Finding the Biggest Number: A Journey Through the Infinite | Warp Talks 001

The what? No. Well, I guess after this talk they'll become like a special edition one, you know? But as of right now, they're just normal cards and normal cubes. Okay, we ready to get started? How are we feeling? All right, so this first talk, as Gabby introduced, the idea here is that we want to do these monthly. We'll select and take submissions and volunteers. We'll also potentially get to invite some external folks. But for that to, you know, for us to build up to that, we likely want to start just in-house ourselves and talk about some cool topics, right? So the one thing that I've been really fascinated with ever since my very early foray into this subject is finding big and big numbers. And so you might think, okay, like, what does that mean? Can you just not come up with a number and just like add 1 to it, right? So that might be one idea. But what I wanna do over the next hopefully 30, let's say 30, 35 minutes is, and I'll try to keep things condensed. I'll make it in a way that it's very accessible so we can all kind of participate and understand these ideas, right? But the goal here is to kind of convey this very central idea that in order to find these really, really big numbers, we actually have to come up with really insane and somewhat crazy ideas in math. And it will take us through this entire journey of almost a century, century and a half worth of mathematics, some of the most central questions in there, some of which are still unanswered. So we'll point those out as we go along. But to start with, the reason you all have a pen and a card in your hand is because we're going to start with a little competition. And here's how it's going to go. And the remote folks, feel free to participate. We'll take your word that you're not lying to us. You know, you'll have— maybe you'll have to show your card. But here's the goal. The challenge is we will start a 15-second timer, and in those 15 seconds, whoever writes down a finite, concretely defined— so it cannot be some crazy— oh, you can't say infinite, you can't say something that cannot be understood by most people. So finite, concretely defined. Largest number, and you get 15 seconds to do it. So I'm going to start the timer. You have to write down a number. It has to be concrete. It has to be definable. Just can't say anymore. 15 seconds. The biggest number wins $100. Ready, set, go! All right, we're done. That's it. Stop. That's it. Stop writing. No cheating. Stop writing. Everyone stop writing. Okay. Okay. All right. All right, um, who thinks they have a big number? All right, what's your big number? I got 100 quintillion. Ah, can we see it on the card? Yeah, it's not that many. It's pretty small on the grid. So that's 1 with how many zeros? That's how many zeros? Uh, 1, 2, 3, 4, 5, 6 sections of 3. 18 zeros. 18, okay. And you actually wrote 18 zeros? I did, yeah. Okay, so Dean's big number is actually quite big. It's 10 to the power of 18, right? 1 quintillion, 100 quintillion. These numbers get a little bit crazy. So that's a pretty big number. Who thinks they have a bigger number than that? I got 35 zeros on here. You wrote quite fast. So this is Dean. Then we've got Kendall with 35. All right, so that's the current winner right now. Who has a bigger number than that? I wrote it as 10 to the power of 10 to the power of 10 to the power of 10. So you created the tower. Okay. So I'm going to just kind of place it over here. How many? 10. 10. Okay, so we— this is a, this is a really, really large number that we'll talk about in a second. But who has a, who has a big number? I think mine's bigger than that. All right. I have 9 to the 9 the same way. How many times? 18. 18. Okay, that, that's gotta be bigger. So yours was 10 times? Okay. 9 to the 9 to the 9. Now what's interesting about doing 9s is when people do 9, there's a funny thing. You could just do 10 and like powers of 10 and you could just write more, right? That's extra writing. Well, actually with 10, yeah. I think if you could do 2, maybe 2 or something. Yeah, that's actually good. That's really based. Okay. Who has a— who thinks they have a bigger number? Anyone has a bigger number? Google. That's not bigger than any of these. No. So Google is— what is a Google? Define. Tell us. I don't know what it is. You wrote like a number. You have to write a concrete number. A Google is just 10 to the power of 100. Google is 10 to the power of 100. That's it. These numbers are way, way, way bigger. Okay. Anyone else have a A number that is bigger than— Yes, that beat us. A power tower of 9. Now we need to see if it's Adam or the guest. Okay. We're calling it. Adam's the winner. Adam wins this one. All right. This is the winner. Not surprised. Okay. So, so $100— SB, $100 to Adam. He wins the prize. Okay. He's like, that's his job. So— That's what I do all day. Write numbers like that. That's what you do behind the scenes. Um, so some, some observations. All right, guys. Okay. Thank you for participating. That was really fun. So you start to see some patterns, right? You start to see some observations that people try to reach for 10 to the powers. These are really, really big. You can just, of course, increase the number. But then the first kind of like interesting number is googol here, which Wyatt presented. Googol is 10 to the 100 because this dwarfs everything here, right? This is obviously bigger than that. But then the first big leap sort of comes by this idea that you can do not just 10 to some number, but you can do 10 to some number to some number to some number. And this is evaluated from the top down, not from the bottom up. So what that means is if you were to do— I lost it. If you were to do 3 to the 3, this is what? 27. 