Two Infinities... And Beyond (Infinity Part 2)

Summary

The Rest Is Science — "Two Infinities (And Beyond)"

Podcast: The Rest Is Science
Hosts: Professor Hannah Fry & Michael Stevens (Vsauce)
Date: March 30, 2026
Episode Theme:
A mind-bending journey through the concept of infinity — from ancient discomforts to modern mathematical revelations — tracing how thinkers have wrestled with the infinite in space, time, and numbers. The hosts illuminate why infinity is so unintuitive, how it divided philosophers and theologians, and the seismic shift brought by Cantor’s theories that revealed not just infinity but many different (and bigger) infinities.

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Main Theme & Purpose

This episode explores the history and paradoxes of infinity:

• What does it really mean for something to be "infinite"?
• How did ancient and medieval thinkers grapple with this concept?
• How did Cantor's work explode the simple notion of infinity, revealing "sizes" of infinite sets?
• Why does this matter to our understanding of the universe, mathematics, and even our childhood curiosity?

The hosts break down paradoxes, philosophical turmoil, and mathematical breakthroughs — all with characteristic wit.

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Key Discussion Points & Insights

The Universe: Is there a bound?

• Children's universal question: "What's beyond the stars? Does the universe end or go on forever?" (00:03)
• The logical tangle: If it ends, what lies beyond the boundary? And what bounds that?
• "What bounds the boundary, right?" (Hannah, 00:46)

The Greek view & Zeno’s Paradox

• Zeno’s Paradox: You can keep cutting distances in half forever — does this mean movement is impossible?
• Greeks tolerated infinite processes (never-ending counting, dividing) but denied actual infinity—no room for truly unbounded, physical infinities.
• "They were okay with that process... But in terms of actual infinity... No, absolutely not. It's finite." (Hannah, 04:24)

Medieval Philosophy: Mathematical vs. Metaphysical Infinity

• The clash over "Is God finite?"
• Thomas Aquinas' compromise: mathematical infinity doesn’t exist (numbers, space), but God is metaphysically infinite (perfection, not quantity).
• Michael: "Could God put an infinite number of marbles in a bag?" (07:03)
• Hannah: "God can have as many marbles as he damn well pleases, frankly." (07:18)

The Harmonic Series and Nicole Oresme

• The Ant and the Rubber Rope Problem:
  • An ant crawls on a stretching rope; does it ever reach the end?
  • The key: If you think in percentages traversed, the total always increases, never retracing — mirrors the harmonic series of fractions (½ + ⅓ + ¼ + ... ).
• "If you add a half and a third and you add successive numbers to it, you're never going to get a smaller number." (Hannah, 10:30)
• Unlike Zeno’s paradox, the sum diverges—it gets ever larger without bound. The process creates infinity, not just the act of dividing.

Infinity as Heresy: Bruno’s Fate

• Giordano Bruno insisted the universe itself must be infinite, because a truly infinite God would create nothing less.
• "Why would he build this tiny, cranky little universe...?" (Hannah, 13:03)
• Theological fallout: If there are infinite worlds, what happens to original sin, salvation, etc.?
• The Roman Inquisition's brutal punishment:
  • "They put a nail through his tongue so he couldn't speak to the crowd, and then they burned him alive." (Hannah, 15:46)

Galileo, Hilbert’s Hotel, and the Strangeness of Infinity

• Galileo realizes there are as many square numbers as natural numbers—paradoxically "the same number and more" (18:25)
  • "How can there be more than and the same number at the same time?" (Hannah, 18:57)
• Most thinkers tried to ignore such paradoxes — until Cantor arrived.

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Cantor Steps In: The Many Sizes of Infinity

22:02 — Post-break, the hosts dive into Cantor's fundamental revelations about infinity.

Understanding Number Types

• Natural numbers – the counting numbers: 0,1,2…
• Rational numbers – fractions/ratios (1/3, 2/7), which seem "more numerous."
• Michael’s diagonal numbering: Any infinity matchable 1-to-1 with the naturals is the same “size.”
• "You can write them in a clever way that allows you to list them sensibly...knowing that you're never missing any." (Hannah, 36:12)

Infinite Fractions between 0 and 1

• Hannah: "How could it possibly be that there's the same number of numbers between 0 and 1 and...between naught and everything?" (28:34)
• Counter-intuitive but true, due to countability.

Real Numbers: Uncountable Infinity

• Real numbers include irrationals (π, √2): not just fractions.
• Cantor’s diagonal argument:
  • However you try to list all real numbers (say, decimals between 0 and 1), you can always construct a new real number by changing the nth digit in the nth listed number—a number not in your list, no matter how you try. (33:49–35:41)
• "There is no way of doing an exhaustive list." (Hannah, 35:20)
• "So the number of real numbers, even just between 0 and 1, is larger. Literally there are more of them than there are whole numbers..." (Michael, 36:01)

Two Infinities: Countable vs. Uncountable

• Aleph-null (ℵ₀): The "size" of all naturals, rationals, evens, odds—countably infinite sets.
• The Continuum: The "size" of the reals—strictly bigger than Aleph-null.
• "Some infinities are bigger than others. It's such a wild idea." (Hannah, 38:08)
• "Infinity plus one is infinity. Yes, it is. But that doesn't mean there's nothing bigger, because there is." (Hannah, 38:18)

