▶︎ 0:00 Hi, I'm Joel David Hamkins, and I'd like to tell you today about the paradox of the largest tweetable number. Of course it's called x.com now, but I want to use this old terminology of tweeting and Twitter and so on. And there's a limitation when you make a tweet of 280 characters, that's the traditional limit, and you could tweet numbers. For example, you could just fill the tweet with digits.
▶︎ 0:25 So if I make a tweet, say here's my tweet, and I could just fill it with digits like 28765 and so on. I could just fill the tweet with digits and I would be tweeting a certain number. What's the largest possible number that you could tweet in that way? Well, of course, we could tweet a much bigger number than that. If we used nines instead of all those other digits, we could just fill the tweet completely with nines, and that would be an enormous number that we would be tweeting. It would be one less than 10 to the 280.
▶︎ 1:04 But we can tweet much larger numbers than that. For example, we could write a description of a number in the tweet. I could write one centillion, and a centillion is 10 to the 303. So that's a number that's much bigger than filling the tweet with nines, with the digit nine. So this is a bigger number. What is the biggest possible number that you can tweet? So really this paradox, the paradox of the largest tweetable number is a kind of exploration of how is it that we can describe enormous numbers with a very small description.
▶︎ 1:43 We can describe much bigger numbers than this. For example, maybe I hear someone from the back saying, "What about a googol?" I could write a googol here. A googol is this traditional term for the number that's 10 to the 100. But actually, we've already tweeted bigger numbers than a googol than that just by filling the tweet with digits. Yes, with 280 digits, it was already bigger than a googol.
▶︎ 2:09 Well, if I want to tweet even bigger numbers, maybe someone has the idea of using mathematical operations, for example, a factorial. I could say a googol factorial. I could in fact put a lot of factorial symbols on the tweet. I could just fill the whole tweet with exclamation points after a googol, and that would be the factorial of this enormous number. And so in this way, we can begin to tweet some truly enormous numbers.
▶︎ 2:40 There's a certain argument that I want to make about the paradox of the largest tweetable number, and that is, first of all, we can observe that there's, in principle, only finitely many tweets that one can make. Certain characters are allowed. Maybe any Unicode symbol, any character that you could type on your computer is allowed as a character in a tweet, almost any. And there's 280 possible characters allowed in the normal limit. And so if N is the number of characters that you can type, then the total possible number of tweets that you could make is N to the 280.
▶︎ 3:16 Actually, the reality is a little more complicated than that, because if you look into it, then in fact, some characters in the Unicode character set are control characters that cause accents on previous characters, and the actual Twitter algorithm allows two of those characters to count as only one symbol because it's putting an accent on and so on. So the actual limit is not quite exactly N to the 280. But let's just take that as an approximation.
▶︎ 3:46 The fact is that there's only finitely many possible tweets that you could make, and some of those tweets describe numbers. Some of the tweets describe your breakfast and some of the tweets describe your vacation in Athens. But some of the tweets describe numbers. For example, I could write a tweet that said the number of grains of sand in the Sahara Desert, and that's going to be a certain number, or the number of stars in the Milky Way galaxy as of this moment. That's going to describe a certain number. Or I could describe some mathematical formalism that would fit in a tweet, such as the formalism here, say a googol factorial, factorial, factorial.
▶︎ 4:24 The factorial of a number, of course, is when you take the number and multiply it by itself minus one and minus two, minus three and so on, all the way down to one. So then we have finitely many tweets, possible tweets, and some of them describe numbers. So therefore, there's only finitely many numbers that we could possibly tweet, and so therefore there must be a largest number that we could tweet.
▶︎ 4:48 So that's the argument that I want to discuss. And then the question is, is it a good argument? Is there really a largest tweetable number, or what is this number? Could we hope to discover what it is?
▶︎ 5:01 So I actually ran this contest about a few years ago. I made a tweet and I said, "Okay, twits, who's gonna tweet the largest possible number?" And I got many, many submissions right away. And actually I offered a prize in my tweet. I said, "The prize is going to be a million dollars." In my tweet I promised to give a million dollars, but there was a little asterisk with a little footnote, because the footnote condition was that the winning prize amount would be a million dollars divided by the value of the winning number, the largest possible tweet that was submitted.
