider the set of sets of sets of sets of natural numbers, and that's even bigger. And so what Kantor was forced to conclude was that not only are there different infinite sizes, there are infinitely many different infinite sizes that these orders of infinity get bigger and bigger and bigger without end.
00:32:14
Speaker
And that's really what gives his work um ah a lot of its interest because what he then did was started to explore the ways in which you can do proper formal mathematics with these different infinite sizes.
00:32:35
Speaker
You can assign measures to them just as we assign measures to different finite sizes. um And that means that there are infinite numbers and you can add these numbers and you can multiply them. There's a whole branch of mathematics here, which is known as transfinite arithmetic.
00:32:59
Speaker
um And it has fascinating results. And that was what Kantor established, a whole whole branch of mathematics, basically.
00:33:12
Speaker
Yeah, so I think if people want to look up the proof, if they search for the diagonal proof or diagonal argument of Kantor, as you say, it's a very visual proof.
00:33:24
Speaker
But I think you know the the other way of kind of getting to the result is, like you say, thinking about the sets of sets or the different... Once you have a set of numbers, um the power set of that, which is all the different ways that you can arrange that's its members into different sets.
00:33:44
Speaker
that gets you to a bigger infinity. um And then when you're at that bigger infinity, you can do the same thing again. You can say, I'm going to take um the power set of this infinity, and that gets you up the the other level.
Absolute Infinite and Theological Parallels
00:33:56
Speaker
ah Actually, my mind's going back to sort of, again, i've got I'm very full of childhood analogies at the moment, probably because I have a young son. But you know if it if you're in the playground and someone says, oh, um you know you're it times 10, and then you take them back and you say, you're it times infinity,
00:34:13
Speaker
And then they say, oh, well, you're at times infinity plus one. I mean, the the real response should be, ah you know, actually power set of infinity of yeah of the countable numbers or something.
00:34:25
Speaker
So there is actually a a valid response there. If there's any um young children listening to this, they they can get one up on their friends by using something like that. Yeah.
00:34:36
Speaker
But ah again, I'm totally with you. I mean, I think there's something very sort of... ah childlike about this fascination that we have with these results and there's something childlike about the the way in which they they pan out. you know Just when you thought it it couldn't get any bigger, there's a way of making it even even bigger.
00:34:57
Speaker
and But I wonder if perhaps I can now use this as an excuse to talk about the thing that I mentioned much earlier, which was um how Cantor himself thought that this related to some of his religious beliefs um because he was religious and he did believe in God and he believed that ah God had a kind of um infinitude. um
00:35:29
Speaker
And he also took it that some of his formal work on the infinite um was in its own way a kind of vindication of ah talk talking in such terms, you know, being being prepared to talk about an infinite being.
00:35:51
Speaker
ah But ah the connection was not... um ah straightforward as that. um And if I can, I'll try and explain why it was unstraightforward.
00:36:12
Speaker
So let's go back to the the point that you that you just made, that um one of the things that's going on here is that we have to appreciate that any set um is smaller than its own power set.
00:36:32
Speaker
So the set of natural numbers is smaller than its power set. The power set of the natural numbers is smaller than the power set of the power set of the natural numbers, and so on indefinitely.
00:36:48
Speaker
And this is why we have to acknowledge bigger and bigger orders of infinity. But supposing somebody comes along and says, well, hold on a sec. um What about the set of all sets?
00:37:04
Speaker
Supposing you sort of throw a big lasso around all of them. um How can that be smaller than its own power set?
00:37:15
Speaker
ah Because ah its power set is just another set of sets. But what we started off with was the set of all sets.
00:37:30
Speaker
um So we've already included all those. So something's gone badly wrong here. ah And Cantor was well aware of ah of this this puzzle.
00:37:47
Speaker
um And by the way, I mean, some of your viewers, most of your viewers, I'm sure, will have heard of something called Russell's paradox. ah And basically, we're skirting Russell's paradox. I mean, we can talk a little bit more about the connection if if you want to. But suffice to say that this um question, what about the set of all sets, is in effect a variation on um Russell's paradox.