Ordinals & Going Beyond

• Ordinal numbers: Describe position/order, not just quantity (e.g., 1st, 2nd, Omega, Omega+1...).
• Even after “infinity” (Aleph-null), there's an "after"—e.g., the person who finishes a race after an infinite number of runners is Omega+1.
• Michael: "What if I can't move them? ... The number of people who ran the race is still Aleph null, but what place do I get? ... It is Omega." (41:00)

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Notable Quotes & Memorable Moments

• On imagination vs. reality:

  • "I can always go...in my imagination with this piece of meat up here, this, like, wet squirting computer. I can go now. Forever. Beat that." (Michael, 00:48)

• Infinity as contradiction:

  • "How can there be more than and the same number at the same time? ... It doesn't make any sense." (Hannah, 18:57)

• Cantor’s impact:

  • "He was, as an individual, solely responsible for grappling infinity. And yet he died believing he was a failure." (Hannah, 48:20)

• On resistance to new ideas:

  • "[Kronecker] said that he was a scientific charlatan. ... He said he was a renegade. He said he was a corrupter of youth. ... I love the idea of, you know, young teenagers hanging out on street corners reading Cantor's diagonalization proof and suddenly becoming, getting up to all manner of things." (Hannah, 44:08)

• Hilbert’s defense of Cantor:

  • "No one shall expel from the paradise that Cantor has created." (Hannah, 48:25)

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Key Segment Timestamps

• “What bounds the boundary?” – Opening paradox: 00:19–00:48
• Greek discomfort with infinity: 03:45–05:14
• Ant on the Rubber Rope / Harmonic Series: 08:08–11:53
• Bruno, Infinite Universe, and Theology: 13:31–16:07
• Galileo’s realization: 17:13–18:25
• Introduction to Cantor and countable infinities: 22:03–29:10
• Cantor’s diagonal argument: 33:49–35:41
• Aleph-null and the hierarchy of infinities: 38:08–40:56
• Ordinal numbers and ‘Omega’: 41:00–42:24
• Cantor's tragic end and legacy: 43:26–48:25

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Conclusion & Teaser

• The episode ends with a poignant tale of Cantor’s struggles and underappreciated genius, and a promise to return in the next episode to climb even higher through infinity’s strange landscape.
• "Finally, parents, you will be able to win the challenge against your children of who can name the biggest number." (Hannah, 48:41)

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For anyone curious about the mysteries of infinity, this episode walks the line between math, philosophy, and cosmic curiosity, led with humor and awe by two passionate science communicators.

Transcript

[A]

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

[B]

And I'm Michael Stevens.

[A]

I think there comes a point in every child's life where they ask, what's beyond the stars, right? How big does it get in that regard? I mean, they're kind of similar to every human who's ever lived, right? Where does it end? Does it end, Michael? Does it end?

[B]

And what if it doesn't?

[A]

And what if it doesn't? But what if it does? Because there's like a strange thing going on here. If you say, no, it's finite, the universe is finite, well, then, well, then it has to be bounded by something. And then. And then what about that thing? Is that infinite or is that finite? Because if that's finite, then that has to be bounded by something. And then so on and so on and so on and so on.

[B]

I know if there's a boundary, what bounds the boundary?

[A]

What bounds the boundary, right?

[B]

And this is all happening in our heads. Like, no matter how rational you try to be about explaining that something ends, I can always go. I disregard that. And it continues in my imagination with this piece of meat up here, this, like, wet squirting computer. I can go now. Forever. Beat that.

[A]

So that is what we are talking about today. We are talking about whether there is a bound to the universe, whether there is a bound to our thoughts, whether infinity actually exists, or whether it is all a complete figment of Michael's squishy wet computer in his head. This episode is brought to you by Cancer Research uk.

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[A]

So okay, in the last episode, we were talking about Zeno's Paradox. We were talking about all of the strange consequences that happen when you start considering infinity. These ideas were extremely troubling. I mean, to to humans, ultimately, throughout history. The Greeks in particular, really disliked this idea of infinity, but they sort of got around it. They managed to make themselves feel a little bit more comfortable about it by saying that there was a difference between a process which went on forever, which in theory didn't actually come to an end. So counting, for example, or taking the steps, a sphere, you can kind of carry on going and going and going, but that's a process that doesn't have an end. And they were okay with that. They were like, all right, we can sort of accept that that type of infinity exists. But in terms of actual infinity, this idea of something that is, like, unbounded, that literally exists, like the infinity of space, for instance, that they were like, no, we just don't. I just don't think that that can happen. Time can be infinite, sure. But space? No, absolutely not. It's finite.

[B]

Yeah. So time is like an infinite process. The clock keeps going. Time, there's always going to be a future. But all of it isn't, like, right here in front of us or in front of anyone. So there's no actual infinities out there to worry about. But we can imagine something that doesn't end. It's a nice, safe. And I think a lot of people today might find that actually Pretty acceptable.

