This Toothpick Contains Everything Ever Said (Infinity Part 3)

Summary

The Rest Is Science – "This Toothpick Contains Everything Ever Said"

Hosted by Professor Hannah Fry and Michael Stevens (Vsauce)
Release Date: April 6, 2026

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Episode Overview

In this mind-bending episode, Hannah Fry and Michael Stevens dive deep into the concept of infinity, exploring its strangeness, mathematical rigor, and the dizzying philosophical questions it raises about reality, the universe, and even our own existence. The conversation weaves playful analogies with rigorous logic, tackling everything from encoding all human knowledge onto a toothpick, to the question of whether the universe is infinite or simply very, very big (and maybe shaped like a donut). The duo also considers the implications of infinity in mathematics and cosmology, and whether we might all just be living in a simulation after all.

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Key Discussion Points

1. The Toothpick Analogy: Every Possible Statement Encoded (00:35 – 03:29)

• Encoding Knowledge on a Toothpick:
  Michael introduces a physical toothpick and demonstrates in theory how—assuming “infinite precision”—you could encode every possible statement or piece of information on it. A code is used where each letter is numbered (A=01, B=02, ..., Z=26) and marked as a decimal along the length of the toothpick.
  • Michael: “Everything that could ever be said is on this toothpick.” (01:12)
  • The idea is: with infinite precision, infinitely long decimal codes could contain everything—“the entire Encyclopedia Britannica on a single notch on this one toothpick.” (03:00)
• Infinity at Any Scale:
  This segues into a discussion of how infinity isn’t just about the very large but also the very small—layer upon layer of possible knowledge or arrangements, all potentially encoded in an arbitrarily tiny space.

2. The Hierarchy of Infinities: Alephs, Beths, and Beyond (05:15 – 19:44)

• Cardinalities and the Names of Infinity:
  Hannah and Michael review basic types of infinity:
  • Aleph-null (ℵ₀): “the smallest size of infinity, how many counting numbers there are.” (05:42)
  • Beth numbers: A different hierarchy—“the number of real numbers there are” (Beth 1), then power sets leading to larger and larger infinities.
  • Michael: “If I have three things... I can fill a basket... Basically, there’s two to the third power number of ways to do that. And as it turns out, two to the power of Aleph null is Beth 1.” (11:08)
• Well-Ordering vs. Power Sets:
  Two different ways to generate larger infinities:
  • Well-ordering things (like lining up infinite marathon finishers—Omega, Omega+1, Omega+2…).
  • Taking all possible combinations (power sets) of prior infinities.
• The Boss Infinity – Aleph Omega:
  There’s an “ultimate boss” infinity, Aleph Omega, which is the first Aleph larger than all Beths below it, and then inaccessible cardinals—“the jump from Aleph null, the smallest infinity, to an inaccessible cardinal is like the jump from zero to infinity.” (19:04)
  • Hannah: “If you get to that, you’ve hacked into the mainframe at that stage.” (18:57)
• Theoretical vs. Practical:
  Even if these sound like playful inventions, “hard, credible, rigorous mathematical proofs demonstrate, irrefutably, beyond any doubt that one of these numbers is larger than the other.” (16:15)

3. Do Infinities Exist in Reality? (20:05 – 25:59)

• Applied vs. Pure Mathematics:
  Discussion of the difference: applied mathematicians test math on real phenomena, while pure mathematicians “search for keys without ever caring about what locks they fit into.” (21:18)
  • Fry: “A lot of the time, these wild ideas... end up sometimes 150, 200 years later to be absolutely foundational to what we need to understand the world.”
• Unreasonable Effectiveness:
  Why do pure math discoveries end up describing the universe so well? “Hundreds of years later, someone finds that lock in the real world and they go, holy crud, math got there before we did.” (22:22)
• Is Any Physical Thing Infinite?
  Returning to Aristotle, who “thought you could have an infinite process but not something that contains infinity.” (24:50)
  • If infinity exists for real, maybe it’s in the cosmos—is the universe infinite?

