Data & The Universe (Understanding Infinity Part 3)

What happens when you finally reach the end of forever? And what exactly is the difference between Omega and Aleph-null? Professor Hannah Fry and Michael Stevens (VSauce) return for the final (possibly not) chapter of our infinity series, unpicking the frankly absurd mechanics of multiple infinities. They reveal how mathematicians don't just measure the endless void, but actively organise it, exploring the bizarre gap between the sheer size of a boundless set of numbers and its actual order. Following Georg Cantor's mathematical meltdown, this episode tackles the mind bending rules of the infinite, why Aleph-null is just the baby of the infinite family, and how jumping to Omega requires a complete rewiring of human logic. ------------------- For more information about Cancer Research UK, their research, breakthroughs and how you can support them, visit ⁠⁠https://cancerresearchuk.org/restisscience⁠⁠ Cancer Research UK is a registered charity in England and Wales (1089464), Scotland (SC041666), the Isle of Man (1103) and Jersey (247). A company limited by guarantee. Registered company in England and Wales (4325234) and the Isle of Man (5713F). Registered address: 2 Redman Place, London, E20 1JQ. ------------------- Find The Rest Is Science all over the internet by ⁠⁠clicking here.⁠⁠ ------------------- Video Producer: Adam Thornton + Oli Oakley Video & Social: Bex Tyrrell Producer: Simona Rata Senior Producer: Lauren Armstrong-Carter Head Of Digital: Samuel Oakley Exec Producer: Neil Fearn Learn more about your ad choices. Visit podcastchoices.com/adchoices

Speaker names marked ? are inferred from speaking time and the show’s known hosts — they are not voice-verified.

• 0:01Hannah Fry?

  Welcome to The Rest Is Science. I'm Hannah Fry.

• 0:03Michael Stevens?

  And I'm Michael Stevens.

• 0:04Hannah Fry?

  And we have been talking for a long time about infinity. But we're still talking about it.

• 0:10Michael Stevens?

  We're still talking about it. I don't know. This series might become endless. Or maybe third time's the charm. We'll see.

• 0:17Hannah Fry?

  Maybe third time's the charm.

• 0:18Michael Stevens?

  I think they will end because we have decided we're done. They certainly won't end because we've covered all the numbers.

• 0:25Hannah Fry?

  No, no, certainly not. The thing is, there's just so much delight to discover. There's so many strange weirdnesses once you allow your mind to wander into the depths of infinity.

• 0:35Michael Stevens?

  Not only are there a lot of strange weirdnesses, but there's a lot of different ways to talk about each strange weirdness. And sometimes I'll find a whole new way of describing a transfinite number. And I'll be like, that is so much clearer than what I've ever heard before. I'm so glad I heard it. But I wanted to start with an object, okay? To kind of like cleanse our palates. So we're going to talk about something that's physical and real. Are you ready, Hannah?

• 1:02Hannah Fry?

  I am, absolutely.

• 1:03Michael Stevens?

  I have here a toothpick. You see that?

• 1:07Hannah Fry?

  Not where I thought you were going, but sure.

• 1:09Michael Stevens?

  This is a toothpick, but guess what?

• 1:11Hannah Fry?

  Tell me.

• 1:12Michael Stevens?

  Everything that could ever be said is on this toothpick.

• 1:20Hannah Fry?

  Oh, tell me how.

• 1:21Michael Stevens?

  Here's how you do it. Here's how you do it. All you need to do is put a little simple code together. And we can stick to the English language if we want. We can say, okay, the letter A is 01. The letter B is 02, C is 03, and so on. That brings us to 26, for 27, 28, and so on. We can add in punctuation, numerals, whatever. Perfect. Now, let's think of something that could be said. How about the word cat? Okay, in our code, cat would be C-A-T, which is 03, 01.

• 1:56Michael Stevens?

  Two zero. T is the twentieth letter. Okay?

• 1:59Hannah Fry?

  I'm impressed how quickly you remembered that, but go on.

• 2:02Michael Stevens?

  You know, I know that because I cannot forget the fact that S is the nineteenth letter. Like that, to me, it's just, maybe it's because my last name is Stevens and as a kid I'm alphabetized a lot. I'm like nineteen.

• 2:13Hannah Fry?

  I one hundred percent would have just panicked and gone through and sung the alphabet at that moment in time, but you're a better man than me. Carry on.

• 2:20Michael Stevens?

  Anyway, so here's the word cat, right? Zero three zero one two zero. Now, let's put a decimal point in front of that and then all we have to do is measure from the tip of this.

• 2:32Hannah Fry?

  I've just got it. The penny's just dropped.

• 2:35Michael Stevens?

  We're gonna go from the beginning of the toothpick, we're going to go point zero one zero three two centimeters to my right. And we'll put a little mark there. And then I can give this to someone. Later on, they can take it. They can grab themselves a ruler. Let me just get my slap bracelet ruler. You gotta be fashionable while you're measuring. And they could line it up and they could look and they could say, okay, so that mark is at, oh, it's at point oh three oh one two. Well, that says cat. So in that fashion, I could, with more and more precision, add more and more characters to the point at which I could put the entire Encyclopedia Britannica on a single notch on this one toothpick.

• 3:19Michael Stevens?

  Therefore, everything that has ever been said, that will ever be said, how you will die and every way that you won't die, it all is on this toothpick right now.

• 3:29Hannah Fry?

  I really like that. I like the idea that you are containing it within one single notch.

• 3:35Michael Stevens?

  Right?

• 3:35Hannah Fry?

  There's something very nice about that.

• 3:38Michael Stevens?

  And you can do this because I'm assuming that we have infinite precision. And I think that was a nice, like, kind of swerve because so far we've been ballooning up more and more, bigger and bigger. But you could also get smaller and smaller and closer and closer and keep finding more and more and more.

• 3:59Hannah Fry?

  This episode is brought to you by Cancer Research UK.

• 4:02Michael Stevens?

  If you wanted to type out the entire human genome, you would have to type at 60 words a minute for eight hours a day for about 50 years. Okay, that's the scale of the DNA rulebook inside each one of your cells, telling it when to grow, when to divide, and when to stop.