27. But if you were to do 3 to the 3 to the 3, that's going to be 3 to the 27, and that's roughly 7.6 trillion. Because it's going to be 3 to the 3 to the 3, 3 times 3 times 3 times 3, 27 times. That's going to be 7.6 trillion. So if you were to write like 100 to like 1 million, would that be like the biggest sum? It would be really big, but here's the key thing. This style of creating a big number is a new idea, right? This is the way— this is a way to reach really, really big numbers really fast because The next level up from here would be 3 to the 3 to the 3 to the 3, and that just means 3 to— 7.6. This thing, 7.6 trillion times. And that's a really, really big number. And we have only used what, 4 digits here so far. So this is called a power tower, and it's a way to create really big numbers. But before we go into these really big numbers, we'll start with something very concrete. And so where I want to start is there's this famous fable from ancient times, and the fable is between a king and a clever sage. And the sage goes to the king and says, I want just a grain of rice from you, from the king. And so, but he says, I'm going to do it on a chessboard where on the first square, first checkerboard, I'm going to have 1 grain of rice. Then on the second, double, so 2 times that. On the third, 2 times that, 8. And the king is like, well, you're starting with 1 and you're just doubling, how big could that be? So the king agrees. And turns out the king is really going to be down on rice because by the end of this thing, there's 64 on this. So you can kind of imagine you reach 2 to the 64 if you're just doubling Times 2 times 2 times 2. This is a number that is roughly 10 to the 19. And one cool thing about this number is that this is kind of how many grains— this is actually bigger than how many grains of sand are on entire Earth. So by just doing this little chessboard exercise, you've created a number that is bigger than the grains of sand on entire Earth. Okay. So that's a big number, but we're going to go past that. 10 to the 19th from 2 to the 64th. It's like an approximate type of thing. So, so it's like more, more precisely, it's actually 1.8 times. It's more like 1.8 times this. So it's a little bit like an approximation. Okay. Where we're going next is, I think all of you have— some of you have Rubik's Cubes, right? Great. Who has like a 3 by 3? Who has one of these like this? He does. We do. Okay, great. So I want you to take whoever has the 3 one, I want you all to just shuffle it. Shuffle it really nicely, give it a good shuffle and go from there. All right. We feel like we shuffled it? Make it kinda random, like you sort of have to break out the patterns. Give it a good shuffle. Is it shuffled, Tyler? I hope so. Okay. How are we feeling about the cube? So, so what's interesting about, um, if you have scrambled a 3 by 3 Rubik's Cube, In a sense, you've created a number that is about this big. It's 4.3 times 10 to the 19th. So this is roughly how many ways are there to scramble a 3 by 3 Rubik's Cube. So if you recognize the 10 to the 19th is sort of similar to that doubling thing that we were doing. This one, because it's 4.3 versus 1.6, actually bigger. So the grain exercise we did, we already kind of exceeded that by scrambling a cube. And why is this such a big number? It's because each corner, you can imagine it has all these different colors it could have. Then it has 12, like, cubes and corners it could be in, and you can multiply all those things and divide by real moves you can actually do. This is roughly the number. Trust me on that. You can Google on it as well. Source, trust me. Now interestingly, Adam here has a 4 by 4 Rubik's Cube. And I would like you to give that a shuffle while I write down what that actually means. 4 by 4 temporary. It's too big. Okay. So just by increasing one, additional cube, so going from 3 by 3 to 4 by 4, we arrive at a number that is many, many, many, many orders of magnitude bigger than the last scramble. So this number is in this range. And one interesting thing that I will note here is that I mentioned how there's about 10 to the 18 grains of sand on Earth. The number of seconds that have passed since the Big Bang is actually just on the order of 10 to the 17. So this is how much time in seconds has passed since the beginning of the universe. Some of you know Big Bang's considered the beginning of the universe. We don't quite know what happened before that. We don't quite know if time even is defined before that. But this is how long before what is widely agreed to be the beginning of the universe. in seconds. So what, at this point, I just wanna pause for a second and appreciate this fact that very quickly with just like household objects, we have defined the concepts of numbers that go far beyond the observable scale of the universe itself. Would you say this has never happened before? Never happened. This, what you have created is a scramble, if it's a true scramble, you haven't like sort of done it in a— you've done it in an actual randomized way, the odds of that ever happening on Earth by all these players who have played would be very vanishingly zero, effectively zero. And yes, and in fact, this point— Brainiac moment. So this point is made much more concrete that we have created a very unique thing by the deck of cards that are in front of you. So for people who are— who think they can shuffle cards or decently at it, why don't we pick up the cards in front of you, and I'd like you to give it a good shuffle, like a shuffle that you're like, this is a good shuffle, there's no way that anyone could play tricks with this, you know what I mean? Sarah knows. Are you good at shuffling? You got a good shuffle? What have you done, Adam? I'm giving it to Vegas Shuffle. No one's counting these cards. Okay. Oh, cute kid. All right, everyone's got a shuffle. All right, so. Okay, may I? All right, so we've got a good shuffle here. Curtsy Maddie. And I'm going to just pick out the first card here. It's— this is what we've got. And the second card. All right, so anyone want to just look at their first 2 cards just to see if there are any similarities? So completely different, completely different, completely different, completely different. So just in the first 2 cards, we kind of appreciate the fact these are different shuffles, right? Now what I, uh, what I want to sort of get into next is what