[A]

Yeah, I think so, too. But. But then also things like Zeno's paradox, where you're cutting something in half and half and half and half, half and half and half and half and half and half and half and half and half forever. And finding that you get to infinities that way that, you know, Aristotle was like, it's no big deal, right? Just don't do the cutting. The process of cutting is where the problem arises. And if you don't do that process of cutting, then you don't find infinities. Infinity doesn't actually exist for most of the. Most of the time since that moment. That's kind of how people have thought of infinity. I think it is exactly as you describe it, right? This sort of, like, comfortable thing. Then it came to the medieval period, right? You know, when you got this resurgence of philosophy, there were a couple of people who were not totally happy. Not totally happy with Aristotle's way of things as being a full stop on the end of the sentence, particularly as it related to God. Because here is the thing, okay? If God is the most perfect being, and perfect means being finite and bounded, well, then that means that God must be limited, right? But if God is limited, well, then. Then he's not God. How can you possibly have a finite God? It doesn't make any sense. So Thomas Aquinas, this is in, like, the mid-1200s. He and his peers were like, okay, look, right, let's separate this out. Let's have a kind of mathematical infinity that has to do with size, and that doesn't exist. And we're okay with that. We're back to Aristotle now that you don't get physical infin. Infinities, but a sort of metaphysical infinity, which God is. That's to do with absoluteness and perfection. That's fine. We're sort of happy with that. So there's like, separation of maths and godlike quantities.

[B]

Okay, so what does that mean that they're separating out, like, an abstract infinity from the mathematical meaning, like an amount? Like, if you were to ask Aquinas, could God put an infinite number of marbles in a bag?

[A]

Well, I think the first answer would be that God can do anything. Michael. I think. I think God can have as many marbles as he damn well pleases, frankly.

[B]

Look, I agree. I'm not questioning how many marbles God's allowed to have.

[A]

But I think that putting marbles in a bag, there's sort of a process of that, right? That's. That is. That kind of brings you Back to the Aristotle thing. Around about the same time, there are other people who are thinking about this more mathematically and don't feel totally comfortable with the idea that infinity doesn't exist, that it isn't a thing. There was someone called Nicole Orsme who was looking at the harmonic series. Now, we actually mentioned this, although we didn't call it this in the last episode. We were talking about an ant or a stretchy piece of rope which does actually have a solution. Do you want to remind us of that puzzle?

[B]

Okay, so here's the puzzle. An ant is crawling along a totem rubber rope. It's important that it's rubber because it's going to be stretching. But before we stretch it, we're watching. And this ant moves at a uniform speed of 1cm per second. So 1cm of rubber crosses his body every second. But then we start stretching the rope and we stretch it much faster at a rate of 1 km per second. So after 1 second, the rope is now 2 km long. Two seconds later, it's 3 km long, and so on. The question is, will the ant ever reach the end of the rope?

[A]

This is the thing. I mean, how can it possibly reach the end of the rope? How can it ever get to the end when the distance that it has to travel increases every single second? This feels like a situation where you could carry on forever and the ant would never get towards the end. But what Nicole Orsme was looking at was a version of the same problem. It's called a harmonic sequence, and it's where you add fractions of numbers, numbers together. So you can add a half and a third and a quarter and a fifth, and so on and so on and so on, which, if you translate it to the ant problem to get it exactly, I mean, it's traveling, what do you say, one centimeter and one kilometer.

[B]

One, one centimeter per second is, is how much rubber the ant is covering, right?

[A]

And the, the band is stretching 1km a second. So these numbers, the fractions you're going to be dealing with, are way smaller from the off. But the, but you're adding these progressively smaller fractions together. So it's, it's, it's also a harmonic sequence. And what bought is that. Okay, if you think of this, I'll do the analogy in the ant form. Okay. If instead of thinking about how far the ant is moving and how far the ant has to go, if instead you think about what percentage of that band the ant has traversed, the ant never goes backwards, right? The Ant never decreases. The fraction of that band that it has traveled, it only ever goes up.

[B]

That insight really helped me with this problem because you can take your. Your piece of rubber and you can make a mark one third of the way across. Now, if you stretch it and you make it twice as long, that mark is still one third of the way across the band, right? Because it's being stretched ahead and behind.

[A]

Exactly, exactly. That fraction, as it were, of the band is, is, is unchanging. And actually, that's kind of similar to what Osmi did in terms of the harmonic sequence. He said, look, if you add a half and a third and you add successive numbers to it, you're never going to get smaller. You're never going to get a smaller number. But what that means is that you have this sequence, this sequence of fractions, even though they get smaller and smaller and smaller and smaller over time. The fraction that the ant is adding to its total is getting smaller and smaller over time. Nonetheless, it is constantly increasing, right? It's constantly increasing, but it's increasing in a way that is not like with Zeno's paradox. With Zeno's paradox, you approach the total, you approach one, and you never quite get there. But with this, with a harmonic sequence, you carry on increasing. You go over one. Effectively, the ant will reach the end. It may take an unimaginable amount of time, but it will get there and

[B]

it will be a finite amount of time. It could be a lot of time. But yes, the proportion left that the ant needs to cover is getting smaller. Even though in, in like, literal terms of length, that might seem large, the remaining amount of rubber for the ant to cross eventually hits zero.

[A]

So this is a problem with the Aristotle idea that, okay, infinity doesn't really exist. It's just something that appears when you chop stuff up. You know, the process of chopping stuff up, you're creating this illusion of infinity that isn't really there. Soon as Osme proved that harmonic sequences that these series will diverge will carry on getting larger and larger and larger and eventually themselves will become infinite. Well, now that same logic doesn't work. It's not just the process that creates infinity. It's actually adding these numbers up, they're getting smaller, will itself approach infinity. It's sort of like. It just doesn't fit that comfortably with the Aristotle idea.