• 4:21Hannah Fry?

  And different tissues read that same rulebook in different ways. So a skin cell doesn't behave like a lung cell.

• 4:27Michael Stevens?

  And cancer can begin when those instructions change. Not one dramatic moment, but through small gradual edits over time.

• 4:36Hannah Fry?

  Now, cancer isn't one disease. It is more than 200 types, shaped by where those changes to the rulebook happen and how cells respond.

• 4:46Michael Stevens?

  Cancer Research UK is the world's largest charitable funder of cancer research, backing studies across all types of cancer.

• 4:53Hannah Fry?

  Work that takes years of very careful, steady progress to deliver each breakthrough.

• 4:59Michael Stevens?

  For more information about Cancer Research UK, their research, breakthroughs, and how you can support them, visit cancerresearchuk.org/the rest is science.

• 5:10Speaker 02

  This episode is brought to you by Indeed. You're ready to level up your business, but the pressure to find the right person to make that happen is intense. Indeed-sponsored jobs can help carry the load. It boosts your job post to help it reach quality talent fast. Cut through the chaos with Indeed-sponsored jobs. Get a $75 sponsored job credit at indeed.com/podcast. Terms and conditions apply. Hiring now? Then this is a job for Indeed-sponsored jobs.

• 5:45Hannah Fry?

  Right, picking up where we left off, which was an eye-wateringly mind-bending series of definitions, which started with finishing a marathon after an infinite number of people had finished a race. We had defined one type of infinity that is the one that everyone sort of knows and is comfortable with, which is the counting numbers. And I think, did we give it its real name? Did we call it aleph null?

• 6:12Michael Stevens?

  Yes, yes, we did. Aleph null is the name for the smallest size of infinity. How many counting numbers there are, positive integers. Zero, one, two, three, four, five, and so on. But then we talked about an infinity that is demonstrably larger. And it's the number of real numbers that there are. Not just the counting numbers, but also include in there all of the rationals, right? All the fractions, but also all the irrationals. All the real numbers we showed using diagonalization, which is a very lovely word. And you should go and look at part two if you missed it. Because how that was demonstrated by Cantor is beautiful.

• 6:52Michael Stevens?

  But you can show that there are literally more real numbers than there are positive integers.

• 6:59Hannah Fry?

  Because that second one has a name as well, right? It's actually called, it's called Beth one.

• 7:04Michael Stevens?

  Yes, it's called Beth one because Beth is the second Hebrew letter in the Hebrew alphabet. Whereas aleph is the first. So in the episode in part two, we called it the cardinality of the continuum, which is also what it is. But to make it shorter, and you don't sound as smart. It sounds like you're just talking about how many friends you have named Beth. Beth one. Beth null is equal to aleph null. Beth one is declared to be the cardinality of the continuum. And then the Beth numbers keep going up. Each one's a larger infinity than the other. And they're defined by each being a power set of the one before.

• 7:44Michael Stevens?

  Where a power set is every possible combination of all the elements of a set you can make. I think it's worth discussing this because this will show us how we get into bigger infinities beyond the two we covered last time.

• 8:00Hannah Fry?

  Okay, so imagine you've got all the decimals. You've got everything that could possibly live on a toothpick. And an infinitely long toothpick. Now, to get to bigger and bigger infinities, we're not just taking the toothpick as it stands, we're thinking about how you could sample numbers from that number line, right? How can you order them? How can you put them together in different and distinct combinations?

• 8:27Michael Stevens?

  Yes. And so the power set of the natural numbers is just every possible combination of numbers you could pull out of there. It includes all the evens, that's a set. It also includes just 1 and 3. Now, using diagonalization, we can very quickly show that the power set of the naturals, all the different ways there are to make sets out of all the numbers, is larger than the naturals. And you can easily do this by just writing out the naturals along a sheet of paper, 1, 2, 3, 4, 5, and then imagine creating a whole bunch of sets right underneath. And you just write a checkmark for yes and an X for no it's not in the set. So we could do this by saying, okay, one set would be all the odd numbers.

• 9:12Michael Stevens?

  So we go, 1 is included but not 2, 3 but not 4, and so on and so on. Another set could just be 5 and 7 only. Another set could be all the numbers except 5 and 7. You just fill this in. You just checkmarks and X's for whether they're in there. Now, do the diagonal thing, where you start up in that corner and you go, okay, my first set was only odd numbers, so 1 was included. Well, I'm going to generate a new set where there is no 1. Now you go down diagonally, and you say the second member of our next set was in there, 2 was in there, so I won't put 2 in mine. Now what about 3 in the third set? It was not in there, so it will be in this new set I'm making. And you keep going like this, and you create a set that is different in every position from every set that you could make.

• 9:57Michael Stevens?

  So, power setting allows you to quickly turn a group into an even larger infinity. And these, if you do this to the natural numbers, this gives you, Beth Knoll? Which is just all the naturals. Beth 1 is the power set of the naturals. Beth 2 is the power set of Beth 1 things. Beth 3 is the power set of Beth 2 things. And so on and so on, for as long as you want to go. Now, we're getting really big here.

• 10:21Hannah Fry?

  It's like samples of samples of samples. The analogy I like to think about this is, I think it's slightly easier to imagine when it comes to words. I mean, you were talking about your toothpick right at the beginning. We've also spoken about Babel's, the Library of Babel on this, on this show before. If you think you've only got 26 letters, right? But actually, if you take the group of that, that 26 letters and then you sample them to make words, right? There's all lots of different combinations of ways that you can make samples from, from that original, original group, right? And then if you take samples of those samples, right? And then samples of the samples of the samples, all of human language, everything that's ever been written, can be constructed just from that original 26 letters, you know?

• 11:05Hannah Fry?

  And, and I think that that really demonstrates how quickly, once you start being like, oh, we're allowed to have combinations of, of words. Oh, we're allowed to have combinations of sentences. Oh, we're allowed to have combinations of, of so on and so on and so on. You can get bigger and bigger and bigger and bigger sizes of infinity as you go. But it's not strictly analogous because the original set is infinite. Plus, there's also re-sampling, et cetera. But it just, I think, is a way for you to latch on to this idea that you can build bigger and bigger and bigger things just by taking samples.