is this number, right? So if we were sampling— if we were scrambling a Rubik's Cube and sampling a possibility space of this, one way to think about this, by the way, is if you shuffle a dice, in a way you're sampling out of a 6-sided thing, right? So the possibility is 6, you pick 1 out of that. So if you think about scrambling a cube, It's kind of like you're picking out one thing out— it's a 4.3 times 10 to the 19-sided die that you just shuffled, and you picked one thing out of it. Shuffling a card deck is actually even crazier because that number— anyone know what that number is, by the way? 52 factorial. Nice. So this number is 52 factorial. So, so what what is fifty-two factorial? It is simply fifty-two times fifty-one times fifty da da da da da until one. That's that's this notation factorial. It just means from there up until one minus like less than less than less than all the way to the one or other way. 1 times 2 times 3 up until that number. This number turns out is about 8 times— 8.1 times 10 to the 67. So this number is already way bigger than anything we have seen so far in this set. Uh, we, we will talk— we'll— we're not touching these yet. Some spoilers have been previewed here. We're not testing them yet, but in the general order of things, this is way bigger. So what that means is anytime you shuffle properly a deck of cards anywhere, anyone on Earth who has done that has created an entirely unique set out of a possibility that is 8.1 times 10 to the 67th white space. So no 2 shuffled decks of cards ever would have lined up. The possibility of that happening— think of it this way. If you had all of population of Earth shuffling cards, proper shuffles, right? That's the big assumption. It's a completely random shuffle. If you took all the population of the Earth shuffling cards, you would run out of time if you were doing one shuffle since the beginning of the universe. You would— it would not be sufficient if you took all of humans and you started them at the beginning of the universe and had them shuffle one at a time. You try to multiply these numbers, you still won't come close to this. Question. Yeah. Would you say the highest concentration of unique moments happens in a casino? Whoa. Well, if— That's a really good point. Like in a physical space sense, probably, because you're constantly shuffling these things and creating all these games. So that's a good point. Yeah. Okay. So we've seen some— so we've seen some big numbers, right? And these are our physical reachable numbers, right? We can kind of reach them in some physical sense. But they still far outweigh anything that we could actually fit in this universe or the scale of the universe. For example, the entire number of particles in the observable universe is only 10 to the 80. So all the— at the atomic scale, The observable universe just means— so the entirety of the universe that we can actually observe, because beyond a certain point, and this is a vast amount, right? Beyond that point, the universe is expanding so fast that the light will never reach us. So we call it the— if you've ever seen, it's like a circular spherical area that is the observable universe. It contains about 10 to the 80 atomic particles. And we have started approaching it by just shuffling a deck of cards. Kind of crazy. But what happens next is even crazier. So what if you think these are big numbers? Journey through the infinite, Nicole. I'm going to have an existential crisis. If you think these are really big numbers, and they really are, We're about to make them look like 0 next to the numbers we're about to reach. So the first hint comes from this thing. This was the first leap in the paradigm that we saw in this room already that someone came up with. And you just did that thing, but a little bit faster and better, right? So you won, but this thing is really interesting. This is called a power tower. So it just means you start with a number and you take the power and you take the power and you take the power, right? And so this thing becomes really crazy really fast because you have to evaluate this thing from top down. So this becomes 2 to the 2 to the 2 to the 2, which is like, okay, this, if you collapse this, it's like 4, and 2 to the 4, that's like 2, 20, 64 or 32. So that's 32, and 2 to the 32 is a really big number. And if you do this with 3, I already kind of showed you that that blows up really, really fast. And the interesting thing with this is you can kind of go even beyond this, right? So here's how to do it. So the Power Tower is better than the factorial? Power Tower is incredibly faster than the factorial. Yes, way faster. When you say faster, what do you mean by that? To get big. Okay. That— no, that's actually a really good question, um, because the way you have to think about it is back to my initial exercise, which is if you had a 15-second timer How much— what is the concrete number you could define quickly that would be really, really big? And so what I mean by that is in finite steps, how quickly could you blow past the previous classes of numbers? And so it's a good way to think about the next class of numbers, which is if you were to reach this class, you sort of like Take a big leap from the last class. And so everything we've been doing, kind of, we're building up these sort of classes of numbers where the only way— there's not good ways to reach the next class from the below. You can only kind of define a new idea and then reach it. Because actually what happens is if you try to reach these numbers from below, you will run out of space in the physical universe if you were to write like a tiny 0. And you're trying to like, OK, I'm just going to do this thing and write lots of zeros. I'll actually write the entire observable universe. I will fill it with atomic-sized zeros. And you would just not reach anything above. You still wouldn't— You wouldn't reach a power tower that is going into like a— it's hard to tell because comparing is very hard. You have to evaluate. Maybe like a 10, right? Like a 10, it would probably exceed that. Can I ask a