00:38:17
Speaker
But anyway, what about the set of all sets? I mean, let's go let's go back to that question.
00:38:23
Speaker
And Cantor was well aware of this question and was well aware that it you know it it was there was something threatening about it. It seemed to undermine all that that he'd been doing.
00:38:36
Speaker
But actually, and this is part of Cantor's genius, this is part of the reason that he's such a a brilliant mathematician, um he He refused to be phased by it.
00:38:52
Speaker
He didn't let it undermine all his work. what What he did it was he said, look, you eventually reach collections like the collection of all sets um that are so big
00:39:15
Speaker
that they don't have a measure any longer. there's there is There's no such thing as an answer to the question, how many sets are there altogether? um We've now reached the point that we can't any longer assign a measure.
00:39:35
Speaker
um And Kantor, I mean, there are various bits of terminology that he used as sometimes he talked about inconsistent totalities.
00:39:48
Speaker
But another bit of terminology that he used in this connection was he sometimes talked about the absolute infinite. And what he was doing was gesturing at this kind of infinity that escaped even his own formal technical apparatus.
00:40:13
Speaker
ah And this was the connection that he saw between his mathematical work and his theological beliefs, because he thought that the kind of infinity that God enjoys, the kind of infinitude that um we have to ascribe to God, is of this more radical kind.
00:40:40
Speaker
the absolute infinite, the sort of infinite that's beyond measure. So when he when he um said that he believed God to be infinite, and when he was doing formal work on infinite collections,
00:40:58
Speaker
ah those two kinds of infinity were not directly related they were indirectly related through this connection with the absolute infinite.
00:41:17
Speaker
Right. Yeah. and Let me try to read that back because I think it's a really powerful... Okay, good. so Because i'm I mean, I'm never very sure how um much sense this makes when people are encountering it for the first time. so I think it's just a ah very powerful idea and it probably bears repeating. um so So something like, I mean, we have this very intuitive and idea that the power set, I mean, it is it's clear that the power set is bigger than the set um And that makes perfect sense. right You take the natural numbers. um
00:41:53
Speaker
It's clear that you can create lots and lots of more collections of those. You can put them into more buckets than there are natural numbers. um Or maybe that's not clear, but it's it's certainly provable.
00:42:04
Speaker
um But it does feel normal. Certainly when you start with finite numbers, ah it's easy to see how that works. There's three people in a room. ah You can you know put two together or the other two in various different ways or have them each on their own.
00:42:18
Speaker
um
00:42:21
Speaker
So you know and all well and good starts very simple. um But then you kind of reach this point where your sets are so big and you that they seem to suggest that the maths is is breaking is breaking down.
00:42:40
Speaker
um
00:42:42
Speaker
and And I guess Cantor's response to this is, well, the math has reached something that it can't decide, but it's it it's maybe pointing towards a place where math doesn't go, but there's still something there, right?
00:43:01
Speaker
um yeah Because another response would just be, well, there's nothing there, and perhaps that's a legitimate one, but it's hard to see why there wouldn't be anything there because what's wrong with ah being able to arrange things into different sets? You might say, okay, well, it makes sense to arrange the natural numbers into...
00:43:17
Speaker
and I can arrange those into different sets, but yeah and what's wrong with arranging the sets of the sets that I've made into different sets? you know um But you know maybe there is.
00:43:29
Speaker
The other thing I wanted to say just makes perfect sense is that yeah it is intuitive um that there should be more kind of ways that things could be than there are and i guess that's loosely linked to this idea that you you have a state of affairs you have a uh you know a set of things but you could arrange them conceptually in many more ways than than they actually are um or i did just want to mention as well that uh russell's paradox in in case people um aren't aware of it uh it's very simple to summarize it's just
00:44:04
Speaker
the The classic statement is, you know, there's this barber on an island and he's only going to shave, he's going to shave all and only those people who don't shave themselves. And the problem is, well, again, does he shave himself or not? Because if he does save himself, well, he shouldn't be shaving himself because he said he's only goingnna shave those people who don't shave themselves.