• 11:38Michael Stevens?

  That's exactly right. If I have three things, you know, let's say 1, 2, and 3, I can fill a basket with those numbers in more than three ways. I can put 1 in the basket, just, just 1. I can put just 2, just 3. I can put 1 and 2, 1 and 3, 2 and 3. Basically, there's 2 to the third power, number of ways to do that. And as it turns out, 2 to the power of aleph null is Beth 1, okay? The power set of all the natural numbers. So, okay, we've got Beth 1, Beth 2, Beth 3, Beth 4. We can go up really high. All the Beths. All the Beths. What about the Alephs? Because we've got Aleph null. What's Aleph 1? Hmm. It's more.

• 12:24Michael Stevens?

  If it exists, it should be more, but how do we describe it? My favorite way to get to Aleph 1 is to go back to our race story. So, we actually ended the last episode by talking about this question of if an infinite number of people finish a race, and then you cross the finish line, what place do you get? And mathematicians have a number for that. They call it omega. You got omega-th place. And the symbol used is the lowercase Greek omega, which looks like a really fancy W. So, I can just start asking about all kinds of races. What if an infinite number of people finish the race and then 2 more do? And then you do?

• 13:07Michael Stevens?

  Well, now, the place you got is omega plus 2. You were the second after the infinite number of people. Omega plus 4 becomes a place you can get. Omega plus 5. Omega plus omega, which is 2 omega. That would happen if an infinite number of people finished the race, and then another infinite number of people finished the race, and then you finally crossed the line. But here's the really important thing, and we talked about this a lot in the last episode. In all of these cases, the total number of people involved in the race was still just Aleph null. Mm-hmm. And this gets back to like Hilbert's hotel. If an infinite number of people finish the race and then you do, you got omega-th place,

• 13:52Michael Stevens?

  but the number of people who raced was still just Aleph null. Because I can take that scenario, an infinite number of people and then you, and I could just have you actually take first place. And I would move first to second, second to third, and now, no one got omega-th place. Hmm. So, it's the same number of people. In fact, we could even have an infinite number of people finish the race, and then another infinite number, and then you. How do I do it? I just have you, uh, wait, and then I have all the even numbers go, and then all the odd numbers go, and then you finish. Boom. Still Aleph null number of people finishing the race, though. That's right. And so, all of these omega numbers, they come after each other, but none of them describe more things. But...

• 14:35Michael Stevens?

  But now imagine this. Imagine every possible way a race with aleph null people could finish. All the multiples of three, followed by all the multiples of four, followed by everything else. All the prime numbers followed by everything else. Like you, this goes on and on, right? There's so many ways to have a race with aleph null people. Let's say that every way aleph null people could finish a race, finish the race. And then you do.

• 15:08Hannah Fry?

  So it's like a race of combinations, effectively?

• 15:11Michael Stevens?

  It's a race of combinations of infinite things. And all of them, every possible combination of aleph null things finishes. Every, every ordered arrangement finishes. And then you finally cross the finish line. What place did you get? Well, we have a name for it. It's easy, it's easy to come up with a notation. It's called omega 1, which is omega with a little 1 subscript. But here's why this is so important. How many people will you need in order for the last person who crosses the line to get omega 1 place?

• 15:46Hannah Fry?

  Because it's not aleph null.

• 15:47Michael Stevens?

  It's not aleph null. It can't be, because if it was, it would have already crossed the finish line. Because remember, we've said that every way you can arrange aleph null things already finished the race. And then you did, but you haven't crossed yet. So omega 1, getting omega 1 place requires having a brand new amount of infinite people. And we call that aleph 1.

• 16:12Hannah Fry?

  Yeah. I mean, look at this. We've got all these different types of infinities now. This is, this is, look, this is all ammunition for when a kid tries to play with you, right? Tries to add numbers and add numbers, be like, okay, sit down. You're Buckle up, baby. I've got a, I've got a story for you. But this is it, right? It's like, this is this strange untamable beast. And okay, maybe there's a little bit here of cheating and just being like, let's call it Aleph whatever. Let's give it a name. But what we're, what's going on behind the scenes of everything you're describing is that these aren't just people coming up with these ideas and then like throwing them out there like they're spitballing.

• 16:56Hannah Fry?

  This is like hard, credible, rigorous, mathematical proofs that demonstrate irrefutably beyond any doubt that one of these numbers is larger than the other, that, that these are not the same size infinities.

• 17:13Michael Stevens?

  Yeah. And I love that there's, we've described two ways to do it. One is by arranging things in a well-ordered way, like how a race would work. And the other is by just combining them into all kinds of different groups of different sizes. Power setting and ordering. And those give us bigger and bigger Aleph numbers. The, the power setting gives us bigger and bigger Beth numbers. And we actually aren't quite sure how much bigger they are than each other.

• 17:41Hannah Fry?

  Yeah.

• 17:41Michael Stevens?

  So if you're ever in a, in a competition with a child trying to name the largest number, you can choose Alephs or Beth.

• 17:50Hannah Fry?

  Just off you go.

• 17:51Michael Stevens?

  Now, here's an interesting and important thing to know if you do get into a competition with a child who's trying to name a bigger number than you. You can keep saying Aleph one, Aleph two, Aleph three. But be careful, because if the child says Beth one, the cardinality of the continuum, there's no way to prove whether you're naming a larger number than them if you're saying Alephs, unless you name Aleph Omega.

• 18:20Michael Stevens?

  Because that's the only Aleph number that we can prove is actually larger than the cardinality of the continuum.

• 18:28Hannah Fry?

  That's the boss. That's the boss.

• 18:29Michael Stevens?

  That's the boss, Aleph Omega, which is an Aleph number that comes after a countably infinite number of Alephs. Aleph Omega is a good one to say.

• 18:39Michael Stevens?