[B]

This was in the 1300s.

[A]

Yeah, this is in the 1300s. So this is around the same time as Thomas Aquinas. There's like, people are revisiting this idea of Infinity a few hundred years later. So, okay, People are like, well, let's just sort of not really worry about that too much. Let's just say that infinity is there, but it's God. It's. God is infinite, and that's fine. But then it starts to get violent. Then people start getting really upset about this. Okay? Because at that point, Giordano Bruno, who was born in 1548, he's like, okay, I just don't. I don't buy the idea that you can't have infinite space. I don't buy the idea that you can only have God as being infinite and nothing else. Because we're. All right, let's say God is infinite, great, infinitely good, infinitely powerful, fine. Why would he build this tiny, cranky little universe that has got these edges to it? It's sort of an insult to his power. You know, if he was a perfectly infinite creator, if that's what we're saying God is, well, then, you know, his power could only be satisfied by creating an infinite cosmos. Yeah.

[B]

Why not allow it?

[A]

Why not? I mean, that seems like a perfectly sensible thing. This is about the same time as Copernicus is changing our view about what's revolving around what and the star being the center of the solar system. And I mean, I agree. What is wrong with it? The Roman Inquisition, they disagreed. They thought there was quite a lot wrong with it. They thought this wasn't a flash through to God's power. Because their argument was, if you're saying that there's an infinite universe and that all of these stars out there are other suns that have other planets, there's an infinite number of alien civilizations. Do they all fall into original sin? Are we saying that God has to incarnate Jesus and be crucified in an infinite number of places and planets in order to save all of them? So something just like completely breaks the uniqueness of the Christian salvation story.

[B]

Even if God did that, which he could, why didn't he tell us about it in the Bible?

[A]

Should have mentioned it, you know, Come on, Matthew, Mark, Luke and John. What were you doing, guys?

[B]

I guess they were earthlings, you know, they just didn't know that he went to Vargas 12 next and was like, guys, I'm going to turn not water into wine, but I'm going to turn some blue blips into gleeps. And that's their miracle over on.

[A]

That's their miracle, you know, or. Well, right. Or maybe they didn't have wine at all on Vegas 12 and then that was the real miracle.

[B]

That was the real miracle. He brought the wine.

[A]

Lucky for us. Lucky for us that wine already existed on this planet. Thing is, okay, I don't know if you know anything about the sort of the Roman Inquisition, but I wouldn't say that they were particularly kind when they were confronted with an idea they didn't agree with. So Bruno, he's like, look, guys, I just. I'm not backing down on this. So. So they made him spend eight years in the dungeons of the Roman Inquisition, arguing his case, like, refusing to recant his. His belief in the kind of plurality of. Of different worlds in 1600, that the Church declared him a heretic and they tied him to a stake. And then. This is horrible. I'm sorry, but they put a nail through his tongue so he couldn't speak to the crowd, and then they burned him alive. I mean, they really didn't like the idea. They really didn't like the idea, did they?

[B]

And the idea they didn't like was that the universe was infinite, Right?

[A]

That the universe was. That infinity was a physical characteristic rather than something that belonged to God alone.

[B]

Wow.

[A]

I mean, isn't that dramatic? Just the very idea of it I find absolutely extraordinary.

[B]

It's extraordinary, and it makes me feel actually kind of sad that I live in such a soft time where we can do not just one podcast about infinity, but two. And, you know, people are like, nah, I'll listen to that. You know, maybe tomorrow after work. If I was doing this 400 years ago, people would be like, nailing my tongue. So I couldn't do it. The knowledge would be so powerful today. It's like, nah. Yeah, you're right.

[A]

Hey, look, Michael, movie. There's two nails waiting for us in the afterlife.

[B]

That's what maybe there are.

[A]

And one and two. Thank you.

[B]

We gotta find the topic that. The mathematical topic today that if we were to discuss it, the Royal Society would nail our mouths shut. There must be one out there.

[A]

I can. I can see the YouTube title already. Michael. Yes, Math so Dangerous they Nailed their tongues.

[B]

Right, okay, see, now we're talking good titles, like the Most Dangerous Mathematical Idea Gone Tongue Nail. I don't know if you guys remember that from, like, 2012.

[A]

Amazing, amazing. The thing is, is that, you know, nailing someone's tongue, burning up the state, it doesn't make this idea go away. You know, 1638. So 40 years later, almost. Galileo Galilei, he's thinking about numbers and he discovers something really unsettling. He's thinking about square numbers, right? So, you know, 1 4, 9, 16. And then he's sort of counting them and he's like, okay, well, you know, one is the first, four is the second, nine is the third, and so on and so on. And then he's like, well, hold on. You could write them as a list, label them with a number, the first one, the second one, the third one, and then this list goes on forever, right? There's an infinite number of square numbers

[B]

and there's an infinite number of whole numbers. So you don't run out of either. You can just.

[A]

You don't run out either. So. Well, then this is a very uncomfortable idea because this concept of infinity, as we were describing in the last episode with Hilbert's Hotel, it doesn't follow the normal rules, right? It's not doing the things that normal numbers are doing. But again, like this other idea of infinity sort of pops its head up and people kind of ignore it and say that it goes away.