question to follow up on that? Yeah. What is the point of creating numbers this big? Very good question. Because mathematicians have a lot of free time and they can do all this. No, no, here's the thing. This line of inquiry, so where we're kind of building in hopefully the next 10, 15 minutes is each of these things, they tie into really, really crazy powerful ideas that are very central in mathematics. So what's happening is in order to define the next big numbers, we kind of have to like come up with a really cool new idea, which is unprecedented so far. So we're inventing a new type of math, new concept, new theorems, and we're building up. And then those things often are being used in a lot of other places. So in some ways it is a theoretical fun exercise, but the only way to do it is it kind of spans— it took us a lot of time. to try and understand the scale of these things. Fun fact, like if you go back to kind of the early Greeks, there were some people who thought the grains of sand on Earth must be a number that's like infinite. So they didn't have a good sense of defining big numbers. So they kind of took it as, oh, it's just like something that we can't ever count. But the people who tried counting it, in fact, some of those people were very like— they were considered heretics at the time. Like, oh my God, you're trying to like define the number. It's like a crazy fool's errand. But those people who tried that pursuit of, okay, what if you actually try to define it? They invented new math and they kind of became the foundation of a lot of these things to come later, right? So we'll see that pattern throughout. Next thing I want to talk about is, so there's this book that sits in my office, a set of books that some CS folks might recognize. It's Art of Computer Programming by Donald, very weird last name, Knuth. Knuth. Apparently that's how you say it. But Bill Gates has a famous quote, I think that's on the spine of the book as well, that if you think you're a good programmer, you should really read through Art of Computer Programming. And if you can go through all of it comfortably, then certainly send me a resume. So I think that's like a quote. Bill Gates has a famous quote, obviously a very smart person. But this is a very, very dense set of computer science textbooks. And Donald Knuth is known to be one of the greatest mathematicians, computer scientists of our times. So he came up with an idea which allows us to go one level above, or not just one level, but in the next class of numbers than this thing. And it's a similar idea, right? We sort of were like, okay, what if we do exponentiation? So someone, the first person tried this thing, which is really just multiplication many, many times, right? And then the next one was this power tower, which is not only you are doing exponentiation, you're taking that and doing it over and over in a hierarchy, in a ladder. Now the next idea here is the way to define this is called— it's called the up arrow notation. This is where things get a little bit crazy, so I'll kind of like keep this concise, but the core idea is simple. You start with 3 arrow 3, and this just means 2 to the 3. Okay. Straightforward, which is just 27. But then you define 3 up arrow up arrow 3, and this means 3 227. So you took the previous number and you substituted up here, right? Just 2 times. And then you try to do 3 up arrow up arrow 3, and this becomes 3 to the 3 to the 3. And remember, this was like about 7.6 trillion, right? So this is— 7.6 trillion is the height of this tower. And that's 3 up arrow, up arrow, up arrow, 3. Arrows take longer to write. No, but hold on. We just said in these 3, we've already reached a tower which we will never be able to write. Because we're saying the length, the height of this tower is 7.6 trillion because it's building from the previous one. Go ahead. So in the time when they were uncovering this, I can get the understanding that they can conceptualize and build the equation, but how do they actually solve for it? What do you mean, like evaluate these numbers? Okay, so we understand how to do the power of. Yeah. Do we actually solve that it's 7.6 trillion though? Uh, yes, this one you can write down. So, so this number 3 to the 27, you can plug this into Like Google, and Google spit back the value. Yeah. But okay. Yeah. Yeah. So that actually there's there's some crazy stories of there's some crazy stories of like very early mathematicians hand cranking through. really, really crazy long calculations. So there are— there is some stuff like that. Now, okay, what is the next one? Uh, hard to say, right? But you get the idea. It would be this thing, 3. The number of arrows here would be these many, right? So you would have to do 3. 3 to the 3. 3, up 3, something like that, and it's the next level. So this is where it becomes— you okay? This level is where it becomes really insane to even try and define, right? But it's just doing the same operation. It's putting the number of arrows that many times. Okay. So can you do this thing like ad nauseam? Yes, and that's how, that's how we build it. What is what the number that I'm about to present to you next? It is called Graham's number, Graham. And this number for a while held for for decades. It's it's written as G64. This number held the record in mathematics for many decades that this was the largest number, largest finite concrete number to be ever used in a real mathematical theorem. So like any— not just, okay, we're trying to like come up with large numbers, but this was actually used in a real mathematical theorem, and this number held the record. And so the way this number is defined is that g1 is this. That's the first one. g2 is g1 I have to make sure. G2 is simply 3 G1 times up arrows 3. So you see what just happened? Yes. So I defined G1 to be this number which was already crazy, and then I said G2 is 3, but the number of arrows itself is G1. And then 3. Okay. So could I just note w1 is g64? Uh, you, you could. That's a really good point. You could. But just note that once we arrive at a new class of number— so