  If you want to go any beyond that, you've got to just change the rules up, right? What are we doing? We're power-setting or we're well-ordering things over and over again. But we can, and this is what makes math so incredible. You can just say, "All right, but what if you do that forever?" Is there something beyond what you could reach with power-setting? And we don't have to find it. We can just say that it exists. And that, whatever that first, that least number is that you cannot reach through power-setting, that is called an inaccessible cardinal.

• 19:19Hannah Fry?

  Now, I think if you get to that, that's sort of, you've gone beyond the boss, right?

• 19:24Michael Stevens?

  You've gone beyond the boss.

• 19:26Hannah Fry?

  You've hacked into the mainframe at that stage, I think. That's a...

• 19:31Michael Stevens?

  That's right.

• 19:32Hannah Fry?

  To mix my analogies.

• 19:34Michael Stevens?

  You have done a jump. The jump from aleph null, the smallest infinity, to an inaccessible cardinal is like the jump from zero to infinity. In that, you cannot take a finite number of things and ever get to aleph null. No matter how many times I add up them or multiply them or whatever, I never get there. In the same way, no matter how many times I arrange and power-set an infinite number of things, I will never get to an inaccessible cardinal. So, the inaccessible cardinal is inaccessible for that reason. But yet with our minds, we can say, "It's there. I'm imagining it. Deal with it."

• 20:14Hannah Fry?

  It's like the brick wall at the end of The Infinite Garden, isn't it, really? Like, you don't know what's there. You don't know what it looks like. You don't know anything to do with it, but you're like, "There's something."

• 20:24Michael Stevens?

  Yeah. And you know what? In my head right now, I'm imagining something on the other side.

• 20:31Michael Stevens?

  Look how easy that is to do. And it's mathematically rigorous.

• 20:35Hannah Fry?

  This, I think, makes me, because I'm sure that there are some people who are listening to this who are just absolutely enjoying the ammunition that we're giving them for the next time that this conversation of naming big numbers comes along. But I think that there must be other people who are listening to this thinking, okay, but what is the point of any of this stuff? I mean, this is just like, this is just imaginary fiction land that you're describing. And I think that what this does is, well, okay, two things. First, I think it calls into the question of whether infinity is real or imagined, full stop, whether there ever actually is something that is genuinely infinite. I mean, we're kind of going back to Aristotle here, who sort of thought that there wasn't.

• 21:16Hannah Fry?

  But I also think that what it does is really distinguishes the different types of mathematics. So my background is I was always an applied mathematician, right? And what applied mathematicians do is they take the real world and they they take the equations that are used to describe it, and then they bend them and they break them and they twist them and they push them to see how far that they can get. Pure mathematicians do exactly the same thing, but the world that they're working with is one that is constructed entirely within these kind of rules, right? Okay, so let's start off with something that we know, the list of numbers. Now let's push it, push it further, push it further still and further and further and further. And now let's bend it and break it and twist it and see just where the bounds are and what are beyond those bounds still.

• 22:02Hannah Fry?

  And it turns out that actually, a lot of the time, these wild ideas that these pure mathematicians have about the way that you can bend and twist and shape numbers end up sometimes 150, 200 years later to be absolutely foundational to what we need to understand the world. The pure mathematicians are the ones who are searching for keys without ever caring about what locks they fit into. And it just so happens that very, very often, they prove to be extremely useful.

• 22:33Michael Stevens?

  Isn't that weird? There's that famous paper, The Unreasonable Effectiveness of Mathematics. You're right, these pure mathematicians are sitting around on podcasts talking about, but what if, but what if an infinite number of people finish before you? And it's entertaining. They've found like the keys to a lock that.

• 22:50Hannah Fry?

  May or may not exist, yeah.

• 22:51Michael Stevens?

  You're right. Hundreds of years later, someone finds that lock in the real world and they go, holy crud. Math got there before.

• 23:00Michael Stevens?

  We did.

• 23:01Hannah Fry?

  Yeah, and it's happened again and again and again. So, you know, maybe it will be the inaccessible numbers at some point when we are, I don't know, exploring the outer realms of the universe, end up, maybe there's one of them secretly hiding out there and it turns out to be the thing we needed.

• 23:19Michael Stevens?

  That's right. That's right. I mean, physics only can go so far, right? Below the Planck length, physics as we know it today doesn't really operate. But for all we know, inaccessible cardinals might describe the actual physical processes that happen down at those scales, right? But even if it doesn't, even if all of this is like completely impractical, it's still so amazing that we can go there anyway. It makes me, when I'm reading this stuff, it makes me go, do we even belong in this universe? We keep leaving it in our minds.

• 23:53Hannah Fry?

  My favorite description of pure mathematics, right? Which is essentially what this episode has been about so far, is that it is a portal to the playground for the soul. I really like that so much. I like that so much.

• 24:07Michael Stevens?

  I think that really captures it well. Exactly.

• 24:10Hannah Fry?

  But we're going to come back to a sort of anchoring in reality, kind of, after the break, because we are going to be asking whether anything actually really is infinite, or it is just a figment of our imagination.

• 24:33Speaker 02

  This episode is brought to you by Indeed. You're ready to level up your business, but the pressure to find the right person to make that happen is intense. Indeed sponsored jobs can help carry the load. It boosts your job post to help it reach quality talent fast. Cut through the chaos with Indeed sponsored jobs. Get a $75 sponsored job credit at indeed.com/podcast. Terms and conditions apply. Hiring now? Then this is a job for Indeed sponsored jobs.

• 25:10Hannah Fry?

