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

14:31Michael Stevens: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. And that's their miracle.

14:47Hannah Fry:That's their miracle. You know, well, right. Or maybe they didn't have wine at all on Vargas 12. And then that was the real miracle. That was the real miracle. he brought the wine. This is 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 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.

15:27Hannah Fry:In 1600, 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.

15:44Michael Stevens:They really didn't like the idea, did they? And the idea they didn't like was that the universe was infinite.

15:49Hannah Fry:Right. That the universe, that infinity was a physical characteristic rather than something that belonged to God alone. Wow. I mean, isn't that dramatic?

16:04Michael Stevens: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, 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. Yeah, you're right.

16:33Hannah Fry:Hey, look, Michael, maybe there's two nails waiting for us in the afterlife.

16:37Michael Stevens:That's what, maybe there are. And one and two. Thank you. We've 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.

16:52Hannah Fry:I can see the YouTube title already, Michael. Yes.

16:57Michael Stevens:Math's so dangerous, they nailed 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

17:15Hannah Fry:The thing is that, you know, nailing someone's tongue, burning up the stake 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, OK, 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?

17:56Hannah Fry:There's an infinite number of square numbers.

17:59Michael Stevens:And there's an infinite number of whole numbers. So you don't run out of either. You can just keep going. You don't run out of either.

18:07Hannah Fry: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.

18:25Michael Stevens:I want to meditate on the freakiness of that. So it was at 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. How can there be more than and the same number at the same time?

19:02Hannah Fry:Like how? It doesn't make any sense. It doesn't make any sense. Yeah. 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 most of that millennia, right? That a few people here and there had come up with these little glimpses of insight, 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.

19:51Michael Stevens:This episode is brought to you by Cancer Research UK. Cancer drugs aren't developed overnight. They start as ideas in the lab, then move into testing to check they're safe and work effectively.

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21:51Michael Stevens: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 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 so there are these little ideas these little

22:13Hannah Fry: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.

22:37Michael Stevens: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. Cause you have to go all the way back to talking about what is a number.

23:11Michael Stevens: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.

23:41Hannah Fry:Okay.

23:42Michael Stevens: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.

24:10Hannah Fry:That literally goes beyond.

24:12Michael Stevens: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. One, two, three, four, five, six, seven, eight, nine, ten. Negative five 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.

24:58Michael Stevens: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, 10, and so on.

25:44Michael Stevens: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 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, 4, 5, 6, 7, 8, 9, 10.

26:28Michael Stevens: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 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 one over two.

27:08Michael Stevens: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 one over one, which is 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.

27:41Michael Stevens:So that means there are the same number of fractions, rational numbers, as there are natural numbers. It's just like Galileo's squares, but, but wait, 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 0 and 1. That's right. I'm talking about all fractions, but you don't have to.

28:21Michael Stevens:Just between 0 and 1. And so how could it possibly be that there's the same number of

28:26Hannah Fry:numbers between 0 and 1 and between 0 and everything? How is that possible?

28:31Michael Stevens: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. Give me a break. Exactly. And 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.

29:10Michael Stevens: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 two. These are numbers that you cannot represent as a ratio between two numbers. You can get pretty close with pi, like 22 sevenths 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.

29:49Michael Stevens: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?

30:08Hannah Fry: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, 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 going to tell me there's the same number. That's what you instinct would say.

30:33Michael Stevens: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.

31:07Michael Stevens: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 talked about this before as well. So I'll try.

31:20Hannah Fry: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.

31:33Michael Stevens:It wins because it is so mind blowing for so little work. It's not like, okay, well, you need to understand piatic numbers and the monster group. No, it's just like numerals.

31:42Hannah Fry:We're going to try and explain it. We're going to try. We're going to try and explain it to you.

31:46Michael Stevens: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, there's clearly more of them, but we can, we can just look at between zero and one. 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.

32:35Michael Stevens: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.

33:13Michael Stevens: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 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 knock it out of there. Just knock it out of there. And so you write down that new different number.

33:46Michael Stevens: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 gonna write down six. I'm just gonna 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.

34:23Michael Stevens: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.

34:36Hannah Fry: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 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.

35:18Hannah Fry: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.

35:30Michael Stevens: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.

36:01Hannah Fry: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 can pair 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.

36:33Hannah Fry: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 that you're not going to miss any out. So there are more than there are natural numbers.

37:12Michael Stevens: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. 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.

37:57Hannah Fry: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.

38:17Michael Stevens:Yes, it is. But that doesn't mean there's nothing bigger because there is. That's right. And 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 addition in the 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.

39:03Michael Stevens: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. 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.

39:46Michael Stevens: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, 507th 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.

40:29Michael Stevens:The number of people who ran the race is still Aleph Knoll. 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.

40:45Hannah Fry: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.

40:55Michael Stevens: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.

41:42Michael Stevens: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.

42:13Hannah Fry:The thing is, there is actually further that 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.

42:31Michael Stevens: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-knoll, and a bigger one, the cardinality of the continuum. And yet, we haven't even begun.

42:53Hannah Fry:We haven't even begun.

42:55Michael Stevens:And we've 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. Well, it's two infinities and beyond. 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.

43:15Hannah Fry:It's, it's, 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 Georg 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.

43:55Hannah Fry:Really? So for Starsis, I mean, people did not like it, like really, really did not like it. He had this arch nemesis, Leopold Kroniker, who's responsible for the Kroniker delta symbol, if you've ever come across that.

44:07Michael Stevens:Every day, don't remind me.

44:09Hannah Fry: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. charlatan. He just didn't believe his proof. He said that he was a renegade. He said he was a corrupter 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 diagonalisation proof and suddenly becoming, getting up to all manner of things.

44:47Hannah Fry: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 the infinity that he was trying to grapple with well it certainly didn't cause them but i 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.

45:29Hannah Fry: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?

45:37Michael Stevens:Where in Germany is it?

45:38Hannah Fry:It's called the Halle Mental Asylum, the Nervan Clinic. It's still active, by the way. It's still a hospital, a 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 is 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.

46:24Hannah Fry:Right. He he also believed he'd uncovered this, this really great conspiracy that Francis Bacon was 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.

46:49Michael Stevens: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 pool? Are we sure he wasn't the king of Spain? Because I believe it after learning what he discovered.

47:05Hannah Fry:I think we at least needed to do a controlled trial, you know? That's the very least that we could have given him. 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. 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 show