4. The Shape and Size of the Universe: Bubbles, Donuts, and Flatness (25:03 – 33:43)

• Cosmic Visibility and the Ant on a Rubber Band (25:44):
  • The universe is expanding, not from a point like shrapnel, but “space itself” is stretching.
  • Are we in a bubble with boundaries, or a space like the Pac-Man donut where “you can go forever but are nonetheless bounded”?
  • Fry: “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… but nonetheless it is contained within a finite surface.” (29:17)
• Have We Seen the Backs of Our Own Heads? (30:00):
  • If the universe loops, distant objects could repeat—but “we haven’t found any evidence that the universe wraps back on itself.” (30:18)
• Current Consensus: Flat and Possibly Infinite:
  • Space appears flat, not curved like a sphere or a donut, suggesting it might “just go on forever.” (31:46)

5. Philosophical Consequences of an Infinite Universe (33:06 – 37:02)

• The Pigeonhole Principle and Cosmic Doppelgängers:
  • If the universe is infinite, then—even given a finite number of arrangements of matter—every possible arrangement must occur somewhere, including duplicates of our entire solar system, and copies of ourselves.
    • Michael: “There’s another me there and another you there. And we’re doing this podcast… except my shirt’s blue.” (33:43)
• Uncomfortable Outcomes:
  • Both infinite and finite-bubbled universes have “equally uncomfortable” implications, from infinite duplicates to universes with radically different laws of physics. (37:02)

6. On the Smallest Scales: Discrete Universe and Planck Length (38:05 – 41:14)

• Can Space Be Infinitely Divided?
  • The Planck Length (10^-35 meters) is the scale at which “our equations break,” but we don’t know if that’s fundamental or a limitation of current physics.
  • Michael: “The closer you get, the more and more detail you see. Because I don’t see how the Pythagorean theorem could hold true if things became discrete at a certain level.” (39:37)
• If the Universe Is Discrete, Circles Don’t Exist:
  • At the smallest scale, if space is pixelated, “perfect spheres cease to exist.” (40:24)
    • Fry: “Either circles and infinity exist for real, or neither of them do.” (40:47)

7. Are We Living in a Simulation? (41:43 – 45:35)

• Simulation Theory:
  Computer scientists are comfortable with discrete space—it’s how computers work. If we found indivisible spatial units, “we might have found some good evidence that we live in a simulation.” (42:10)
  • The idea is: in games, reality only renders what you see—maybe the universe is the same.
• Quantum Mechanics Tie-In:
  The “collapse of the wave function” could make sense as a simulation trick: only render reality when observed. (44:09)
  • Michael: “Saying that we live in a simulation is kind of a good bet… overall it’s more likely that I would find myself in a simulation than in a real universe.” (44:35)
  • Ultimately, both prefer to believe in a real, unsimulated universe—though “there is some suspicious stuff going on.” (45:23)

8. Looking Ahead and Final Thoughts (45:43 – 47:26)

• Answers about the universe’s size are beyond reach for now, but new breakthroughs in particle physics may come during our lifetimes.
• Fry: “Either infinity doesn’t exist, it’s all a figment of our imaginations… or there are some really uncomfortable things that are the conclusions that we must draw.” (46:30)
• Michael’s poetic summary: “We founded infinity like we invented it, but someday we will find that it was always there.” (46:50)
• Celebrating mathematician Georg Cantor, the pioneer of infinite sets. (47:06)

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Notable Quotes & Memorable Moments

• “Everything that could ever be said is on this toothpick.” – Michael, 01:12
• “The jump from Aleph null, the smallest infinity, to an inaccessible cardinal is like the jump from zero to infinity.” – Michael, 19:04
• “Pure mathematicians are the ones who are searching for keys without ever caring about what locks they fit into.” – Hannah, 21:18
• “Hundreds of years later, someone finds that lock in the real world and they go, holy crud, Math got there before we did.” – Michael, 22:22
• “Pac Man lives on a donut.” – Hannah, 29:17
• “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…” – Michael, 33:43
• “Either circles and infinity exist for real, or neither of them do.” – Hannah, 40:47
• “We founded infinity like we invented it, but someday we will find that it was always there.” – Michael, 46:50

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Essential Timestamps

• Physical metaphor for infinity and toothpick encoding: 01:12 – 03:29
• Defining different sizes of infinity (Alephs and Beths): 05:42 – 11:43
• Infinity in races and ordering—Omega and Aleph 1: 13:53 – 15:43
• Philosophical/practical use of higher infinities and Cantor’s leap: 16:43 – 19:44
• Flat vs. curved universe; donut analogy: 28:27 – 31:46
• Pigeonhole principle and cosmic doubles: 33:06 – 34:45
• Simulation hypothesis tie-in: 41:43 – 44:35

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Tone and Style

Lively, playful, and mind-expanding. The hosts oscillate between accessible analogies (toothpicks, Pac-Man, card shuffling), mathematical rigor, and wide-eyed philosophical curiosity. There’s frequent friendly banter and encouragement for listeners to relish these “strange weirdnesses,” whether for winning number-naming competitions with kids or for pondering the limits of reality itself.