  Okay, we're back. One of the other things that we discussed in the last episode was how Aristotle handled infinity. He was sort of of the opinion that you could have an infinite process. So you could, in theory, have something that went on forever, but you can't actually have something that is that contains infinity. You can't actually have something that's infinitely small or something that is infinitely large. And then we were talking about how, you know, during the intervening millennia, people started wondering about space. And I think that really, if infinity is for real, if it really, really exists, rather than just as a figment of our imaginations, then space is definitely

• 25:52Hannah Fry?

  the place to look. And cosmologists have done exactly that, right? Looked quite hard to see whether there are bounds on the universe. Yeah, they've been looking for a wall. They've been looking for, they've literally looking for a wall. Okay. So the thing that's worth saying, first of all, is that we are, we're sort of trapped in this kind of spherical bubble of light, because light has this strict speed limit. So there's, we can only see objects whose light has had enough time to travel to us since the beginning of the of the universe. And the universe is 13.8 billion years old. So common sense, you would think that you can look at, you're looking back 13.8 billion years, effectively. But actually, it's much bigger because that story that

• 26:34Hannah Fry?

  we had, the puzzle that we had in the first episode about an ant on a rubber band, stretching, that is the best possible way to think about the way what is happening to the universe. Yeah, surprise, you are an ant on a rubber rope. You are. You are. Except that you're not, you're not sort of moving along it. You're kind of, if you like, from your perspective, you're kind of staying still and then the entire universe is expanding around you. And that's a significant difference. That's a significant difference. That is a significant difference, because it means you're not going to catch the end, for one thing. Exactly. But it also

• 27:07Michael Stevens?

  You will eventually be completely alone.

• 27:09Hannah Fry?

  You will eventually be completely alone, exactly.

• 27:12Michael Stevens?

  Eventually, every, every single star in the universe will have shifted so far away from us that we no longer see it and the skies will be dark. It won't happen in my lifetime.

• 27:21Hannah Fry?

  No, no. I mean, we're talking trillions of years here.

• 27:25Michael Stevens?

  Trillions of years.

• 27:26Hannah Fry?

  Trillions of years.

• 27:27Michael Stevens?

  Our sun won't see this happen.

• 27:29Hannah Fry?

  No, but there is something quite nice about this, I think nice maybe, this perfect moment of sort of like cosmic visibility, you know?

• 27:37Michael Stevens?

  Yeah.

• 27:38Hannah Fry?

  But like, oh, aren't we lucky? Back in the day, my great, great, great, great, great grandparents, when they could see the universe. Whereas now you're just a speck in the middle of nothingness.

• 27:49Michael Stevens?

  Yeah, there will be a day where you look out the windshield of your planet and there's just, there's no stars in the sky.

• 27:54Hannah Fry?

  Nothing.

• 27:55Michael Stevens?

  And, you know, and we weren't born so early that like galaxies hadn't formed yet. We've got beautiful stuff to look at and so we're, we should be grateful that we're alive now.

• 28:04Hannah Fry?

  Lucky us, lucky us. But that thing about the rubber band, it sort of works in another way too. Partly because it, it really makes that conclusion quite obvious. But the other thing that's worth saying about it is that it's, it's really easy, I think, to imagine the big bang as though it was, I don't know, like a hand grenade going off and sort of spraying stuff outwards. Where the kind of edge of the universe is like the dome of the shrapnel as it, as it blows up. But you shouldn't think about it like that because it's space itself that is expanding. It's the rubber band itself that is, that is expanding. And the stuff just happens to kind of be going with it. So given that that's, that's sort of where we are, there's, there's essentially three potential

• 28:49Hannah Fry?

  options for what's going on at the edge of the universe, right? Either there is a wall, there is like a boundary. Then the question is, well, what do you, what happens when you meet it? You know, can you, if you, if you can stick your hand through it, well, then that's empty space. That's still part of the universe. Doesn't count. Or if, if there's true nothingness, like sort of, you know, you put your arms through it and then there's nothing on the side. There is like a genuine boundary. Well, I don't know, my brain can't handle that. I don't think many scientists' brains can handle it. Sort of, it doesn't really feel like it, what does that mean? What does it mean that there's an edge to it?

• 29:22Michael Stevens?

  Right. So, so wait, this is like that question, if it's expanding, what is it expanding into?

• 29:27Hannah Fry?

  Right. And there is, there is a way that it can be expanding and uh not have a fixed wall on the edge, right? There not be kind of like a fixed boundary. And this is sort of the Pac-Man universe, essentially. Cause if you imagine with Pac-Man, okay, if you're, you're kind of going along, you can travel left to right. You can go off the edge of the screen on the left-hand side and then you reappear on the right-hand side. And that is because the Pac-Man screen is effectively, it's playing really on the shape of a donut. If you can imagine the screen, you take it, you, you kind of pull it away from, from the computer and you wrap it around itself. So it's kind of running around in a circle. So you now have effectively a tube.

• 30:10Hannah Fry?

  But this Pac-Man can also go off the top of the screen and then come back around the bottom. So you can take your tube and bend it round and it's essentially is operating on a, on a torus, on a donut. You're, you're really playing Pac-Man on the surface of a donut. Pac-Man lives on a donut. Okay. So, so it could be that the universe is actually infinite in the sense that you can carry on going and going and going and going and going forever. The process goes on forever, but it is nonetheless bounded. So there's no edges in Pac-Man. You're not kind of like walking into any walls.

• 30:42Michael Stevens?

  Right.

• 30:43Hannah Fry?

  But nonetheless, it is contained within a finite surface, right? In a kind, in a finite amount of area.

• 30:48Michael Stevens?

  And we've tried to find out if we live on a donut like Pac-Man. We've looked, if, if, if we do, then we should be able to look far enough out that we eventually see the backs of our own heads.

• 31:00Hannah Fry?

  Right. You should be able to see the backs of your own heads. Or you look off in the telescope in one direction and you're like, oh, hello, that's an interesting like star constellation going on. I've never seen galaxies right off in the distance. It's very interesting. And then you turn around, you go the other way. It's like, hang on a second. That's the same one.

• 31:15Michael Stevens?

  It's the same stuff from the other side. Yeah.

• 31:17Hannah Fry?

  It's the same one.

• 31:18Michael Stevens?

  We have not, we have not yet found that to be the case.

• 31:22Hannah Fry?

  It wouldn't be the backs of our own heads though, actually. It would be the backs of, I mean dinosaurs heads, actually longer, right?

• 31:28Michael Stevens?

  Uh, that's true. Yeah. Uh, well, my minor, minor point, but yeah.

• 31:35Hannah Fry?

  Hang on, that's my house. But before.

• 31:38Michael Stevens?

  Hang on.

• 31:38Hannah Fry?