hold on, just to finish this point. You can keep doing this and you will arrive at, um, can we put on the second thing? So this is g— this is Graham's number. So you started with the, the bottom layer, which was this first number I told you about, 3, 4 up arrows, 3. And then the next step you said, I'm going to take that first number, that many arrows, and then 3. And then in the third step you say that many arrows, then 3. You do this 64 times, you arrive at Graham's number. Here's a fun fact about this number. If you try to Fit the information of this number, like how many digits in your head? Your head would collapse into a black hole. You know what that? Great, great point, Nicole. This is a okay. Series B valuation. Yes. It's the— it's, it's Ramps' next raise. You have to add up? Yeah. Okay. All right. So that's Graham's number. We will, we will come back to this question of why define these numbers towards the end as well. But I will present just one more craziness and then we will start to close with something equally crazy but slightly different from what we're doing right now. So once we arrive at Graham's number, I want you to just kind of note that any of these numbers are basically zeros. Like everything we've seen up until now is basically zero compared to Graham's number. There is no amount of like number writing or whatever operations you could do to arrive at Graham's number. You just can't. That's grants. But the next one is something called— can you go to the next one? So this one I'll try to do very quick. It's a— it's a— OK. So I have to— so here's the craziest sequence, which is 3. Tree 2 equals 3, and tree 3. So the number we're talking about is tree 3. So this is a sequence that goes like this: tree 1 is 1, tree 2 is 3, tree 3 is a number that is. So much bigger than Graham's number that we have not been able to concretely pin it down yet. Tree thing. So what is, what is this tree thing? Now, as I, as I promised you, in order to come up with the bigger number, you have to come up with a bigger idea. So Graham's number, we were able to reach Graham's number by this underlying idea that you take an operation and you apply that operation recursively over and over to it, right? And you keep rising that thing. This is a number that is a very different kind, but turns out— and it took mathematicians a while to kind of prove that this is a finite number, it's not infinite. Like, this is a thing that has an end, but it is so much bigger. It is dwarfing anything that grams or any numbers we've come across. So I'll very quickly explain this, what this is, and then we'll go to a more interesting thing because we'll be kind of done with finite big numbers. Tree 3 is just kind of this. You basically start with— it's a game, and the game is if you just have one color, what kind of a tree can you make? You just have one tree. It's like a tree is just little nodes with things. You have seeds and you kind of make this, right? Trying to make a forest. If you have 2, you could do— so that's Tree 1 is one. If you have 2, you could do This, you could do this, or you could do this. This game ends the moment you have a tree that is part of the sec— bigger tree. So as soon as you have this, that's part of that, right? Like it's, it's a copy, it's a subset of that. This game ends. So you just have tree 2 is 3. Now here's where it gets crazy. The moment you go from 2 to 3— And we're talking about distinct colors, right? Yes. Yes. So you have 3 colors in this case. And you try to play the same game. Those are some of the first few values. And you can use up to 3 colors, but those are actually just using 2 colors so far. So you can add the third number. This sequence, if you keep doing it— and the game is the game will end once you have found a tree That is part of another tree that you've already seen. So, uh, that just means like this thing is part of a bigger tree, right? This number, I mean, it's like, it's a monster. We don't really know. Uh, there's no way to pin it down, but we did prove that this is a finite number. So, um, everything that we've seen so far is a finite number. But there's a way to compute it, right? Which is all of these numbers, you can define them, and— or you can define a series of steps on how to build that number, right? I defined to you like this thing, it will run very long, but if you had an omnipotent computer and you just ran it for very long, it will stop and you'll have the number, right? Like all die out. The universe may have imploded, who knows? But it is a finite thing. And there exists a way to say, given enough time, a computer will just do it. And so all these numbers are called computable because they're— they can be computed like that. Everything we've seen so far. Now, where I want to take us, and then I promise this is Sort of where nears the end of this part is that there's something called a busy beaver function. And this is— can we go to the next one? So busy beaver, it goes like this. BB1, BB2, BB3, BB4, BB5. BB5, and I'll tell it BB6. OK, so this function goes kind of like this. This is 1, 6, 21, 107. So this is 1, this is 6, this is 21, this is 107, this is 47 million— where did it go? 47,176,870,176. 