[B]

I want to. I want to meditate on the freakiness of that. So it was it Galileo who really kind of first wrote about this in a way that people heard of. Because it's frightening, I can imagine, to go, all right, well, look, square numbers. I know what those are. But not every number is square. Seven is not square. Twelve is not square. So clearly there are, you know, more numbers than there are square numbers. But then Galileo was like, you can't prove that, guys. I think there's the same number of them.

[A]

How can there be more than and the same number at the same time? Yeah, like how? It doesn't make any sense. It doesn't make any sense.

[B]

Yeah.

[A]

I mean, no wonder so many people have struggled with this because it doesn't make any sense. This is essentially how things stood for for most of that millennia, right? That a few people here and there had come up with these little glimpses of insights, Galileo being one of them. But, you know, ultimately most people just wanted to brush infinity to one side rather than stare it directly in the face. Until the hero of our story, Gale Cantor, which we'll come to after the break. This episode is brought to you by Cancer Research uk.

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[A]

And we're back.

[B]

We're back. So as Hannah said before the break, for a long time people tried to brush infinity out of the way. But. But in the second half of this episode, we're gonna let it get brushed right into our brains. And what Cantor did is that he did a lot of this brushing and he found in that detritus infinities that were bigger than others.

[A]

So there are these little ideas, these little moments of insight. Galileo in the 1600s. But for most of that entire millennium, people are like, do not want to face infinity. They know there's something weird going on about it. But nobody is turn to look at it absolutely directly until the great hero of our story. Michael.

[B]

Yeah, one of the things Cantor starts doing is really precisely thinking about numbers. And this is honestly, this is what got me into math again. My whole childhood I skated through math by remembering formulas. You know, oh, to find the derivative, you do this and you do that to the exponent and blah blah blah and you add, add C plus C. I didn't know why, but you have to do that. Talking about calculus here. But then when I started reading about infinity, I was like, oh my gosh, this is like math that a child could understand because you have to go all the way back to talking about what is a number. And I want to preface this by going back to part one. So last week we talked about is infinity a number? And. And we got into this little micro debate where you say it's not. And I'm like, it is. It does number y things. It's an amount. And then I watched my episode on infinity from years ago, and at 1 minute and 18 seconds, here's a direct quote from me. Infinity is not a number. After that I say it's a type of number.

[A]

Okay, okay.

[B]

And so I stand corrected. By myself, I agree that infinity, the word infinity does not designate a number. It designates a kind of number. And so to show what Cantor did, I want on to focus first talk about some types of numbers. And this is worth it, guys, because we are going to be using these tools to build ourselves a ladder that reaches infinity and then finds other infinities that are even more endless that literally goes beyond, literally goes beyond forever. The numbers that we're all most familiar with are called whole numbers. These are numbers that don't have a fractional part. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Negative 5 is a whole number. But let's just focus on a different kind of number called the natural numbers. Natural numbers are just positive whole numbers. Funny enough, some people say 0 is included and some don't. I do include it. So 0, 1, 2, 3, 4, 5, 5 and a half. No, not a natural number. 6. Love it. Very natural. Clearly the number of natural numbers that exist is infinite. There's no end to it. I can always add one to the biggest natural I've thought of and have one that's even bigger. And this goes on and on and on. But the natural numbers, as we've already hinted at, do not represent every possible number. Five and a half, for example, one third. Those are called rational numbers because they are represented as a ratio between two numbers. The ratio between 1 and 3 is 1/3. Now, natural numbers are also rational because two can be written as a fraction. Two over one. Okay, now you might start to think, ah, I'm starting to see potentially some different sizes of infinity because we've got the, the natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on. But then we have the rational numbers, which includes all of those and then some more. A half, a third, 11.2, right? These are all rational numbers and there must be more of them because they include all the naturals and then some. However, that's not True. Imagine for a moment writing yourself a grid of every possible rational number. This is a very fun activity to do at home. Write down every possible fraction. And you can do this just on a, on a square grid, because a fraction only has two numbers in it, an X and a Y. I can write 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 along the top and I can then number down 1, 2, 3, 4th, 5, 6, 7,8, 9, 10. And I can combine them all. I can get every combination. 1 over 1, 1 over 2, 1 over 3, 2 over 1, 2 over 2, 2 over 3. They're all going to be there. And this is, this is infinite, right? This thing, it goes, it goes off to the right forever and it goes down forever.

[A]

So you've got two, two dimensions, both

[B]

of which are infinite in two dimensions. So clearly it's bigger than the one dimensional number line of just the naturals. Or is it? There's this thing called moving diagonally that will blow your mind. So start in the upper left corner with your, your first fraction, which might be 1 over 1, then move to the right, and now we're at 1 over 2. But then rather than continuing on to the right, which you'll never end, you'll, you'll keep going forever, just go down to the left to the fraction below. One over one diamond, two over one. Okay, now you're actually snaking back and forth and you're shading in this entire square and each time you hit a fraction, give it a natural number like you're numbering them. Okay? You're number one, move over your number two, go diagonally down your number three, move over and you go back and forth and you number them all. And because there's an unending amount of natural numbers, you're never going to run out of them to label a fraction. So that means there are the same number of fractions, rational numbers as there are natural numbers.

[A]

It's just like Galileo squares, but, but

[B]

way, but more complicated, but more complicated. So first of all, that, that is mind blowing, isn't it? Because it seems like, yeah, you're right, Galileo, there should be more numbers that aren't square than numbers that are square. And yet that's not true. It definitely seems like there should be more rational numbers than whole numbers, and yet that's not true.