  Vegetation existed.

• 31:41Michael Stevens?

  It's like, you know, when someone shares their screen during a Zoom call and we're like, whoa, we live on a donut. And everyone else is like, I'm getting sick. It wouldn't be like that. You're right.

• 31:49Hannah Fry?

  No, it wouldn't be like that.

• 31:50Michael Stevens?

  The light we see is light that left.

• 31:52Hannah Fry?

  The delay would be substantial.

• 31:54Michael Stevens?

  Billions and billions of years. So it wouldn't quite look the same. But point is, we haven't found any evidence that the universe wraps back on itself.

• 32:03Hannah Fry?

  Right. Because if it wraps back on itself, then it curves. And if it curves, there should be some hints in the way that things are laid out. You know, we know that on a flat surface, the, the internal angles of a triangle add up to 180 degrees, right? For example. So you can look out into space and, and run these kind of tricks. Like, are there any sort of signs here and there that, uh, that, that triangles don't make sense? Does it look like we're sitting on the surface of a sphere? Does it look like we're sitting on the surface of a spat- uh, of a saddle? Or on a donut? And nothing. There is nothing. Everywhere you look, no matter how accurate you make your measurements, it's flat, flat, flat, flat, flat. We are on a piece of paper, effectively.

• 32:46Michael Stevens?

  It could be that we just can't look far enough yet.

• 32:50Hannah Fry?

  Right.

• 32:50Michael Stevens?

  To, to see the curvature.

• 32:52Hannah Fry?

  Right. It could be that we are just so small, just so tiny. Because actually, if you think about, we're on the curved surface of Earth. And yet, when you're standing here, it looks pretty flat, to, to, flat to all of us.

• 33:04Michael Stevens?

  Yeah.

• 33:04Hannah Fry?

  Just because we're too small to see it. So it could be, you're right, that the universe is just so big that we, uh, just can't see it. But, I mean, thus far, no evidence. So now, if you put those two things together, that it's not curved, and we think there's no wall, well, then the only other conclusion is that it's, is that it just goes on forever. It's just flattened, goes on forever. And maybe that's it. Maybe there really is infinity. But there's some pretty uncomfortable conclusions if that's the case. Because if there is a genuinely infinite universe, well then, if you throw a loop, throw a dome around any amount of stuff, right? So let's just say you take our solar system. There is a fixed, finite amount of stuff in our solar system.

• 33:50Hannah Fry?

  Finite amount of quarks, of atoms, whatever it might be, right? It's way, way less than aleph null. It's finite. And there is a finite way that you can combine all of that stuff. I mean, the number is going to be absolutely unimaginably large, right?

• 34:06Michael Stevens?

  But it's finite.

• 34:07Hannah Fry?

  It's finite. And so if you have a genuinely infinite universe, you have an infinite number of rolls of the dice, effectively. And so there's a thing called the pigeonhole principle. If you do something enough times, even if it has a very, very, very small probability of happening, it will eventually come true. And so what that means, the only conclusion you can make is that if the universe is infinite, there must be another solar system out there that precisely, down to the quarks, mirrors our own.

• 34:42Michael Stevens?

  Which means there's another me there and another you there. And we're doing this podcast exactly the way it is, except my shirt's blue. And there's another solar system where my shirt's orange, but we're having the exact same conversation. Let's say the likelihood that all this matter would be arranged like it is at this moment is one in a quadrillion. If you've got a non-million volumes, then you wouldn't be surprised to find that a couple of them match.

• 35:11Hannah Fry?

  Exactly right. I mean, we were talking about the number of ways to shuffle a deck of cards a few episodes ago in our finite numbers episode. It's huge. It's so, so, so gigantic. But if you did carry on going forever, shuffling cards, shuffling cards, eventually you would find the same shufflings more than once. And if you carried on going forever and ever and ever and ever, you would end up basically with an infinite number of every one of those infinitesimally unlikely combinations. It's very uncomfortable.

• 35:44Michael Stevens?

  Yeah, you do it long enough, you will have an infinite number of times you shuffled the cards into. Perfect order. And if we consider our, you know, solar system a perfect order, then there's an infinite number of them out there.

• 35:57Hannah Fry?

  Which is very uncomfortable.

• 35:59Michael Stevens?

  And this bothers me a lot.

• 36:01Hannah Fry?

  Me too.

• 36:02Michael Stevens?

  Because it makes me think, well, who am I? If there are countless other me's out there doing the same thing, what makes me special? I guess I'm the only Michael doing it here?

• 36:15Hannah Fry?

  Yeah. Yeah. I mean, this kind of gets on to like,

• 36:20Hannah Fry?

  I think there's like a lot of things of.

• 36:21Michael Stevens?

  So do you, do you believe that? I mean, I don't see how I can't accept it. I, it, it is the best picture of the universe as far as I'm concerned. And yet.

• 36:31Hannah Fry?

  I don't.

• 36:32Michael Stevens?

  I want to know what you think.

• 36:34Hannah Fry?

  I don't know. I think I'd probably marginally prefer the idea that we're too small and that we are curved. And that we are contained within a finite space. I mean, that, that also has extremely uncomfortable conclusions, by the way. One of the, one of the descriptions of the universe that looks like that is that we're, we're a bubble effectively. If you imagine that there is a, a boiling ocean, right? It's sort of like a pan of water. And the, and the way that you get these bubbles that appear effectively from nowhere as the water is boiling, the kind of the, the, the physical, the physical fluid rips apart, it's called cavitation, rips apart and sort of becomes filled with gas. And the idea behind the bubble, bubble multiverse

• 37:19Hannah Fry?

  is that a big bang is just one of those bubbles appearing. It's the way that sort of energy, um, energy creates, you know, matter and in this sort of like little explosion within this, this wider fluid. Now, the, the slight problem with that theory is that each of these bubbles could have completely different rules of physics, you know? It could be that one of them, gravity, the speed of light, the mass of electron, you know, any of those things is just totally different in all of these different bubbles. We had just have no idea. And so that, I mean, the conclusions there are, I would say, equally uncomfortable. You, you, you can't, you don't have to sit long with this stuff before it starts getting, um, back to Michael with a slightly different colored shirt on.