6, 8, 7, 0. And this we will leave to God to figure out. But what is this thing and why is this one the finite final step in the exercise we've been doing? This is a function that is uncomputable, which means that there is no way to create an algorithm, a set of steps, any set of steps that any computer could run to determine this number. So what that means is like I— the first few values are actually kind of easy. So 1, 6, 21, 107. This was only proven out in like 1984 or something. And then this BB5 was only proven to— it's 47 million something something. It doesn't look that big. But we know that this is a function that grows faster. This is a sequence that grows faster Than anything that is computable. So anything that we've seen, even tree, which was crazy, or tree to the tree to the tree to the tree, or tree of tree of tree, all these things are finite computable things because you can kind of define them. You can list out the procedure to build them. This is the thing that is beyond that. I won't try to get into why this is beyond that, but I'll give you a slight hint. This thing comes from a very deep idea in computer science, which is that you cannot build a godlike machine that can tell you whether a given program, whether any given program will ever stop. So if you have a program, it can go in an infinite loop, right? Like you might have seen a spinner. So sometimes, you know, you go to a loading screen, it's an infinite spinner, it's not gonna run, it's gonna crash your computer. Turns out That is a property of computers that is— you can't get rid of. If you— you cannot define, predetermine a way where given any program, can you say this program will stop or not stop? So it kind of relates to that. And the reason this function has to grow faster than that is because if you were able to find out the values of this, you would solve that problem, which is called the halting problem. And for Or if you ever saw The Imitation Game, the movie, Alan Turing. Alan Turing, who invented basically all of computers, is the father of modern computer science. He showed us that halting problem, as it's called, is impossible. It is— it cannot be done. You cannot build a machine that will tell you, given any program, will it stop or will it just run forever? So this is a function that is built using that halting problem. And it just means like given a machine like that, what is the most number— what is the busiest machine you could come up with? Like it just does the most amount of work. It prints out like the most. And these 1, 2, 3 just means kind of like how complex your program is. So you can have like 2 states, 3 states, 4 states. And the more complex the program, the more things you could do. And here's some crazy things about this number. We know that BB is already bigger than Graham's number. So 6, we can't pin it down, we can't define it, but BB is already going to be bigger than Graham's number. Remember how much work it took to Graham's? So next time you ask me, hey Rahul, you have 15 seconds to write the biggest number, I can just do BB? You could just do BB— I would recommend doing BB just to be safe. But, uh, yes, now you know. Anytime you're in the— we have reached the end of the finite numbers being big. There's just nothing that gets bigger than this. Anything you could define, any function, anything, it will always grow slower than this. This is the upper bound. This is the ultimate limit. For now? No. Well, in this document, yes, kinda. But I want to switch gears. Okay, so I'll share just one thing, which is there's a funny joke that's like BB6. We just found out, we finally defined BB5 2 years ago, and it took a distributed group of researchers, math researchers who collaborated all together. They ran all these distributed compute programs to solve all these things. A lot of these hundreds and potentially thousands of people came together online to prove out that BB5 is this. We don't think— the best researchers in this field think that we will never find BB6. We'll never probably find BB6. Yeah. I mean, seriously, genuinely, why can't you ask like an AI? Well, so the joke goes, the joke is that we'll let the superintelligence that precedes humanity worry about BB6, and then God can worry about BB7. So, but here's a fun fact, right? Like, fun fact is we, we have good reasons to believe that BB643— so just this number, just this number— is independent of all of mathematics. So what that means is all of the math that we have come up with, that we could ever come up with, will never be able to allow us to figure out what this number is. But we can do 644? No, no, not after that. Like, like, this is like the first— no, no, like, just by the time you reach this, it is what's called independent of all of, uh, there's the, the axiomatic system of math. And at that point, you just, uh, you will not be able to prove or disprove whether— what the actual value of this It will just be outside of the realm of mathematics. It's kind of crazy, right? That there are these like firewalls on the knowledge of what we can know. And it's a very deep area. So the point of like defining these bigger numbers is actually it starts to run into the limits of knowledge itself. These things have— there's a fundamental idea here that is stopping us from going beyond. There's a fun thing, which is if you could solve BB20, you would solve a 300-year-old problem in maths called the Goldbach conjecture. What is that? So Goldbach conjecture, won't define it, but it's hundreds of years old, never been solved. Most people believe it will never be solved. But if you knew the value of BB20, you could run a program that would just run for BB20 number of steps and it will tell you, is Goldbach conjecture true or not? So there are problems like that. There's something stopping. It's kind of weird. Like, I come at these things from physics point of view. What's stopping it? But there's something about the nature of our reality that's like, no, no, no, you can't do that. Remember when I told you, like, if you try to imagine Graham's number in your head, it would become a black hole because the information density in your brain would be so much that it's bigger than the limit of the universe and you'd form a black hole. And that's just nature's way of telling, don't try that. So I would recommend not trying to fit Graham's number in your head. But it turns out you can't do this thing. There are these fundamental limits. So, okay. I will— we're almost at the end, and I know you guys have been so patient with putting up with all the crazy numbers, but I promise you this last part is going to blow your minds, and then we're done. Just to recap so far, multiple power power factorial Gram tree beaver. Yes, perfect. And I will, I will send the notes so far. Perfect. Now here's, uh, here's the next step of this. Hey, Colonel. Every— everything that we've seen so far, even though we reached, uh, something that was uncomputable, those were still finite