00:44:21
Speaker
So you get this kind of self-referential problem, which is very analogous, as as you say, to the problem sets. um But one doesn't conclude from Russell's paradox, oh, God, right? So, you know, on the one hand, um there's something very appealing about Cantor's proof because we've reached this, you know, grandiose state where math seems to be breaking down and yet we don't know why um On the other hand, you know, maybe there's just like a prosaic explanation, which is, well, you know don't go there.
00:44:52
Speaker
ah It broke down much earlier. um But where, I don't know. Yeah, well, um good. So so i i I like the way that you've put this. And and um I mean, if I can, if um
00:45:12
Speaker
ah grind an axe of my own at this point, I mean, i Again, this is perhaps a nice excuse for me to do so.
00:45:24
Speaker
I mean, my own my own view for for what it's worth is that we have to be very, very sceptical about whether...
00:45:38
Speaker
we can grasp the infinite, whether we can really properly make sense of the infinite, um whether we can tame the infinite.
00:45:49
Speaker
um And I think there are all sorts of reasons for for denying that we can. Now, um often, if if you express a view like that, often um mathematicians in particular,
00:46:10
Speaker
turn round and just present Cantor's work. And they say, you're you're just um you're just being absurd. um ah This is mystery mongering for the for the sake of it. um What Cantor did was to show that we can make sense of the infinite. you know Here's this really rigorous, formal mathematical work that's unassailable,
00:46:39
Speaker
um work of of the very highest caliber in which exactly what he's doing is showing how you can measure infinite sets and do calculations with infinite numbers and and blah, blah, blah.
00:46:54
Speaker
So what are you talking about saying that we can't make sense of the infinite? ah And so my response to that is not to challenge Cantor's work,
00:47:07
Speaker
um which is absolutely unassailable, and I agree is work of the very highest caliber. I mean, this is mathematical craftsmanship of of of the greatest kind.
00:47:20
Speaker
So that's not in dispute. I mean, let's not quarrel with Kantor. You need to be a very foolhardy kind of philosopher to think that you could quarrel with Kantor.
00:47:32
Speaker
That's not the issue. The issue is what we mean by the infinite. I mean, this takes us right back to to square one. I mean, the you know, the opening question that you put to me, what is our idea of the infinite?
00:47:47
Speaker
And um if you ask, most people what they understand by the infinite, you're going to get a variety of different responses.
00:47:59
Speaker
But I think there's a very, very good it's only a question of time before concepts like endlessness, unlimitedness, and immeasurability get a mention.
00:48:22
Speaker
you know so but At some point, somebody is likely to appeal to one or more of those concepts. um i mean perhaps i mean if If they're more theologically minded, perhaps they'll start talking about perfection.
00:48:40
Speaker
and or omnipotence or something like that. But ah certainly if it's somebody that's more mathematically minded, as I say, there's a good chance they'll they'll talk about what goes on forever, what's endless, what's unlimited, what's immeasurable.
00:48:59
Speaker
And indeed, those are the sorts of words that you're liable to find in a dictionary if you look up the infinite in a dictionary. So let's go back to Cantor's work.
00:49:13
Speaker
Where do those concepts apply?
00:49:18
Speaker
They apply to the whole hierarchy. I mean, it's not true that the set of natural numbers exhibits endlessness or unlimitedness or immeasurability.
00:49:35
Speaker
Because what Kantor does is show that we can measure it. He gives a precise um numerical definition to how big it is.
00:49:47
Speaker
And you don't have proper endlessness there because um the the argument that you sketched earlier about each set being smaller than its own power set
00:50:00
Speaker
The endlessness is emerging further up the hierarchy. I mean, the endlessness is the endlessness that you get as you travel higher and higher and higher up this hierarchy of power cells.
00:50:11
Speaker
And it's not an and unlimited either. It's limited in size. That's precisely the point. um There are fewer natural numbers than there are sets of natural numbers.
00:50:24
Speaker
The really unlimited, the really measure the really endless, looks as if it's what emerges when we're dealing with the absolute infinite, what Kantor himself wanted to call the absolute infinite.
00:50:45
Speaker
And it's a real question what kind of grasp we have of that. And for Kantor himself, it was a real question, what kind of grasp we have of that. ah This ties in again with his religious convictions.