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Conclusion

If you want to think deeper about the nature of reality, the mathematics that describes it, and whether every fact, conversation, or even you yourself might exist somewhere else in the universe—or on a toothpick—this episode is a playground for the mind and soul. And as the hosts suggest, maybe infinity is something we dreamed up…but perhaps it was there all along, waiting for us to notice.

Transcript

[A]

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

[B]

And I'm Michael Stevens.

[A]

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

[B]

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.

[A]

Maybe third time's the charm.

[B]

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

[A]

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

[B]

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?

[A]

I am, absolutely.

[B]

I have here a toothpick.

[A]

You see that? No, I thought you were going. But sure.

[B]

This is a toothpick. But guess what?

[A]

Tell me.

[B]

Everything that could ever be said is on this toothpick.

[A]

Uh. Oh. Tell me how.

[B]

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. For 27, 28, and so on. We can add in punctuation, numerals, whatever. Perfect. Now let's. Let's think of something that could be said. How about the word cat? Okay. In our code, cat would be cat, which is 03-0120. T is the 20th letter.

[A]

Okay, so I'm impressed how quickly you remembered that. But go on.

[B]

You know, I know that because I cannot forget the fact that s is the 19th 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 19.

[A]

I 100% 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.

[B]

Anyway, so here's the word cat, right? 030120. 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.

[A]

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

[B]

We're going to go from. From the beginning of the toothpick. We're going to go.01032 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 got to 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 0.03012. What 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. 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.

[A]

I really like that. I like the idea that you are containing it within one single notch. Right, right. There's something very nice about that.

[B]

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 can get smaller and smaller and closer and closer and keep finding more and more and more.

[A]

This episode is brought to you by Cancer Research uk.

[B]

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.

[A]

Okay?

[B]

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[A]

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[B]

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[A]

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[B]

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

[A]

That takes years of very careful, steady progress to deliver Each breakthrough.

[B]

For more information about Cancer Research uk, their research breakthroughs and how you can support them, visit cancerresearchuk.org TheresTestisscience.

[A]

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?

[B]

Yes, yes we did. Aleph Null is the name for the smallest size of infinity, how many counting numbers there are positive integers, 0, 1, 2, 3, 4, 5 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, if you, if you missed it, because how that was demonstrated by Cantor is beautiful. But you can show that there are literally more real numbers than there are

[A]

positive integers because that second one has a name as well. Right? It's actually called, it's called Beth1.

[B]

Yes, it's called Beth1 because Beth is the second Hebrew letter in the, 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, 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. Beth1. Beth Noel is equal to Aleph Noel. Beth1 is declared to be the cardinality of the continuum. And. 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, 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.

[A]

Okay, so imagine you've got all the decimals, you've got everything that could possibly live on a toothpick and, 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?

[B]

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 check mark 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. So we go, one is included, but not two, three but not four, 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. Just. You just fill this in. You just check marks 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, our first set was only odd numbers, so one was included. Well, I'm going to generate a new set where there is no one. Now you go down diagonally and you say, the second member of our next set was in there. Two was in there. So I won't put two in mine. Now, what about three 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. So power setting allows you to quickly turn a group and into an even larger infinity. And these, if you do this to the natural numbers, this gives you Bethnull, 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.

[A]

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 letter, you know, and I think that that really demonstrates how quickly once you start being like, oh, we're allowed to have combinations of words. Oh, we're allowed to have combinations of sentences. Oh, we're allowed to have combinations 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. It's not strictly analogous because the original set is infinite, plus there's also resampling, et cetera. But it just, I think, give. It's a way for you to latch onto this idea that you can build bigger and bigger and bigger things just by taking Samp.

[B]

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 one in the basket. Just, just one. I can put just two, just three. I can put one and two, one and three, two and three. Basically, there's two to the third power number of ways to do that. And as it turns out, two to the power of Aleph Noel is BETH one. 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.

[A]

All the Beths.

