The Rest Is Science - Two Infinities... And Beyond (Infinity Part 2)

Episode Date: March 30, 2026

Why were the ancient Greeks absolutely terrified of the infinite? How did a boundless mathematical concept start bitter historical feuds? And what happens to reality when you realise that some infinities are actually bigger than others? Professor Hannah Fry and Michael Stevens (VSauce) plunge back into the mind-bending history of infinity, tracking the spectacular panic it caused across the centuries. From individuals trying to mathematically contain it, to others wrestling with its endless quantities, they explore how the greatest thinkers clashed over the universe's most impossible idea. The chaos truly peaks with Georg Cantor, the man who completely broke maths by proving that infinity comes in different sizes. Building on part one, where Hannah and Michael desperately tried to figure out if this strange beast is an actual number or just a brilliant hallucination, this second episode looks at the human cost of counting forever. ------------------- For more information about Cancer Research UK, their research, breakthroughs and how you can support them, visit ⁠⁠https://cancerresearchuk.org/restisscience⁠⁠ Cancer Research UK is a registered charity in England and Wales (1089464), Scotland (SC041666), the Isle of Man (1103) and Jersey (247). A company limited by guarantee. Registered company in England and Wales (4325234) and the Isle of Man (5713F). Registered address: 2 Redman Place, London, E20 1JQ. ------------------- Find The Rest Is Science all over the internet by ⁠⁠clicking here.⁠⁠ ------------------- Video Producer: Adam Thornton + Oli Oakley Video & Social: Bex Tyrrell Assistant Producer: Imee Marriott Producer: Simona Rata Senior Producer: Lauren Armstrong-Carter Head Of Digital: Samuel Oakley Exec Producer: Neil Fearn Learn more about your ad choices. Visit podcastchoices.com/adchoices

Welcome to the rest of science. I'm Hannah Frye. And I'm Michael Stevens. 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? And what if it doesn't? 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, the universe is finite. Well, then it has to be bounded by something. 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. I know. If there's a boundary, what bounds the boundary? What bounds the boundary? Right? 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. 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 where there is all a complete figment of Michael's squishy wet computer in his head. This episode is brought to you by Cancer Research UK. If you wanted to type out the entire human genome, you would have to type at 60 words a minute for eight hours a day for about 50 years.

Okay, that's the scale of the DNA rulebook inside each one of your cells, telling it when to grow, when to divide, and when to stop. And different tissues read that same rule book in different ways. So a skin cell doesn't behave like a lung cell. And cancer can begin when those instructions change. one dramatic moment, but through small, gradual edits over time. Now, cancer isn't one disease. It is more than 200 types shaped by where those changes to the rulebook happen and how cells respond. Cancer Research UK is the world's largest charitable funder of cancer research, backing studies

across all types of cancer. Work that takes years of very careful, steady progress to deliver each breakthrough. For more information about Cancer Research UK, their research, breakthroughs, and how you can support them, visit cancer research, UK.org forward slash the rest is science. Square knows that in hospitality, efficiency is everything. That's why their system lets you take payments. Track sales, handle inventory, manage staff, send invoices, and keep up with finances all in one place.

Fly through orders with zero mistakes. Get the data you need and keep everything working together. So you're ready for whatever's next. Learn more about their customizable plans That's squareup.com This episode is brought to you by Nespresso. Hear that, that's your next obsession. Every coffee, a new world.

Every sip, a new taste. This is the new Nespresso. One touch, endless possibilities. Iced, flavored, long, short, because some days call for that espresso kick and sometimes a smooth silky latte just wins. It's exceptional but effortless. Like actually effortless.