00:51:02
Speaker
um He, at one point, said something like... um I can't remember exactly how the quotation went, and in any case, um relying on an English translation, but something like the absolute can only be acknowledged, it can never be known.
00:51:25
Speaker
um and And this is a way of admitting that there's a grasp of the absolute that we lack.
00:51:41
Speaker
And so, as I say, my my own view for what it's worth is that we have to think very seriously about whether we have any grasp on the on the truly infinite. And indeed, Cantor's work, so far from putting pressure on that scepticism, fuels that scepticism.
Language, Mathematics, and Infinity
00:52:04
Speaker
There's almost like a progressive breakdown of our... sort of capabilities of of understanding. and so At some point, language, or everyday language, loses its grip on the infinite, but maths at least holds onto to it and is able to give us a formal description of um this ascending hi hierarchy of infinitudes.
00:52:25
Speaker
But then you reach, like you say, the the sort of limit of that is something that Kantor admitted even maths can't grasp. um and and we're we're all at sea.
00:52:36
Speaker
um The quote that comes to my mind actually is is is from Niels Bohr, who was, as you know, a physicist, but also a bit of a mystic. And he said, ah yeah it was of atoms where he said, language, you know, we must be clear that ah when it comes to atoms, we can only use language um as poetry.
00:52:56
Speaker
And he meant, well, you know, we at least have a mathematical description of how stuff moves, but we can't put it into everyday language. um and But in this case, there's some point where even the maths breaks down. Yeah.
00:53:11
Speaker
yeah
00:53:14
Speaker
i I think that's, I think, I think that's right. I mean, again, you know, you've, you've always got to be, careful as always in in philosophy or maths or any other discipline for that matter. You've got to be careful to be as as clear as possible and to define your terms as carefully as possible. And of course, you know, there are ways of defining the infinite um whereby um it's just undeniable that Kantor showed that we can do the mathematics of the infinite.
00:53:48
Speaker
And So for example, if you if you define the infinite in the in the way that most mathematicians do, um which is to say if you call a set infinite when it is the same size as one of its own proper subsets,
00:54:14
Speaker
So like you know the fact that the natural numbers is the same size as the set of even numbers, if you say, well, if that if that happens, if a set has that feature, then that's what for the set to be infinite. but Well, fair enough.
00:54:32
Speaker
um Okay, so if that's what you mean by the infinite, then Kantor has shown us how to do the formal mathematics of the infinite.
00:54:43
Speaker
He's dealing with sets that are like that. um And if that's what you mean by the infinite, then he's dealing with infinite sets. But as I say, i mean, if you look up infinite in a dictionary, you're not going to find talk about ah a and sets having as many members as one of their own proper subsets, or at least it would be a very unusual dictionary that would,
00:55:10
Speaker
um proffer that as any kind of definition. What you're going to find in in in the case of most dictionaries are words like endless or immeasurable or unlimited.
00:55:24
Speaker
These are going to be the terms in which the infinite is is defined, or these are going to be suggested synonyms for the infinite. And then it's a real question whether they apply to the natural numbers or not.
00:55:39
Speaker
um And so this goes back to the point that you were making, that Cantor's work shows that that conception of the infinite, which arguably is the familiar everyday conception of the infinite, or one of the familiar everyday conceptions of the infinite, at least one of them, that conception of the infinite
00:56:10
Speaker
seems to resist
00:56:14
Speaker
mathematical treatment.
00:56:17
Speaker
Does it? Does it create problems, though, for other conceptions of the infinite?
Physical Manifestations and Cosmology
00:56:23
Speaker
For example, if we think of, you know perhaps we get some evidence that the universe is infinitely large and there's some cosmological models, which are probably less fashionable now, but that suggest that it is.
00:56:36
Speaker
um And there's even kind of wilder thought experiments from Max Tegmark where, you know, he believes we live in a kind of type of multiverse where it's not just... um the laws of quantum mechanics which are producing branching but but actually all of the physical constants and and possibilities are realised.