numbers, right? Like they are a number that you could potentially write down and it will end. But Growing up, I think like if someone gave you a game of like what's the biggest number, as soon as you learn about something called the infinity, you will probably say infinite. So what is this thing and what like how does that even relate to the things we're talking about so far, right? In the kind of early 18th, 19th century, mathematician named Georg Cantor, he finally asked this question. And he sat down and took this thing that was ridiculed almost at the time that he tried to pin down and make concrete what infinity is. And that area of inquiry turned out to be revolutionizing our understanding of math and all the foundation of math itself, which is basically set theory. So what we're going to really quickly cover is first— what is the first infinity that people come up with that you come across with? What's like the— What's that? What is the first infinity that people come up with? Perfect. Thank you. Natural numbers, right? That is just this thing called n. So natural numbers— You're— that, that's it. Perfect. Um, so we've got— What is a natural number? N, that's just 0, 1, 2, 3, 4, da da da da, on and on forever, right? That is the set of natural numbers, and it is the counting numbers that we use. It's one of the simplest objects we come up with when we're growing up learning math. That's the first infinity. I kind of want you to appreciate one thing, that everything that we saw that was finite, even though we were like, this is mind-boggling and crazy and kept going and going. If you kind of try to compare them from infinity, they're all zero. Like effectively, right? Like you could take the biggest number that we came up with, like BB20 or something, and you could subtract that from, let's say, the set of infinity. You could take it away from it and it would still be infinite. Think about that. And the— and one way to, one way to really see this is imagine you arrive at a hotel that has an infinite number of rooms. So you arrive at a hotel. This, this problem is called Hilbert's Hotel. Hilbert is a great mathematician. Um, you arrive at Hilbert's Hotel, this funhouse crazy infinite hotel. The rooms are labeled 0, 1, 2, 3, 4, 5, 6, on and on and on. It never stops. It has infinite rooms. So then every room is occupied. Okay, so we've got someone in every room. One person arrives, shows up at the door at Hilbert's hotel. It's like, Hilbert, can you fit me? And at first Hilbert's like, well, you know, I've got— every room is occupied. I know I have infinitely many rooms, but everyone's occupied. I can't fit you. But then the guy goes, actually, why can't you just Shift everyone by 1. So every person, just tell them to go to the next room. And so boom, boom, boom, boom. It's like in 1 step you freed up the first room and now you can fit this new person, right? So even though if you have infinitely many rooms, every single one is occupied, new person shows up, you can fit them. Cool, right? How about if In the next step, an infinite number of people show up. Can you still fit them? Yes. You just have to replace by infinity. Yes. How? In the front? Well, I mean, I don't know. Yeah. So here's a way to do it. So a bus arrives, and this bus is Hilbert's bus. It has infinitely many, infinitely many new people. So infinitely many rooms all occupied, infinitely many new people come. Just, you can still fit them. How? Just tell each person to shift, skip by 1. They're never getting any sleep. They're just constantly gonna be moving. So effectively you'll say, well, if you're in an even— if you're in 1, uh, go to a— if you're in 2, just go to like the, the next, even multiple. If you're an odd, just go to the next even multiple, and you'll free up— in one step, everyone can do this, and then you'll free up all the gaps in the middle, right? Because everyone's gone one skipped over, leaving their current room. So you could fit all the infinite people. So this is actually how, um, I remember when I said if you took like Graham's number compared to infinity, it's nothing. Because if, if a Graham's number worth p people arrived in a bus, you could just shift everyone by Graham's number over here and then you just fit them. But it turns out you can actually fit infinitely many just as easily. Just skip them one over. You've got one room, one room, one room, fit them. Okay. So this is really interesting, but this thing that we just defined, which is N0, let's call it this first infinity. So this is the infinity that dwarfs every finite number. It is— it behaves in these weird ways, but there is a way to define and play with this in a very concrete way. That is where, um, we find that there are actually bigger infinities. So, um, I'm trying to think how much of this proof is exciting versus interesting versus not. Okay. So we realize that if we just have numbers, that's just finite countable numbers, right? But then if you say, okay, 0, 1, 2, 3, if you have -1, -2, -3, -4, we saw that this is an infinity and this is infinity, right? But are these things— is this whole thing The same size as just the numbers like 0, 1, 2, 3. Is there infinity in between those numbers? Is there negative infinity? We kind of don't know. Um, so, so these things are exactly the same. Why? Because if you started— if you were to label all these numbers in order, you could just say this is my first number, this is second number, This is third number, this is fourth, this is 5, and you'd be able to pair up 1 to 1, 2 to -1, 3 to 2, and so on and so on and so forth, right? Every number you could just match up with the natural numbers and they'll pair up equally one by one. This is just called a one-to-one correspondence. And what that shows you is that in a concrete sense, even if you have double the infinity of negative integers and positive, like this whole set of integers, it's the same as natural numbers. All right. One more thing after this, and we're almost done. All right. So, you know, numbers that are like this, right? Like rational numbers that are any number by any