▶︎ 5:42 So I can't really afford a million dollars easily on the weekend, and so therefore I immediately followed up my announcement of the contest by tweeting myself in reply, "One million." That was my submission. I tweeted, "One million." I just wrote out the words "one million." And of course, that was very important to, for sort of prudent urgency, because now of course the prize amount would be divided by a million and so I was only gonna be on the hook for about a dollar or less.
▶︎ 6:17 So that's saving my wallet. But nevertheless, I was alarmed because somebody made a submission like this to my tweeting contest.
▶︎ 6:31 What they tweeted was an image of a gigantic number one. And so the question is, this was a larger number than any of the numbers that I had tweeted when I had a million, because this number one is enormous. And so if this would be counted as the winning prize, the largest number, then I would have to be dividing a million by one, of course, and that's quite worrisome for the prize. But does it really count as a large number? It's just the number one, which is not very large on the numerical scale even though it is very large when written there, and of course this is making a kind of conflation between number and numeral. This is a large numeral but it's not a large number. The numeral is the symbol that we use to represent the number whereas the number is the thing itself, the abstract quantity itself.
▶︎ 7:36 It reminds me of my childhood, I read this wonderful children's novel, The Phantom Tollbooth by Norton Juster, and in that novel, there's the City of Digitopolis and under the City of Digitopolis is the number mine where they find the numbers. They mine them out of stone and they found the largest number. It was gigantic, an enormous number three, it was over four meters tall. That's a similar kind of mistake to this tweeting mistake here. So this distinction between number and numeral is sometimes called the semantics, the syntax-semantics dichotomy,
▶︎ 8:20 And there's another way to talk about this. Once I was at high table at Oxford, which is a sort of formal meal at the college, where the scholars discuss various topics, and we were discussing the use of language, and one of my colleagues, Alex Moran, was citing customary practice in the North of England, and what he had said was, "I generally use pants as trousers." That's what he said. Or maybe he said, "I generally use pants as trousers." So the question is, was he talking about the words or the thing?
▶︎ 8:57 And maybe it's important for the Americans in the audience to mention that in British English, the word pants is sometimes used to refer to what in the US we might call underwear or underpants, and so when he said, "I generally use pants as trousers," it might have a different meaning than you think at first. And actually I had taken a discreet glance under the table, and he didn't seem to be wearing pants as trousers at the time, and so I think that he was talking about the word pants and the word trousers and saying that he used those words to mean the same thing as we generally do here in the United States.
▶︎ 9:38 Once my daughter, Hypatia, asked me, "Does everything rhyme with itself?" And I said, "No, no, those two words don't rhyme." Which, of course, was not what she was really asking, so I can just write this out here. This is also the syntax-semantics dichotomy, the difference between using a word and mentioning the word. So what she asked is, "Does everything rhyme with itself?" And of course my answer was answering the question, if we put these two words in quotations, it was answering the mention case where she was mentioning both of the words. No, the word everything, so when we put it in quotes we're mentioning the word rather than using it, doesn't rhyme with this word itself, mentioning this word also. Those two words don't rhyme.
▶︎ 10:38 But actually we have four combinations here of use mention, because I could ask, for example, does "everything," does the word everything rhyme with itself? And you might say, "Well, yes, maybe, maybe every word rhymes with itself." Of course it's a poor poet who makes a poem and the rhyme is a word with itself, but technically I would find it reasonable, especially with a mathematical way of thinking, to say yes, we want to say that rhyming is a reflexive relation, every word should rhyme with itself, and in particular this word everything rhymes with itself. If I say everything and everything, then those are rhyming words. So everything does rhyme with itself.
▶︎ 11:22 But I could arrange it differently like this. Does everything rhyme with itself? No, because the word hippopotamus does not rhyme with this word itself. Those two words don't rhyme, so that answer is no. And then finally maybe the question where we're mentioning, so where we're using both of these words, does everything rhyme with itself? I might say yes, because that's just what we said before. If we interpret everything as meaning every word, yes, every word rhymes with itself. I think that's quite a reasonable answer.