Simply press, brew, and explore. What else? Keep exploring at nespresso.com. 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 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 around 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, 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. 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. Yeah, I think so too. But then also things like Xeno's paradox where you're cutting something in half and half and half and half and half and half and half and half and half and half and half and half and half and half and half and half and half and half and half and half and half and half and half and Aristotle's 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 infinity doesn't actually exist for 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? It's a 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 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 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 this is like separation of maths and godlike quantities. 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? Well, I think the first answer would be that God can do anything, Michael. I think God can have as many marbles as he damn well pleases, frankly. I agree. I'm not questioning how many marbles God's allowed to have. 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 this 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 Orsmey who was looking at the heart. harmonic series. Now, we actually mentioned this, although we didn't call it this in the last episodes, 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? Okay, so here's the puzzle. An ant is crawling along a tot 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 one

centimeter per second. So one centimeter of rubber crosses his body every second. But then we start stretching the rope and we stretch it much faster at a rate of one kilometer per second. So after one second, the rope is now two kilometers long. Two seconds later, it's three kilometers long and so on. The question is, will the ant ever reach the end of the rope? 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 Orsmey was looking at was a version of this same problem. It's called a harmonic sequence and it's where you add fractions of 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 one centimeter per second is how much rubber the ant is covering right and the band is stretching

one kilometer a second so these numbers the fractions you're going to be dealing with are way smaller from the off but you're adding these progressively smaller fractions smaller fractions together so it's also a harmonic sequence

and what austmey realized is that okay if you think of this, I'll do the analogy and 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, that it has traveled. It only ever goes up. That insight really helped me with this problem.

Because you can take 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. Exactly. Exactly. That fraction, as it were, of the band is unchanging. And actually, that's kind of similar to what Orsme 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 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 we. With Zeno's paradox, with Zeno's paradox, you get, you approach the total, you approach one and you, you know, 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 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 like literal terms of length that might seem large, the remaining amount of rubber for the ant to cross eventually hits zero. 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. But as soon as Orsmey proved that harmonic sequences, that these series will diverge, will carry on getting a larger and 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, the getting smaller, will itself approach infinity? It's sort of like it just doesn't fit that comfortably with the Aristotle idea. This was in the 1300s. Yeah, this is in the 1300. 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 the 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 eye.

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, right, let's say God is infinite, great, infinitely good, infinitely powerful, fine. Why would he build this tiny, crampled 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, why not allow it? Why not? That's, I mean, there 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 centre of the solar system. And I mean, I agree, what is wrong with it?

The Roman Inquisition, they disagreed. They thought it was quite a lot wrong with it. They thought this wasn't a flashery 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 somebody just completely breaks the uniqueness of the Christian salvation story. Right. And even if God did that, which he could, why didn't he tell us about it in the Bible? Should have mentioned it, you know? Come on, Matthew, Mark, Luke and John. What were you doing, guys?

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 Gleaps. And that's their miracle over on Vargas 12. That's their miracle. You know, well, right, or maybe they didn't have wine at all on Vegas 12. And then that was the real miracle. That was the real miracle. He brought the wine.

It's just 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 to 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 they made him spend eight years in the dungeons of the Roman Inquisition, arguing his case, like refusing to recount his belief in the kind of plurality of different worlds. In 1600, the church declared him a heretic.

and they tied into 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. Wow. They really didn't like the idea. They really didn't like the idea, did they?

And the idea they didn't like was that the universe was infinite. Right, that the universe was that, that infinity was a physical characteristic rather than something that belonged to God alone. Wow. Hmm. I mean, isn't that dramatic?

Just the very idea of it, I find absolutely extraordinary. 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, eh, 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. Yeah, you're all right.

Hey, look, Michael, maybe there's two nails waiting for us in the afterlife. That's what, maybe there are. And one and two, thank you. We got to 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.

I can see the YouTube title already, Michael. Yes. Maths so dangerous. They nailed it. their tongues. 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. Amazing.