[B]

All the Beths. What about the Alephs? Because we've got aleph null. What's Aleph 1? It's. It's a. It's more. If it exists, it should be more. But how do we describe it? My favorite way to get to Aleph one is to go back to our race story. So we. 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 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 two more do. And then you do. Well, now the place you got is Omega plus two. You were the second. After the infinite number of people, Omega plus four becomes a place. You can get Omega plus five, Omega plus Omega, which is two 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. 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 Noel. 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 place. But the number of people who raced was still just Aleph Noel. Because I can take that, 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. 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.

[A]

Still NF null number of people finishing the race, though.

[B]

That's right. And so all of these Omega numbers, they come after each other, but none of them describe more things. But. 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. This goes on and on, right? There's so many ways to have a race with Aleph Noel people. Let's say that every way Aleph Noel people could finish a race, finish the race, and then you do.

[A]

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

[B]

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 one 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 1th place?

[A]

Because it's not Aleph Null.

[B]

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 1th place requires having a brand new amount of infinite people. And we call that Aleph one.

[A]

Yeah. I mean, look at this. We've got all these different types of infinities. Now this is. Yeah, 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 gonna buckle up, baby, 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 Alif 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. 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 these are not the same size infinities.

[B]

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 power setting gives us bigger and bigger Beth numbers, and we actually aren't quite sure how much bigger they are than each other.

[A]

Yeah.

[B]

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

[A]

just off you go.

[B]

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 1, Aleph 2, Aleph 3. But be careful, because if the child says Beth 1 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, because that's the only Aleph number that we can prove is actually larger than the cardinality of the continuum.

[A]

As that's the boss. That's the boss.

[B]

Aleph Omega, which is an Aleph number that comes after accountably infinite number of Alephs. Aleph Omega is a good one to say if you want to go any beyond that, you've got to just change the rules up, right? Is what. What are we doing? We're. We're power setting, or we're, we're, well, ordering things over and over again. But we can. And this is, 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.

[A]

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

[B]

You've gone beyond the boss.

[A]

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

[B]

That's right.

[A]

You have begun analogies.

[B]

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.

[A]

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 what anything to do with it. But you're like, there's something.

[B]

Yeah. And you know what? In my head right now, I'm imagining something on the other side. Look how easy that is to do. And it's mathematically rigorous.

[A]

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 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. 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. 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.

[B]

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 found, like, the keys to a

[A]

lock that may or may not exist. Yeah, you're right.

[B]

Hundreds of years later, someone finds that lock in the real world and they go, holy crud, Math got there before we did.

[A]

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. End up. Maybe there's one of them sneakily hiding out there, and it. And it turns out to be the thing we needed.

[B]

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.

[A]

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.

[B]

I think that really captures it well. Exactly.

[A]

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. Okay, we're back. One of the other things that we discussed in 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. The. The intervening millennia, people started wondering. 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 the place to look. And cosmologists have done exactly that. Right. Looked quite hard to see whether there are bounds on the universe or whether

[B]

they've been looking for a wall.

[A]

They've been looking for a. They've been 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. It's much bigger because that story that 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?

[B]

Yes. Surprise. You are an ant on a rubber rope.

[A]

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.

[B]

And that's a significant difference.

[A]

That's a significant difference. That is a significant difference because it means you're not going to catch the end, for one thing.

[B]

Exactly. But it also means you will eventually be completely alone.

[A]

You will eventually be completely alone. Exactly.

[B]

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.

[A]

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

[B]

Trillions of years. Trillions of years our sun won't see this happen.

[A]

No. But there is something quite nice about this, I think. Nice. Maybe this perfect moment of sort of like cosmic visibility, you know, 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.

[B]

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.

[A]

Nothing.

[B]

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 should be grateful that we're alive now.

[A]

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 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 arm 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, Right.

[B]

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

[A]

Right? And there is, there is a way that it can be expanding and 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. 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 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. 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 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. But nonetheless it is contained within a finite surface, right? In a finite amount of area.

[B]

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

[A]

Right? You should be able to see the backs of your own head. 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 over there. That's very interesting. Galaxies right off in the distance. It's very interesting. And then you turn around, you go the other way and say, hang on a second. That's the same one.

[B]

It's the same stuff from the other side.

[A]

Yeah, it's the same one.

[B]

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

[A]

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?

[B]

That's true. Yeah. Well, my minor, minor point, but yeah,

[A]

hang on, that's my house. But before, hang on, vegetation existed.

[B]

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.

[A]

The light, it wouldn't be like that.

[B]

The light we see is light that left. The delay would be substantial 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, right?