▶︎ 11:58 All right. I had another entry to the largest tweetable number contest and it was the following. Here was the tweet and the person had simply tweeted the number zero. And you might say, "Well, obviously that's not the largest tweetable number." Except, well, it depends on what order you're using. For example, in ordinary numerical order, zero is not a very big number. In fact, it's the smallest natural number, of course.
▶︎ 12:38 But what if you were thinking of the numbers in a different order, for example, in alphabetical order? Then zero maybe would be the last number, the largest number in alphabetical order. It would come at the very end of the book of numbers if it were arranged alphabetically. And so in that sense, zero is a very large number in alphabetical order. But then of course we would be dividing the million dollar prize by zero, which maybe is infinite. I would really be on the hook then, except my lawyers would argue that a million divided by zero is undefined rather than infinite, and so I'm good.
▶︎ 13:16 Let's get down to the details of describing some very big numbers in tweets. That's really what the paradox of the largest tweetable number is about. It's a reason to think about how can we describe enormous numbers in a very small space. One of the things we might do is, of course we would want to use exponentials, say 2 to the 100 is a pretty big number, or 2 to the 1,000, that's a big number, but we could also do what's called iterated exponentials.
▶︎ 13:53 For example, I could do 2 to the 3 to the 100, and so on, or in general, a to the b to the c. This is called an iterated exponential. And it's, on the one hand, possibly ambiguous, because it has two different meanings. Namely, on the one hand, we could interpret this as a to the number b to the c. This would be the upward associated instance of this iterated exponential. But on the other hand, it could mean a to the b to the c? Both of these are a to the b to the c. But they're not always the same, so which one do we mean if I write a to the b to the c?
▶︎ 14:41 But the difference, the fact that these are not equal is another way of saying that exponentiation is not an associative operation, because it matters which order you exponentiate it. If you do the first two first and then the third versus the last two first and then take the exponent with the first one, you sometimes get a different result. And one way of seeing that is, if you look at this one, you can see that it's the same thing as a to the b times c, whereas this is a to the b to the c, and generally b to the c can be much larger than b times c. And so generally this one is larger than this one.
▶︎ 15:17 And actually, that tells you a way of resolving the ambiguity. So when someone writes a to the b to the c like this, we almost always mean this one and not this one, because we have another way to write this one using the exponentiation rule. I could fill the tweet with iterated exponentials, say 10 to the 10 to the 10 to the 10 to the 10 to the 10 to the 10 and so on, and just fill the entire tweet that way.
▶︎ 15:45 Let's talk about some other big numbers. We already mentioned a googol. Googol, which is 10 to the 100, and if you wrote that out in decimal, it would be a one with 100 zeros after it. Incidentally, I've heard that the company Google, when you google online and so on, is named after this number, but it's spelled differently of course. This is O-L and the company is L-E.
▶︎ 16:12 And there's another number called a googolplex, which is 10 to the googol, or in other words, 10 to the 10 to the 100. So this is an instance of iterated exponentials. And I'd like to understand this googolplex number a little bit more. Actually, I think the center of the Google company is also called Googleplex, so it's funny. This would be in decimal a one with a googol number of zeros after it.
▶︎ 16:50 There's an interesting feature about a googolplex, and that is, it's very easy to describe a googolplex. I just did. It would fit in a tweet, 10 to the 10 to the 100. That's a very easy description. But let's think about the typical number less than a googolplex. A googolplex, as we said, is a one in decimal, it's a one with a googol number of zeros after it, and so the typical number less is going to be random digits, a googol many, approximately, digits. And so, could we hold such a number as an object of thought in our minds?
▶︎ 17:32 Let's just take a typical number and the digits will be essentially random, and so maybe the only way to describe the particular value of such a particular number would be to recite these digits. And then the question would be, well, could you do that? Let's suppose you were really good at reciting digits. How long would it take you to recite the digits of a typical number less than a googolplex? Well, you'd have to recite a googol many digits, which is 10 to the 100 many digits. Let's suppose that you could say a million digits every second.