[A]

The thing is, there's an infinite number of fractions between naught and one.

[B]

That's right, I'm talking about all fractions. But you don't have to just between 0 and 1.

[A]

And so how could it possibly be that there's the same number of numbers between naught and one and between naught and everything? How is that possible?

[B]

I know. I mean, sure, like why, why am I talking about all fractions? We could just do 1 over 1 and then everything that's smaller, 1 over 2, 1 over 3, 1 over 4. Clearly the denominators are just going up by integers. We can label them all with a natural number. They correspond one to one. There's the same number of them. There are just as many fractions between 0 and 1 as there are whole numbers. Give me a break.

[A]

This is. Give me a break. Exactly.

[B]

And say to something like that, ah, that doesn't make sense. I love it, but it does make sense. That's what these demonstrations show. It just feels like we shouldn't have learned that. It's true. Like, we're getting into the how the sausage is made and we liked it, or maybe we didn't. It's, it's scary. It's going to get even worse, though, because there are even other numbers that are not whole numbers and are also not rational. These are numbers like PI, the square root of 2. These are numbers that you cannot represent as a ratio between two numbers. You can get pretty close with PI. Like 22/7 is pretty close, but it's not quite there. The problem rational numbers have is that their decimal expansion never ends. So there is no, like, finite number that we can say, oh, yeah, just take that and divide it by, you know, some other number there. And there it is. When you start including all naturals, all rationals, and then everything else. Now you're describing the real numbers and this is a really big category. How many real numbers are there?

[A]

Your instinct would be, well, we'll just go. I mean, if we've got the same number of fractions as whole numbers, I mean, surely it's the same thing. Like thus far, Hilbert's hotel has shown me it doesn't matter what you do to infinity. Bend it, stretch it, squid some in the middle, double it in size. It's always infinity. There is only one type of infinity. I mean, it's just, you're, you're going to, you're going to tell me there's the same number.

[B]

That's what you say it's going to be the same number. I can always fit more in. In the last episode, Hannah, you showed us that we could fit an infinite number of buses, each containing an infinite number of people, into a hotel that has an infinite number of rooms. But they're all full. We can still fit everyone in. So give me a break. Of course the real numbers are going to be just the same as the, the fractions and the naturals. But then Cantor shows that that's not true. And this is one of my favorite things. I think about this while I'm driving in the car. I, I try to, I try to tell it to my daughter at night as she goes to bed. It's like counting sheep. Grow up. We're counting reels. And I still don't think that audio only, it's going to be necessarily that clear. But Hannah, you can help me too because you've, you've talked about this before as well. Yeah, so, so I'll try.

[A]

So. It's so good. I mean this, this is like if there award for best WTF moment in the history of maths. I think this, I think this one wins. I think this wins.

[B]

It wins because it is so mind blowing for so little work. It's not like, okay, well you need to understand pic numbers and the monster group. No, it's just like numerals.

[A]

We're going to try and explain it. We're going to try. We're going to try and explain it to you.

[B]

So to, to do this, we're going to get back to our sheet of paper and we're going to just write out every real number. And I think it's fine to just imagine randomly writing down numbers. So to make this easy to keep in your head, we're going to only talk about the real numbers between 0 and 1. That's it. Okay, so there's, there's clearly more of them, but we can, we can just look at between 0 and 1. That means numbers like 0.1, but it also means numbers like 0.34297743 forever. Okay, so just randomly fill this out and sometimes you'll just hit a bunch of zeros. And now you've reached a number that happens to be rational, but you're just randomly writing these down. And you don't have to actually write them down, but you can imagine that you've done this and you're writing one below the next so that you can number them. That means you're corresponding each one to a natural. We've got the first real number and then there's this big string of digits, the second real number and we randomly write down a whole bunch of other digits. If there's ever a match, you know, we delete it. So we have all these unique strings of random Digits which represent. We imagine that there's a decimal in front of each one. They all represent every real number between 0 and 1. And you might think, okay, well, obviously I can keep doing that forever, but I can keep labeling them 1, 2, 3, 4. You know, I can do that forever, too. Same number. But then what Cantor said is, guys, let's get diagonal with it. And this is when things get scary. So you go up to that first real number you've written, and you look at the very first digit, the leftmost digit, and write down whatever it is, but you change it. So let's say it's a seven. All right, well, let me pick something that's not seven. How about five? Or I guess you could even just add one to it. We could, we could make this much more systematic.

[A]

It doesn't matter, just as long as you change it in some way.

[B]

Just change it in some way.

[A]

Up a bit. Just muck it up a bit.

[B]

And so you write down that new different number. Then you go to the next real number you've written, and you look at the second digit in that number, and let's say that that digit is 3. Okay, now in this new number you're creating, write down something besides three. How about two? Okay, now you go to the third number and you look at the third digit in the third number and let's say it's a seven. Cool. I'm going to write down six. I'm just going to keep going one below. And as I do this, I continue out, the fourth digit in my new number is different than the fourth digit in the fourth number. The fifth digit in my new number is different than the fifth digit in the fifth number. And I continue this on forever. I have just created a number that differs in one way, at least one way from every single real number that could be written.