[A]

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. We know that on a flat surface the internal angles of a triangle add up to 180 degrees, right? For example. So you can look out into space and run these kind of tricks. Like are there any sort of signs here and there that, 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 spa, 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.

[B]

It could be that we just can't look far enough yet to see the curvature, right?

[A]

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, flat to all of us 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 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 flat. And 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. Finite amount of quarks, of, you know, of atoms, whatever it might be, right is way, way less than our left 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?

[B]

But it's finite.

[A]

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.

[B]

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. The. 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 nonillion volumes, then you wouldn't be surprised to find that

[A]

a couple of them match 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 of the. Every one of those infinitesimally unlikely combinations. It's very uncomfortable.

[B]

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 solar system a perfect order, then there's an infinite

[A]

number of them out there, which is very uncomfortable.

[B]

And this bothers me a lot.

[A]

Me, too.

[B]

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.

[A]

Yeah, Yeah. I mean, this kind of gets onto, like. I think there's, like, a lot of things of.

[B]

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

[A]

And yet I don't.

[B]

I want to know what you think.

[A]

I don't know. I think I 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 a universe that looks like that is this. We're a bubble, effectively. If you imagine that there is a boiling ocean, right. Sort of like a pan of water. And the way that you get these bubbles that appear effectively from nowhere, as the water is boiling, 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 is that Big Bang is just one of those bubbles appearing. It's the way that sort of energy. Energy creates, you know, matter, and in this sort of, like, little explosion within this. This wider fluid. Now, 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 an 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 don't have to sit long with this stuff before it starts getting back to Michael with a slightly different color shirt on.

[B]

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

[A]

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 maybe this is actually the secret to what's going on more broadly across. Across the entire universe.

[B]

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

[A]

because our equations break.

[B]

Essentially, 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.

[A]

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.

[B]

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

[A]

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

[B]

Right.

[A]

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

[B]

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

[A]

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

[B]

Yeah, I don't like it, I don't like that my brain can go, oh, really? There's a, there's a smallest discrete piece of space. Cut it in half. Yeah, look, I just, I just did.

[A]

Just did.

[B]

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

[A]

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 discreet, the fact that you can't cut something up small enough because you get down to a pixel and there's nothing smaller than a pixel, there's nothing smaller than, than a bit that, you know. Quite happy, quite comfortable with that.

[B]

Yeah, well, you know, right. If we, if we did find like 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.

[A]

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 always in your own. How old my reference is, or new, isn't it coming out the new one. Anyway, sorry, I'm just, I'm such a gamer. I'm just really eagerly anticipating you're such a pro gamer.

[B]

Like I can't even keep up. You're talking about Pac Man. What are you gonna do next, Pong?

[A]

Look, I, I might live in Donkey

[B]

Kong country for all we know. Mathematically, your gaming references are superb.

[A]

I, 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 reality is when you are on a game, I mean, you sort of are in your own observable universe. Right? Like where you're turning your camera around, it just hasn't rendered anything outside of it.

[B]

Yeah.

[A]

So, I mean, maybe that is the solution to all of this. Maybe it is the. Just that actually it's all. I mean, that would make the sort of. This is a kind of one for another day, I guess, but the collapsing of the wave function. Another deeply weird and troubling physics problem with our understanding of the universe. That we don't actually know where particles are. Right. And they sort of seems 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.

[B]

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

[A]

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 can be any manner of things.

[B]

Until you look at it, 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, to answer the question. But I still think that we are in a real unsimulated universe.

[A]

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.

[B]

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?

[A]

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 breaks. 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, right?

[B]

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

[A]

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.

[B]

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

[A]

our eyes and particularly Cantor's brain. What an incredible brain that was.

[B]

Particularly Cantor's brain. Yeah, yeah, he took the first leap.

[A]

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

[B]

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

[A]

That was a blast. Well, I hope that. I hope that all of you enjoyed it. I, 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. Just to give you a little sweetener. As ever, you can send us in any thoughts or questions that you have to the rest of scienceolehanger.com and you

[B]

can join our newsletter at the restis.com

[A]

science I would say actually send us email. Sure. But if you put it on comments on YouTube then me and Michael can read them as well, which we do. Oh that's true.

[B]

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 they don't want us to see.

[A]

If you want to send us really, you know, like if you think you've solved Riemann's hypothesis or whatever and you want to like send that, send in 50 pages 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 in the comments on YouTube. Until next time.

[B]

See you later.

[A]

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