00:56:56
Speaker
um Leaving aside perhaps Tegmark's vision but if we just think about a cosmologically infinite universe, but that seems much more palatable than the sort of absolute infinite that that Cantor was getting about. And it and it strikes me that as long as we don't let allow um for sort of infinite recombinations of the subsets and or try to discuss infinite recombinations of the subsets of that infinite space, we'll have something which we can perhaps get something of a grasp on. Of course, it will be immeasurable um and you know and impossible for us to survey in its entirety, but it it doesn't seem as paradoxical. Yeah.
00:57:39
Speaker
yeah Yeah. No, that's a, I mean, I think that's an excellent ah point that you're making. And, you know, once again, i have to defer to you, you're you're the person with training in physics and I'm not, and you know far more about this sort of thing than I do. But, yeah.
00:58:01
Speaker
ah what what What about the idea that the infinite has application to physical reality? um And, you know, you've sketched some of the ways in which that might be true. And certainly, i mean, if you go back to the...
00:58:22
Speaker
to newton and If you go back to the 17th century, before people were aware of all the complications of relativity and and all of that, um it was taken for granted the um space was was infinite.
00:58:48
Speaker
Whether the physical universe inside space extended infinitely with is is was a further question. and i mean, newton newton Newton himself would have said that it that it did. He would have said that the physical universe was was infinite as as well as space. But the popular view was that space, at least, was infinite and that time was infinite as as well,
00:59:15
Speaker
um
00:59:18
Speaker
in both directions, that there was an infinite past and an infinite future. And and um 20th, 21st century ah developments in physics have shown us that it's but it's not quite as straightforward as those guys thought.
00:59:40
Speaker
um But nevertheless, there may still be room for the view there's a kind of infinity and and ah exhibited in the physical world. And maybe there is still room for the view that space is...
00:59:54
Speaker
in infinite, even though it's no longer orthodoxy. And maybe there is room for the view that our universe is just one of an infinity of universes in a kind of pluriverse.
01:00:10
Speaker
ah So I think, you know, we have to take these possibilities seriously. And I think you're quite right to raise the question and then um indeed to press the question and say, so,
01:00:24
Speaker
What does this show us about what kind of grasp we have on the infinite? um And it's ah it's a fair point because this looks as if it is something close to a sort of everyday conception of the infinite. I mean, words like endlessness and unlimitedness and immeasurability do kick in at this point.
01:00:46
Speaker
in a way that they don't with Cantor's mathematical work. um ah Nevertheless, you know, it looks as if we can make a kind of physical sense of what's going on. um And,
01:01:02
Speaker
you know, i so i i so I simply agree with all of that. And and I think it's just a stark reminder of how unstraightforward all of this is um and how you have to,
01:01:15
Speaker
um think about ah the infinite in all its various applications and various manifestations.
Human Curiosity about the Infinite
01:01:23
Speaker
So if the question i mean if the question arises, can we understand the infinite?
01:01:32
Speaker
Let's put it in those very bold terms. Or do we have a grasp of the infinite? um and The only legitimate answer to that question is it depends what you mean, and which is the answer that philosophers are always giving to questions. i know ah But, you know, you've just got to be more precise. um And there are going to be some ways of making it more precise whereby the answer is yes.
01:02:02
Speaker
And there are going to be some ways of making it more precise whereby the answer is no
01:02:08
Speaker
Yeah, I think that's right. And it gives some comfort, at least, that um even though, as you hinted earlier, physicists sort of shy away from infinities. And, you know, there's places where they're bit frustrated, like the center of black holes, where you have something that appears of infinite density or and infinite.
01:02:31
Speaker
And that kind of gets to dismissed dismissed as a singularity. and We can't say exactly what's going on there. But also excites people as well a lot because it's these kind of limits or edge cases that often are where you discover really interesting things. Yeah.
01:02:48
Speaker
yeah physics is really fond fond of boundary conditions as well. I remember like yeah if you have to figure out something as as simple as you know what is the velocity that you have to throw something off the Earth so it escapes the the gravity? Well, the way you run that equation is you have to imagine that it go that you're throwing it to infinity because that's that's the only place that it truly escapes gravity.