• 38:01Michael Stevens?

  Yeah, well, I mean, and most of these like universes, if they're just randomly being assigned constants like speed of light and mass of the electron, a lot of them would be so unstable, they would just collapse right away. I think it's Lee Smolin who has that theory that, well, if this is happening, then there might be some kind of natural selection process for universes. And the ones that last the longest are sort of ones like ours, where it's possible for black holes to form and for intelligent creatures to come about. And I think his whole theory is that a black hole creates another bubble at its singularity, a whole different universe. And that universe is like the one that the black hole was originally in. It's literally like a child.

• 38:43Michael Stevens?

  And so you wind up with many more universes like ours than you do universes where, you know, protons decay in a quarter second.

• 38:50Hannah Fry?

  Exactly. I mean, all of this stuff ends up getting really tied in together because of course, black holes, it's, as far as we know, it's just infinite density. But, but maybe this is actually the secret to what's going on more broadly across, across the entire universe.

• 39:05Michael Stevens?

  And then on the other hand, like, how, how small can we get? Like, we can zoom in and look at molecules and then atoms, and we know that those atoms are made of things, but how granular does the universe get? Even if it's, you know, finite and curved and bounded or whatever, how small does it get? We already talked about the encyclopedia wand or the, the toothpick that contained everything that can ever be said. How much precision is possible? I mean, famously, we've discussed this many times on the podcast, there is a limit to what our current physics can describe. We've mentioned this on the podcast a few times, but like, the limit of physics as we

• 39:49Michael Stevens?

  have it and know it today is anything that happens within or amongst Planck units. The Planck length is like the smallest sensible length within physics as we know it. If, if a photon had a wavelength that was the Planck length, which by the way, is 10 to the negative 35 meters. If a photon had that small of a wavelength, had that much energy, it would turn into a black hole. So the Schwarzschild radius at that length is a photon's energy. And so we kind of go, well, you can't get any smaller than that because it destroys itself, or it's gone, or whatever. But that doesn't mean that that's actually the smallest length that anything happens at.

• 40:34Michael Stevens?

  It's just, we can't describe any further.

• 40:37Hannah Fry?

  Because our equations break, essentially.

• 40:39Michael Stevens?

  Our equations break, but the equations seem pretty good for everything else. So do we need new equations? Do we need a different interpretation of them? I don't know. I mean, it seems like space is going to need to be continuous, like the real numbers. You just, the closer you get, the more and more detail you see. Because I don't see how, like, Pythagorean, I don't see how the Pythagorean theorem could hold true if things became discrete at a certain level. If eventually you look close enough and you just see a bunch of pixels of space, now it's no longer true that a squared plus b squared equals c squared. Because c squared is no longer a continuous straight line. It's a stair step.

• 41:24Hannah Fry?

  Well, there's also, I mean, a lot of things end up, I mean, circles cease to exist. Perfect spheres cease to exist if down, once you get small enough, the universe is discrete rather than continuous. It's very uncomfortable.

• 41:39Michael Stevens?

  Well, and if they cease to exist there, then they never existed. Everything is at every level pixelated.

• 41:46Hannah Fry?

  A figment of our imagination, which means essentially, you know, they go hand in hand. Either circles and infinity exist for real, or neither of them do.

• 41:58Michael Stevens?

  Right.

• 41:58Hannah Fry?

  You know, you can't say I'm not having infinity without getting rid of circles as well.

• 42:03Michael Stevens?

  Wow, yeah, that's true. So where do you fall? Do you believe that circles and infinity literally exist?

• 42:13Hannah Fry?

  I don't know. I sort of think, I don't know. Because I think the idea that the universe is discreet, spatially discreet, is also really uncomfortable.

• 42:23Michael Stevens?

  Yeah, I don't like it. I don't like that my brain can go, oh, really? There's a, there's a smallest discreet piece of space, cut it in half.

• 42:31Hannah Fry?

  Yeah.

• 42:32Michael Stevens?

  Look, I just, I just

• 42:33Hannah Fry?

  Just did it.

• 42:34Michael Stevens?

  Went beyond the universe. Come on. I, I, I, I think that it's, the universe is infinite and space is infinitely divisible.

• 42:42Hannah Fry?

  You know, there are some people, some computer scientists, who are very comfortable with this idea. Because they essentially see everything that we're living in as, as a kind of computer simulation, right? And in computers, the fact that space is discrete, the fact that, that you can't cut something up smaller. Because you get down to a pixel and there's nothing smaller than a pixel, there's nothing smaller than, than a bit. They're, you know, quite happy, quite comfortable with that.

• 43:10Michael Stevens?

  Yeah, well, you know, right, if we, if we did find like a, a, a unit of space that could not be divided. Then we might have found some good evidence that we live in a simulation. That like, ah, there it is. That's the bit of the alien computer that we're running on. Cool.

• 43:25Hannah Fry?

  I mean, also, you would say if, if this was a simulation, that the whole issue of how far out we can see the observable universe and what's beyond it, that's not a problem anymore either. Because it's just that the simulation has just hasn't bothered. You know, if you're running around in, I don't know, like Grand Theft Auto and you just go into a new, you're, you're always in your own, how old my reference is, or new, isn't it coming out, the new one? Anyway, um, sorry, I'm just, I'm such a gamer, I'm just really eagerly anticipating.

• 43:54Michael Stevens?

  You're such a pro gamer, like, I can't even keep up.

• 44:00Michael Stevens?

  You're, you're talking about Pac-Man, what are you gonna do next, pong?

• 44:06Hannah Fry?

  Look, I uh...

• 44:07Michael Stevens?

  We might live in Donkey Kong country for all we know mathematically. Your gaming references are superb.

• 44:14Hannah Fry?

  I exclusively talk about the ones that have made it, cut it across to the mainstream. I refuse all of these niche, these niche games that people chat about. It's just not interesting to me. But the, but the reality is when you are in a game, I mean, you sort of are in your own observable universe, right? Like where you're turning your camera, it just hasn't rendered anything outside of it.