number. It's a fraction. So fractions, exactly. So these are called rational numbers represented by Q. So we ask the question, is this bigger than just natural numbers? Yes. Because we can do so many of these, right? Like these, you're dividing. There are so many of these. Turns out this is also the same as natural numbers. And the way to do that Is, is this really neat way of arranging these numbers, which is you just go like, like this: 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, blah, blah, blah. And like, you just take 1 by 1, and it's like 2 by 1, and this is 3 by 1, and this is 1 by 2, and this is 2 by 2, and so on, right? In this way, you could create all the fractions because it's Every number here on the top, every number here on the bottom, and then what you do is you start counting them like this: one, two, three, four, five, six, seven, eight, nine, ten. Da da da da da. So in this way, you'd be able to just pair up every single fraction with a normal number, with a natural number. So once again, they're the same. Right. So, so far they're all the same. Every bigger infinity that we tried coming up with is actually just the same as natural numbers. All right. But that's not where this ends. And I want to show to you, and this will be, I promise, the last thing. We're pretty much done. What I want to show you is there is a bigger infinity. And in fact, you've already come across it. So the bigger infinity is a— do you guys remember the number line? And you remember the concept of like everything on a number line is like a number, right? So you started with like 1, and then maybe you have fractions, but every continuous point on a line is a number. These numbers are called the real numbers, right? If you took some math, you might remember these are the reals. This just means all the rationals plus not rational, so irrational numbers like square root of two, square root of three, e, pi. All these are irrational. So you take all the fractions we saw, you take all the irrational, and they fall somewhere on this continuous line. Do do do do do. This line is infinitely dense. Turns out this num— this infinity, the set of all real numbers, is strictly bigger than natural numbers. That is, I won't try to do the proof, proof of this. We have run out of time. It's very beautiful, very simple. It's called Cantor's diagonal argument. You can look it up. But what it shows is that if you try to list real numbers and all of them, you will still run out. If you try to pair it up one by one with naturals, it doesn't work. So all the tactics that we tried and we were able to kind of pair them up one by one, it will not work. And it is proved that you can prove that out. This is bigger. So what that actually means is what's interesting about it is it tells you that most numbers are actually irrationals because this is a natural number type of thing. Like this thing is actually so much bigger than the natural numbers. It's kind of hard to appreciate. One way that I found to visualize it is this idea. Take like on this number line, just take this segment. So like you have this segment, right? 0 to 1. That's it. In this line, uh, we have 1, 2, 3, 4, 5, 6. Take like break it in half and then assign To break that in half and then break that in half and break that in half and then half and half, you keep halving. You can take all those little segments and you can match it up to every single number and it will be perfectly matched. So in just like, you can take one little piece of this thing of the reals and you can pair up all the natural numbers. And inside you could keep doing that. You could start like just like this, just like this, as long as you have a continuous line thing. Turns out the density of these things is so much higher than natural numbers, fractions, rationals, everything we've seen. So this is a thing. There's an open question. This is a question that will never be solved in mathematics because we know that it is independent of mathematics. Like similarly, the last question was BB643. We will not solve in our mathematics. The number— the question here is, does there exist an infinity that is bigger than naturals But less than reals. We don't know and we can't find out. Um, but you can keep making bigger and bigger infinities by just doing this operation of take these numbers and take all the ways to combine them and find their combinations. So it turns out this is also a recursive thing where you could keep doing this and it will keep coming up with bigger numbers. So that's about it. Um, and we have found, we have found, perhaps kind of like the, the end of this tunnel where we started with all the finite numbers and some really big ones. We saw that in order to come up with a bigger one, or at least a bigger class of numbers, you sort of have to come up with a new kind of idea, new kind of paradigm. A lot of these things that we covered in this like 40 minutes, these took over 100 years, um, 100 plus, 150 years for people to figure out in various branches of mathematics. It goes through graph theory, uh, exponentiation, math notation, to set theory, naturals, infinities. And these are at the foundation— like, this stuff especially is at the heart of all mathematics because this is set theory. And for those who don't know, set theory is the underlying idea behind of mathematics. Like everything in physics, which I would love for this to be true, not everything can be derived by just a finite number of things. Like there's no grand unified theory that we have found, but in math there is, and that's set theory. So all these questions kind of wrap up with Nicole's question of why do we come up with all these things? Turns out many of these numbers were actually byproducts of studying these various other really central questions in math, and they just pop out of those things. Typically, people weren't sitting down trying to invent big numbers, but now that we have 100 years worth of math behind us, we can just run through and take this tour. So hope that's interesting to you. That's infinity, that's big numbers. Thank you.