▶︎ 18:09 Then we could calculate how long it would take for you to recite the digits of that typical number less than a googolplex. And if you could say a million digits every second, the physicists tell us that the age of the universe is about 13.8 billion years, which is less than 10 to the 18 seconds. And so, if you had 10 to the 18 seconds and for each one of those seconds, you recited a million digits, that's 10 to the 6. That would be 10 to the 24 many digits, but you're needing to recite 10 to the 100 many digits, so there's no way you could do it. You would just be reciting the tiniest fraction of the digits.
▶︎ 18:53 The point I'm trying to make is that we can easily describe a googolplex, it's 10 to the 10 to the 100, but the typical number less than a googolplex is essentially random digits of length a googol, and there's no way that we could recite those digits even if we recited a million digits every second since the beginning of time, since the Big Bang. So, there's no way we could hold that number as an object of thought in our minds. It's simply impossible.
▶︎ 19:22 And what that means is that it's sometimes the case that we can describe enormous numbers but there are much smaller numbers that we cannot describe as easily. The shortest description of those smaller numbers would be much, much larger. And this is a phenomenon that's related to the paradox of the largest tweetable number, because a paradox of the largest tweetable number is all about describing enormous numbers with a small description. And the point is that just because you describe an enormous number with a small description doesn't mean that you can describe all the smaller numbers with such a small description, and in fact, there aren't enough small descriptions to go around for that.
▶︎ 20:00 Let me describe another number now, what I call, let me just put it here, a googolbang. A googolbang means you take a googol and you take the factorial of it, which means you multiply 1 times 2 times 3 times 4 and so on, all the way up to a googol. Or if you start at the top, it's googol times a googol minus 1, a googol minus 2, a googol minus 3, and so on all the way down to 1. In general, x bang means x factorial. It's just a whimsical way of talking about the factorial function.
▶︎ 20:37 I want to compare these two numbers, a googolplex and a googolbang. Which is bigger? It's a fun little puzzle. Which number is bigger, a googolbang versus a googolplex? Well, if you think about it, a googolplex is 10 to the googol. So if we wrote that out, it would be 10 times 10 times 10 times 10, and so on with a googol number of factors. Whereas a googolbang is, because of the factorial, it's a googol times a googol minus 1 times a googol minus 2 and so on, all the way down, times 3 times 2 times 1.
▶︎ 21:23 So the number of terms here is a googol, the same as the number of terms here, except most of these terms are much, much larger than 10. A few of them at the bottom are smaller than 10, but all of the rest of them are much bigger than 10, are bigger than 10, and a lot of them are a lot bigger than 10. And so we can see pretty clearly that a googolbang is much bigger than a googolplex because the size of these extra terms totally outweighs these few little terms at the end that are less than 10. So a googolbang is bigger than a googolplex.
▶︎ 21:58 A slightly more challenging version of the puzzle is to allow iterated applications of these suffixes. For example, I could make a tweet like this. I could write a googolbang plex bang bang plex plexbang. When you take a plex, xplex just means 10 to the x and xbang means x factorial. So I can iterate this. A googolbang plex bang bang plex plexbang and so on. I could just fill the tweet with these adjectives, but now it becomes a little bit of a puzzle to figure out if you have two such submissions, you need to know which one is bigger, and so you would need to compare this one, for example, with a googolplex bang plex bang bang bang plex and so on. So for all these different suffixes, you need to know how to compare the sizes of these numbers, and it's not always so obvious, but if you think about it, in fact, there's an algorithm for determining that.
▶︎ 23:17 There's another adjective we might add and that's called a googolstack. So let's do that one. A googolstack means 10 to the 10 to the 10 to the 10 iterated, an iterated exponential where the height is a googol many 10s here. So this number is actually far, far larger than both a googolbang and a googolplex because these iterated exponentials are growing so rapidly.
▶︎ 23:51 But now, of course, I could talk about the googol, googolplex bang stack hierarchy. I could talk about numbers such as a googolstack, a stack, an xstack just means I do 10 to the 10 to the 10x many times. So I can do a googolbang plex stack or a googolstack stack stack bang plex stack and so on, and I would need to know how to compare these numbers in order to judge the largest tweetable number contest.
▶︎ 24:23 Introduced a certain notation which is extremely helpful, and it's partaking of some of these ideas, and that's his up arrow notation. Let's talk about that a little bit. The basic up arrow, say, two up arrow four, is just referring to exponentiation, so this just means two to the fourth, or in other words, two times two times two times two. In general, A up arrow B, this is the base case of his recursion. It just means A times A times A, B times, so that's the same thing as A to the B.