• 44:34Michael Stevens?

  Yeah.

• 44:34Hannah Fry?

  So, I mean, maybe that is the solution to all of this. Maybe it is that just that, that actually it's all, I mean, that would make, um, the sort of, well, this is a kind of one for another day, I guess. But the collapsing of the wave function, another deeply weird and troubling physics, uh, problem with our understanding of the universe. That, uh, we don't actually know where particles are.

• 44:57Michael Stevens?

  Right.

• 44:59Hannah Fry?

  Uh, and they sort of seem to be in many places all at once, exploring all potential possible futures until you actually observe them. We'll definitely do an episode on this at some point.

• 45:09Michael Stevens?

  And you observe them and then the simulation goes, oh, you gotta render that. Okay, uh, here, it happened to be like this.

• 45:15Hannah Fry?

  Exactly. Because actually, if we're living in a computer simulation, that's fine too, right? If a particle is a blob, then it doesn't make any sense that it can be here and there and there and there. All of those places simultaneously at once until you see it and realize exactly you fix its position. But if it's a computer simulation, that's absolutely fine. It can be any manner of things until you look at it.

• 45:35Michael Stevens?

  It's absolutely fine. So yeah, saying that we live in a simulation is kind of a good bet. Especially because if you really think about it for a long time, you start to go, well, you know, if, if an alien made a simulation, they could make so many simulations and they could fill it with so many people that like overall, it's more likely that I would find myself in a simulation than in a real universe. So you should probably, like, kind of it's like a new version of Pascal's wager. Just, just wager that you're in a simulation and you're more likely to be right. I, I'm gonna go on record by saying that I think we just don't know enough about the universe, the real one that we live in. To...

• 46:18Michael Stevens?

  To answer the question, but I still think that we are in a real, unsimulated universe.

• 46:22Hannah Fry?

  I think, I think that we are too, just because I really want to be. But, but, but I also, I like it as a theory because there is some suspicious stuff going on. There is some suspicious stuff.

• 46:34Michael Stevens?

  There is, there is. And do you think that we'll have answers within our lifetimes? Or do you think that this is all going to remain very theoretical?

• 46:43Hannah Fry?

  I mean, I think for the expanses of space, no. I think that one's going to be way beyond us. But I think that there might be. I do think that when it comes to particle physics, I think, and thus things like Planck length or Planck length. I think that there's a chance, you know? I think there's a chance that we might get some, get a couple of big breakthroughs. Sort of, it sort of feels like there's one. The models that they use are so disgustingly messy that it sort of feels like there's something that should be, that's pushing at the door, waiting to come through.

• 47:14Michael Stevens?

  Right? Yeah, it does. And then we'll like stand it and look from a different angle and go, oh, it was messy, but that's because we were wrong about. And then the Planck length will become a historical unit. And we'll get smaller.

• 47:30Hannah Fry?

  But here, I guess, is the big conclusion that we have. I mean, either infinity doesn't exist, it's all a figment of our imaginations. Or, and maybe also, there are some really uncomfortable things. Really uncomfortable things that are the conclusions that we must draw.

• 47:50Michael Stevens?

  I would put it that way. I would say that we founded infinity, like we invented it. But someday we will find that it was always there anyway. Our brains just jumped ahead of our eyes.

• 48:07Hannah Fry?

  And particularly Cantor's brain, what an incredible brain that was.

• 48:09Michael Stevens?

  Particularly Cantor's brain, yeah.

• 48:12Hannah Fry?

  Yeah.

• 48:13Michael Stevens?

  He took the first leap.

• 48:15Hannah Fry?

  He did take the first leap. Do you know what, Michael? I think, I think we might have come to the end of infinity.

• 48:21Michael Stevens?

  I think we came to our end. I don't know if we've come to its end, but what a blast.

• 48:26Hannah Fry?

  That was a blast. Well, I hope that all of you enjoyed it. I know that I did. Let us know in the comments if there are other mind-bending, weird, crazy topics that you would like us to explore. We touched on a few of them there, just to give you a little to give you a little sweetener. As ever, you can send us in any thoughts or questions that you have to [email protected].

• 48:49Michael Stevens?

  And you can join our newsletter at therestis.com/science.

• 48:52Hannah Fry?

  I would say, actually, send us emails, sure. But if you put it on comments on YouTube, then me and Michael can read them as well, which we do.

• 48:59Michael Stevens?

  Well, that's true. Yeah, we don't have direct access to the email. The producers pull out their favorites and they put them into a rolling Google Doc for us. But the comments, we can see it all, even the stuff that they don't want us to see.

• 49:12Hannah Fry?

  If you want to send us really, you know, like if you think you've solved Raymond's hypothesis or whatever, and you want to like send that, send in 50 pages of proof, send that to the producers. That's fine. That's okay. But if you've got like a nice, quick-witted, interesting thing you want us to know about, chuck it in the comments on YouTube. Until next time.

• 49:31Michael Stevens?

  See you later.

• 49:45Speaker 03

  Here we are getting a statement from Number 10 on that flight with 350 passengers on board.

• 49:54Speaker 04

  The Great Game, the strategic struggle between Britain and Russia.

• 50:01Speaker 05

  We assess, Prime Minister, with high confidence it was Russian.

• 50:08Speaker 06

  Holding the Russians accountable is World War III.

• 50:10Speaker 07

  They took down a British passenger jet containing 350 people.

• 50:13Speaker 04

  Man in the Kremlin is playing, Alex. Got to ask yourself, do you want to play or not?

• 50:19Speaker 08

  You're the Prime Minister. You've done nothing!

• 50:26Speaker 08

  We would need to understand the policy context.

• 50:28Speaker 04

  All the relevant context will be structured as a framework today. And disseminated into an operation order, or FRAGO.

• 50:37Speaker 09

  A FRAGO that says to kill the President of Russia?

• 50:42Speaker 04

  Yes.

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• 51:03Speaker 06

  Where Britain plays its strongest hand. He's the best. Against its strongest foe. Don't do it Duffy! Bear Hunt. Available now wherever you get your podcasts.