01:03:10
Speaker
um and And actually, I guess, Nusson's argument for why the universe was infinite was... related to that, I guess he thought that if all the if it wasn't, everything would kind of collapse in on itself under under gravity.
01:03:23
Speaker
um Of course, we could have a problem even with physically infinite universe if part of the physics required us to think in terms of everything being kind of rearrangeable.
01:03:34
Speaker
um And, you know, it was based on that. So there we might end up with somewhere where our physics is pointing us towards or down the route of Cantor ah towards something which we really would struggle to grasp so um i don't know whether i hope or fear that we might end up with uh that but so that would be a certainly interesting thing but um we're some steps away from i mean that's that's very speculative um But it is a reminder, isn't it, of how um ah in in physical reality, there are all sorts of different ways in which the infinite might manifest itself. I mean,
01:04:14
Speaker
um ah ah a lot of physicists obviously believe that In fact, I think this is the orthodox view that everything began with a big bang.
01:04:28
Speaker
um And there are quite a few physicists that think that it's all going to end with a big crunch, that eventually um gravitational pull will bring everything back down together a in into this tiny, infinitely tense blob or whatever it goes.
01:04:50
Speaker
ah And um your first thought might be, well, that's a striking example of how it's all ultimately finite. Big bang, expansion, contraction, big crunch, end of story.
01:05:07
Speaker
But of course, who's to say that it doesn't then just repeat you know the ah the idea that it might be bang, crunch, bang, crunch, bang, crunch, bang, crunch, going on indefinitely.
01:05:22
Speaker
Yeah. yeah And you have to contend with the you know what's going on when it's um just banging and just crunching. yeah Is there a point at which everything is concentrated in, again, yeah infinite ah density? Yeah.
01:05:37
Speaker
yeah i mean that's right i i just use the phrase infinite blob and infinitely dense blob or or some such phrase and even as i was saying it realized that that was already a hostage to fortune yeah i we're we're running low on time here unfortunately we are Quite finite beings.
01:06:00
Speaker
And yes I'd invite you to ah actually, i I'd like you to um ah give us your thoughts on on that. You know how is it that we as um such ah limited um blobs of matter are able to and motivated, I think, are and fascinated and by these these ideas? um Where does it come from?
01:06:27
Speaker
Well, I do. I mean, I actually do believe ah that you have just posed the $64,000 question. i mean, I think that is the really crucial question in all of this.
01:06:43
Speaker
what What is it about us, finite beings, that we are? um that fascinates us about these questions.
01:06:55
Speaker
And if we can't grasp the infinite, if there is ultimately a sense in which the infinite is beyond our grasp, what what are we even doing saying that we can't grasp the infinite? I mean, isn't that already paradoxical? Isn't that us grasping this thing which is ungraspable?
01:07:14
Speaker
So i think that I think this is taking us right to the to the heart of all of these questions, actually. And again, if I can do a bit of axe grinding, I mean, this is a view that I've tried to defend in a number of places. um i do I really do think that part of what's going on here is that we're exercised by the infinite because we're
01:07:44
Speaker
self-conscious um and in particular as self-consciously finite. I mean, in a way what's going on here is that we're aware of our own finitude and exercised by our own finitude, which takes all sorts of forms.
01:08:06
Speaker
um I mean, you know, there's the simple fact that each of us only occupies a finite amount of space. and will only be around for a finite amount of time. um But there's all sorts of other respects in which we're limited and finite.
01:08:27
Speaker
um And the universe is full of things that are limited and finite. um But what's distinctive about human beings this is that they're self-consciously limited and finite. They're limited and finite, and they're aware of the fact that they're limited and finite.
01:08:49
Speaker
And I think that's part of our fascination with the infinite, because there is inevitably this sense of a contrast. You think, well, okay, we are limited. It seems clear that we're limited, but as opposed to what?
01:09:07
Speaker
ah And does the sheer fact that we have boundaries inevitably mean that we have to reckon with the possibility of that which doesn't have boundaries?
01:09:20
Speaker
um Is everything finite and limited? Is there no such thing as the infinite? ah or are we finite and limited as part of some great cosmic a story that is unlimited in some way?