[A]

That's right. Because. Because. Because your number that you've just written down is basically a diagonal line, right? You've const it by forming this diagonal line down your table, which means that it has to be different from the first of your numbers because. Because you've mucked up that first digit, it has to be different from the second because you mucked up the second digit, it has to be different from the third because you've mucked up the third digit. So whatever, however long you spend constructing your list, however exhaustive you make your original list list, it is so easy by running along the diagonal and just mucking things up as you go to find a number that is not in your List that is a brand new number that has come in from everywhere. Thus, whatever you do with your list, it will never be exhaustive. There is no way of doing an exhaustive list.

[B]

That's right, because I can write this new number and I can go, oh my gosh, I found another one and I can add it back into my list and then I can go diagonal again and come up with another number. So even though my first list had real numbers that corresponded to every natural 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on forever, I can keep adding new things to it. So the number of real numbers, even just between 0 and 1, is larger. Literally there are more of them than there are whole numbers on the entire number line.

[A]

Because the key thing here is that with, with Galileo's square numbers, the reason why you know that are the same number of both, it's because you compare them up. You can say, that's the first, that's the second, that's the third, that's the fourth, and so on, and so on and so on. Likewise with your fractions, you can write them in a clever way that allows you to list them sensibly. That's the first, that's the second, that's the third, that's the fourth, knowing that you're never missing any. And that's how you know that the two are the same, that the number itself, the fraction and the ordering of your list. But what Cantor proved in this theorem that you've just described, so genius, is that it doesn't matter how you try and do it, it doesn't matter what order you put them in. There is no system, there is no sensible way to order the real numbers. The irrational, horrible, ugly numbers muck it up for you. There is no sensible way to order them, that you can list them and assign them a first, second, third, fourth, and carry on knowing that you're not going to miss any at.

[B]

So there are more than there are natural numbers. No, there are an infinite number of natural numbers. So we're now talking about two different endless quantities, one of which is bigger than the other, literally describes more things than the smaller infinity. So we've got two infinities here, okay? One is how many natural numbers there are. These are the counting numbers. And the other is the cardinality of the continuum. That's the name for this. Cardinality means how many. And the continuum means all the real numbers, not just the wholes and the fractions, but all of them. How many of those there are the cardinality of the Continuum is a bigger infinity than the counting one that we've been discussing so far.

[A]

I think this is such a wild idea that some infinities are bigger than others. It's such a wild idea. And it's like, it goes. So, I mean, it goes completely counter to the games that you play with your children. Hey, what's the biggest number you can name? Blah, blah, blah, blah, blah. Infinity. Oh, infinity plus one. Psych. Infinity plus one is infinity. Yes, it is. But that doesn't mean there's nothing bigger, because there is.

[B]

That's right. When we know that if you add one to infinity, you still have the same number of things. Hilbert's hotel shows us that. But when you add in, add is the wrong word to use. But when you squish in, when you squish in, in the, the reals, in between all of these numbers, now you have something that is demonstrably different and bigger. So let me tell you some terminology. I mean, Hannah knows this, I'm telling you, the audience. I'm talking to you guys. The amount of natural numbers that there are, which is the same as the number of fractions that are possible, it's the same as the number of even numbers that there are odd numbers, square numbers. That is an amount, and that amount has a name. It is called Aleph null, and Aleph null is a number. It's an amount of things that you can imagine, and it is the smallest infinity. And it's also annoying that the cardinality of the continuum is kind of just that it doesn't have its own special name because we're not sure where it lies.

[A]

Yeah, of course.

[B]

But we can get bigger than the cardinality of the continuum, the number of real numbers that exist. But to do that, we're going to have to switch to a different kind of number. And I know this sounds, you know, abstract and weird, but it's not. Because the other kind of number we're going to discuss are called ordinal numbers. They're about order. And we use these all the time when we talk about. Oh, the first thing I did was this. The second thing I did was this. 3rd, 4th, 5th, 507th place in the race is what I achieved. Those are all ordinal numbers, numbers, they're about order. And ordinal numbers are going to act a lot different than cardinal numbers, which tell us how many things there were. You can always fill up an infinitely roomed hotel and then add one more person if you, you know, squish, squish everyone forward a little bit you've got a new room. Right, but what if I can't move them? What if, as we said at the end of the last episode, an infinite number of people finish a race and then I cross the finish line? The number of people who ran the race is still Aleph Nol, but what place do I get? And that's when we have to come up with a whole brand new number. It's the very first number name that comes after all the naturals and it is Omega.

[A]

So after you have finished, after this infinite race has finished, then you do plus one. I mean, that's how you trick your children, basically.

[B]

Exactly. But I want to emphasize that it's not just plus. Like more precisely, we're not adding. It's not plus, it's after. So Omega comes after Aleph null things, the smallest infinity. It doesn't contribute to the total. We don't have more. But if we're going to order things, we suddenly do realize that we can't do the Hilbert Hotel shuffling. Omega plus one comes after Omega. Omega plus three, Omega plus four. These are all numbers, but they don't describe more things, they just describe things coming after infinity. Omega is also the answer to the question we brought up in the last episode about, okay, so Hilbert has a hotel with an infinite number of rooms. If another hotel opens up next door and it has an infinite number of rooms, there still aren't more rooms. But how the heck does this new hotel number its doors? Because Hilbert has already numbered all of them. He's bought all the numbers available in the town. There aren't any numbers for the other one to buy. They're going to have to use ordinals. So their first room can just be called Omega. The next room is Omega Plus 2. Again, there are not more rooms. It's not that we can fit more people, it's just that we can put them in an order.