Amazing. The thing is, is that, you know, nailing someone's tongue, burning up at the state, it doesn't make this idea go away, you know,

1638, so 40 years later, almost. Galileo, Galilee, he's thinking about numbers and he discovered, something really unsettling. He's thinking about square numbers, right? So, you know, one, four, nine, 16. And then he's, he's sort of counting them. And he's like, okay, well, you know, one,

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. And there's an infinite number of whole numbers. So you don't, you don't run out of either. You can just keep going. So, so, well, then this is a very uncomfortable idea, because this, this concept of infinity, as we were describing in the last episode at the Hilbert's Hotel, it doesn't follow the normal rules, right? It's not doing the things that, the 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. I want to, I want to meditate on the freaking, 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. 12 is not square. So clearly there are, you know, more numbers than there are square numbers. But then Galileo is like, you can't prove that. Guys, I think there's the same number of them. How can they? How can they? 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. Yeah. I mean, no wonder so many people who have struggled with this because it doesn't make any sense. This is essentially how things stood 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, Gail Cantor, which we'll come to after the break. This episode is brought to you by Cancer Research UK. We often think of beating cancer as treatment, but imagine stopping it before it begins. After years of work, Cancer Research UK scientists are launching a clinical trial of lung vacs, the first vaccine designed to prevent lung cancer.

It builds on TracerX, the world's largest cancer evolution study, which tracked lung cancer cells over many years to uncover the disease's earliest warning signs. Lung Vax is designed to train the immune system to spot these signs early on, destroying 40 cells before cancer develops. So it's not treatment, but preventative, with the potential to stop lung cancer before it starts. The first stage of the trial starts this year, focusing on people at higher risk. It shows what long-term research makes possible. For more information about Cancer Research UK, their research breakthroughs and how you can support them,

visit cancerresearchukuk.org forward slash the rest is science. Amazon presents Jeff versus Taco Truck Salsa, whether it's Verde, Roja or the orange one. For Jeff, trying any salsa is like playing Russian roulette with a flame thrower. Luckily, Jeff saved with Amazon and stocked up on antacids, ginger tea, and milk. Habaniero? More like Habinier, yes. Save the everyday with Amazon. This episode is brought to you by Defender. With its 626 horsepower twin-turbo V8 engine, the Defender Octa is taking on the Dakar rally. the ultimate off-road challenge. Learn more at landrover.ca.

And we're back. We're back. So as Hannah said before the break, for a long time, people tried to brush infinity out of the way. But in the second half of this episode, we're going to 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. 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 turning to look at it absolutely directly until the great hero of our story, Michael. 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, blah. And you add C plus C. I didn't know why, but you have to do that. I'm 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 we got into this little micro debate where you say it's not. And I'm like, it is. It does numbery things. It's an amount. And then I watched my episode on infinity from years ago. And at one 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. Okay. Okay. 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 to 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. that 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, 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 zero is included and some don't. I do include it. So zero, one, two, three, four, five, five and a half. No, not a natural number. Six, 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 one and three is one-third. 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 natural numbers,

one, two, three, four, five, six, seven, eight, nine, ten, 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 rationality. number. This is a very fun activity to do at home. Write down every possible fraction. And you can do

this just on a square grid because a fraction only has two numbers in it in X and a Y. I can write one, two, three, four, five, six, seven, eight, nine, ten, and I can then number down one, two, three, four, six, seven, eight, nine, ten. And I can combine them all. I can get every combination, one over one, one over two, one over two, one over three. They're all going to be there. And this is This is infinite, right? This thing, it goes off to the right forever and it goes down forever. So you've got two dimensions, both 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 first fraction, which might be one over one. 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 keep going forever,

just go down to the left to the fraction below 1 over 1. Diagonally. 2 over 1. 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 number one, move over, you're number 2,

go diagonally down, you're number 3, 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. It's just like Galileo squares, but way, but more complicated. But more complicated. So first of all, 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. The thing is there's an infinite number of fractions between nought and one. That's right. I'm talking about all fractions, but you don't have to. Just between zero and one.