01:09:39
Speaker
And I think, you know, you posed the question, where does our fascination with the infinite come from, given that we're finite? And I think that's just an excellent question. And I think in a way,
01:09:57
Speaker
it's a question which serves as its own answer. um Our fascination with the infinite comes from fact that we're finite. um Let's go back to young children, you know, maybe your seven-year-old son.
01:10:15
Speaker
is he... why um is a um why why you know Why has he noticed at such a young age that numbers go on forever? Well, first of all, he would have been introduced to numbers as a way of counting the stuff around him.
01:10:36
Speaker
um you know No doubt one of the first things that he did with numbers was, I don't know, count bananas in in the fruit bowl or toys that that he had round about him or whatever.
01:10:54
Speaker
um
01:10:57
Speaker
This is a way of engaging with his finite surrounds. But then quickly, quite quickly, he realises But it doesn't stop there, does it it ah does it?
01:11:17
Speaker
this is a kind of self-consciousness that's kicking in. um i mean, that's not necessarily going to be how it finds expression. But i think that I think it is a kind of self-consciousness that's kicking in. He's aware...
01:11:32
Speaker
that um he's been using finite numbers to deal with the finite surrounds that he can engage with as a finite being.
01:11:44
Speaker
um And then he thinks, but it doesn't stop there. And that's a self-conscious thought, it seems to me. Mind you, I've never met your seven-year-old son apart from briefly a little earlier.
01:12:04
Speaker
I may be attributing thought processes to him, which I shouldn't. But if he's the least bit typical of seven-year-old children, then I think this is something like what's going on.
01:12:17
Speaker
Yeah. Yeah. I think, as you say, it's the idea of our own limits leads us to think, well, what what if things are were otherwise? um ah It's quite an elliptical quotation, but i'm there's a wonderful poem by Chen Chen.
01:12:34
Speaker
i think the title might be, or it certainly has the line, when I grow up, I want my i want to be a list of further possibilities. Yeah.
01:12:46
Speaker
Which, yeah, um I mean, as you say, if philosophers are people who never grow up, um
01:12:57
Speaker
that that list of possibilities never gets blown down. Yeah, again, that's interesting you talking about further possibilities because that ties in with something that you said earlier. ah ah I can't remember now exactly how you put it, but you said something like – um you know isn't one of our first um thoughts about the infinite, the thought that um the way things are arranged is just one of the ways in which they can be arranged. I mean, you didn't put it exactly like that, but something to that effect.
01:13:33
Speaker
um And ah yeah ah yes, and again, I think that's the same thing that's that's kicking in because um part of this self-consciousness that I keep going on about self-consciousness and self-conscious awareness that we are limited, we are finite, and the actual is just one of a number of a range of possibilities.
01:13:58
Speaker
um ah You know, here's how things are. here Here you and I are engaged in conversation, but we might not have been. um the The technology might have thwarted us at the beginning of this process, and or things might have panned out differently, or um your seven-year-old son might have come in having had a nasty accident that you just had to attend to that took priority over any conversation with me.
01:14:29
Speaker
um And, you know, there are all these different ways in which things might have been, all these different ways in which the stuff in the universe might have been rearranged.
01:14:41
Speaker
And that's a sense of the infinite as well. The sense, ah again, that self-conscious sense that the finite limitations of how things actually are point to a broader infinite range of ways in which they might be.
01:14:58
Speaker
Yeah. Yeah. I think, yeah, this is um one of those conversations where... ah We could go on and on. ah We could.
01:15:10
Speaker
We absolutely could. Finite time was has thwarted us, hasn't it? we Yeah. We come to the end. yeah Well, thank you so much, Adrian. um Yeah, this has been, know, what a wonderful tour.
01:15:25
Speaker
we've We've covered a lot of incidents. I want to say infinite ground. um And, yeah, it's been lovely. Well, thank you very much. It was a lovely excuse to talk about the infinite, which I'm always happy to do. So thanks very much for inviting me to take part in this conversation.
01:16:02
Speaker
Thank you.