[A]

Now the thing is, there is actually further that you can you. And we will go on this. But Michael, I just had a little pause in the recording and we, we decided that much like infinity itself, we might have to do infinity plus one episodes on this because there's just too

[B]

much to say, I think marinate in what we've discussed and then we can use those tools and an extra one called the axiom of replacement to start growing. Because remember, we've only reached two infinities, the smallest Aleph null and a bigger one, the cardinality of the continuum. And yet we haven't even begun. We haven't even Got these ordinals now that are going to help us climb and climb and climb until we reach. Well, stay tuned and you'll find out.

[A]

Well, it's, it's Two Infinities and Beyond.

[B]

I think that's T W O Two Infinities and Beyond. That's. I don't know if that's a good title for the episode, but it's definitely what we just discussed.

[A]

It's, it's, it's Nerd. Gives me nerd cachet for coming up with that pun. Let me just tell you though, before we leave you on this episode, let me just tell you a little bit more about Cantor. Because this guy, I mean this is like the most mind blowing idea, right? All of these people, all across history, all of these great mathematicians, philosophers, theologians had wrestled with this unimaginable beast of the infinite and none of them had been able to tame it. And yet Gail Cantor was the one who got it within his grasp and made it make way more sense. And he paid actually a really heavy price for this really. So for stasis, I mean people did not like it. Like really really did not like it. He had this arch nemesis, Leopold Crow Kronecker, who's responsible for the Kronecker delta symbol, if you've ever come across that. And he every day, don't remind me non stop. He was this old school mathematician. He was like, no, you know, he's back to the Aristotle things like no, only finite whole numbers are valid. This is not. I don't like this idea that some infinities are bigger than others. So he famously said God made the integers, everything else is the work of man. Just really dismissing Cantor's work. He said that he was a scientific charlatan. He just didn't believe his proof. He said that he was a renegade. He said he was a corrupter of youth. Youth, right. Which I love the idea, I love the idea of, you know, young teenagers hanging out on street corners reading Cantors diagonalization proof and suddenly becoming, getting up to all manner of things. That really pleases me. But the thing is, is that Cantor was somebody who, I think by modern standards we would say that he had bipolar disorder. He had a sequence of really serious mental breakdowns. It wasn't necessarily that he was trying to grapple with or it certainly didn't cause them, but I think it's general consensus that it didn't make it any better and it probably did exacerbate some of his symptoms that he was trying to grapple with these great ideas. But Especially that he was just under so much pressure from all of his peers who just ridiculed his work, ridiculed his ideas. He ended up being in a sanatorium in Germany, which I've actually, I went to go and see, went to go and visit.

[B]

Oh, really? Where in Germany is it?

[A]

It's called the Halle Mental Asylum, the Nirven Clinic. It's still active, by the way. It's still a hospital, specialist hospital. It's absolutely beautiful. I can't describe it to you. It's this just stunning architecture, kind of green ivy up the sides. It is the kind of place that you could imagine going for respite, you know, for a way to sort of find yourself again. It's this psychiatric hospital. While he was in there, he had all kinds of really just incredibly distressing hallucinations. He believed that when he had a chamber pot in his room, and he believed that when he was stirring his urine, he was controlling the weather. Right. He also believed he'd uncovered this really great conspiracy that Francis Bacon secretly had written all of the works of Shakespeare. There were things about him believing that he was the king of Spain. There was just, I mean, really, really difficult hallucinations in amongst all of this unimaginably deep mathematical thinking that you're describing.

[B]

Well, that's. Yeah, I mean, I want to say, like, this guy discovered larger infinities. Are we sure he wasn't also controlling the weather by stirring his poo? Are we sure he wasn't the king of Spain? Because I believe it. After learning what he discovered, I think

[A]

we at least needed to do a controlled trial. You know, it's like that's the very least that we could have. We could have given him.

[B]

Yeah, yeah.

[A]

Unfortunately, the conclusions were the opposite. Whether people were as dismissive of his infinity ideas as they were the fact that he was the king of Spain. You know, he died this really tragic death. He was. He was really malnourished. He was really deeply depressed. He had been stripped of his academic glory by the sort of nastiness of his peers. And he died of a heart attack in 1918. And what was really like this really great shame was that when he died, he. He had fundamentally rewritten our human relationship with this previously untamable beast. Right. He was, as an individual, solely responsible for grappling infinity. And yet he died believing he was a failure. And, you know, the mathematicians that came after him, David Hilbert in particular of Hilbert's hotel fame, he wrote this incredible defense of Cantor's genius. He said no one shall expel from the paradise that Cantor has created. Isn't that beautiful idea?

[B]

Wow. He brought us there. And now we're there.

[A]

And now we're there. And you will be there, too, in our next episode when we take you larger and larger and larger still. Finally, parents, you will be able to win the challenge against your children of who can name the biggest number.

[B]

Finally. Yeah.

[A]

So we'll be back next week with that episode and later this week with our usual episodes of Feel Free Notes. You can catch us then.

[B]

Yeah. And if you'd like to ask us a question, we might answer it in our Thursday episode. Send that to the Rest is science@goal hanger.com until next time. Bye.