And so how could it possibly be that there's the same number of numbers between nore and one and between nore and everything? How is that possible? I know. I mean, sure. Like, why am I talking about all fractions? We could just do one over one and then everything that's smaller, one over two, one over three, one over four. 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 zero and one as there are whole numbers. Give me a break. This is, give me a break, exactly. You can say to something like that, oh, 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 how the sausage is made. And we liked it or maybe we didn't. 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 pie, the square root of. two. These are numbers that you cannot represent as a ratio between two numbers. You can get pretty close with pi, like 22-7s is pretty close, but it's not quite there. The problem of rational

numbers have is that their decimal expansion never ends. So there is no finite number that we can say, oh yeah, just take that and divide it by, you know, some other number. And there it is. When you start including all naturals, all rationales, and then everything else, now you're describing the real numbers. And this is a really big category. How many real numbers are there? Your instinct would be, well, we're just, gosh, I mean, if we've got the same number of fractions as whole numbers, I mean, sure 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, squidge 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 going to tell me there's the same number. Yeah, exactly. 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 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 try to tell it to my daughter at night as she goes to bed. It's like counting sheep grow up. We're counting wheels.

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 talked about this before as well. So I'll try. It's so good. I mean, this is like if there was an award for best WTF. moment in the history of math, I think this one wins. I think this wins. It wins because it is so mind-blowing for so little work. It's not like, okay, well, you need to understand peatic numbers and the monster group. No, it's just like numerals. We're going to try and explain it to you. We're

going to try. We're going to try and explain it to you. So 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 zero and one. That's it. Okay, so there's clearly more of them,

but we can just look at between zero and one. That means numbers like 0.1, but it also means numbers like 0.342-9774-4-3 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 zero and one. And you might think, okay, well, obviously I can keep doing that forever, but I can keep labeling them one, two, three, four, 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 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 make this much more systematic. Doesn't matter.

Just as long as you change it in some way. Just change it in some way. Just mark it up a bit. Just mark it up a bit. 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 three.

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 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. That's right. Because your number that you've just written down is basically a diagonal line, right?

You've constructed it by forming this diagonal line down your table, which means that, it has to be different from the first of your numbers 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, 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. 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, one, two, three, four, five, six, seven, eight, nine, ten and so on forever, I can keep adding new things to it. So the number of real numbers,

even just between zero and one, is larger. Literally, there are more of them than there are whole numbers on the entire number line. Because the key thing here is that with Galileo's square numbers, the reason why you know that there are

the same number of both is 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. Likewise with your factions, 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 geniusly 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 then you're not going to miss any out. 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 holes in 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. 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. 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. 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 the reels 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 alifnol. And alif nol 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. Yeah, of course. 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. Third, fourth, fifth, five hundred and seventh place in the race is what I achieved. Those are all ordinal 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 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 alif 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. 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. 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 alifnol 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 two.

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 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 decided that much like infinity itself, we might have to do infinity plus one episodes on this because there's just too 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, alifnol, and a bigger one, the cardinality of the continuum. And yet, we haven't even begun. We haven't even begun. And we've got these ordnals now that are going to

help us climb and climb and climb until we reach. Well, stay tuned and you'll find out. Well, it's two infinities and beyond. I think that's, uh, 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. It's, it's, it's nerd, it 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 Starz's, I mean, people did not like it. Like really, really did not like it. He had this arch nemesis, Leopold Kronica, who's responsible for the Kronica Delta symbol, if you've ever come across that. And he was this...

Every day, don't remind me. Non-stop. He was this old school mathematician. He was like, no, he's back to the Aristotle thing. It's like, no, only find out how all 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 mad, just really dismissing Cantor's word. 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 corruptor of youth, right? Which I love the idea.

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. That really fluses 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 the infinity 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. Oh, really? Where in Germany is it? It's called the Halle Mental Asylum, the Nervin 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 size. 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 was 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. Well, 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... Are we sure he wasn't the King of Spain? Because I believe it after learning what he discovered. I think we at least needed to do a controlled trial, you know? It's like, that's the very least that we could have given him. Yeah.

Unfortunately, the conclusions were the opposite, where 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 really malnourished. who 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 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 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 us from the paradise that Cantor has created. That beautiful idea. Wow, he brought us there and now we're there. 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. Finally, yeah. So we'll be back next week with that episode and later this week with our usual episodes of field notes. You can catch us then. 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 at goalhanger.com. Until next time. Bye.