▶︎ 25:14 But then we have the next level of recursions. Say A double arrow B. What this means is A up arrow A up arrow A up arrow and so on. A, where we have B times here. We associate to the right, so this is also called tetration or iterated exponential. What it is, is each one of these is an exponential, and so what this means is A to the A to the A to the A, where the height of that exponential, that iterated exponential is B. The double up is tetration, which is the stack operation that we referred to earlier.
▶︎ 26:00 In general, well, I can go one more. The triple up arrow means that I do the same idea except I repeat the double ups here. So, double up arrow A, double up arrow A, and so on. Double up arrow A. What this means is that each one of these is a stack, and so ultimately what this boils down to is a stack of As whose height is a stack of As whose height is, and so on, such that this length is B. Here we were iterating the tetration B times.
▶︎ 26:43 You can see quite clearly these numbers are growing extremely rapidly, using this Knuth notation. These ideas are related to the Ackermann function, which was introduced in the early 20th century by Ackermann. We can unify these definitions with the following recursion. We could define, say, A up arrow with a zero B just means that you're multiplying, and A up arrow N plus one B means that you do A up arrow N, A up arrow N and so on, up arrow NA where the number of terms is B. The next level of iterated up arrows is just determined by iterating the previous level B times. And so, numbers like three quadruple up arrow three and so on, these are just mind-bogglingly huge numbers. It's difficult even to describe them except by using this kind of Knuth notation. But I want to go beyond Knuth, even.
▶︎ 28:02 It's a way of climbing on top of what he did, and that is one can define the strong double arrow, which is different from an ordinary Knuth double arrow. So I want to define the strong double arrow. A strong double arrow B means that I do the B fold up arrow of A with itself, so this is transcending what Knuth did.
▶︎ 28:33 And then, of course, we can have the double strong double up arrow, which means the similar kind of iteration that we did before where the number of terms here is B and so on. And then we can define the N fold iteration, double strong up arrow and so on, and just continue the recursion past the previous levels. In this way we describe some truly huge numbers.
▶︎ 29:06 So, for example, I might make my entry in the largest number contest, so after defining this, then of course we're going to define the versions of the triple strong up arrow and so on, and the quadruple strong up arrow and so on. And so my entry in the largest number contest, just to be definite here, is going to be the quadruple strong up arrow of three with itself, which is a truly vast number. Some people may have heard of a number called Graham's number, which can be described in terms of these double up arrows. And this number is far, far larger than Graham's number. Let me just talk a little bit
▶︎ 29:57 More abstractly about the nature of these tweetable numbers by introducing the concept of Kolmogorov complexity. Maybe you have the idea that when you tweet a number, really what you're doing is tweeting a description of how to compute the number, because we can view these as computer programs in a sense. What I really did was give recursive definitions of these numbers, and so what I'm tweeting is a description on how to calculate the number. And the Kolmogorov complexity of a number or of any string, any finite string of symbols, is the size of the smallest program that will produce that string or that number.
▶︎ 30:35 Let me say it again. The Kolmogorov complexity of a number is the size of the smallest program that will produce it. So, for example, the Kolmogorov complexity of a googolplex is very small because I could describe it so easily, I could write the program that computed 10 to the 10 to the 100. That's a very short program. It would easily fit in a tweet, but the number itself is enormous. Or for example, a googolplexplexplex stack, there's a very small program that computes that number, but the number itself is enormous.
▶︎ 31:12 And so generally, this phenomenon that we mentioned earlier, that a googolplex is very big, but it has very small Kolmogorov complexity, whereas the typical numbers less than a googolplex, you couldn't hold as an object of thought because it would take you more time since the Big Bang even to recite the digits, you wouldn't have enough time. Another way of describing that situation is that those smaller numbers have a very high Kolmogorov complexity because basically the shortest program that would produce them, if the digits were truly random, the shortest program that would produce them would be just the program that hard codes the digits into the program. If the digits were random, you wouldn't be able to compress that information at all substantially, and therefore the program that produced it would be approximately the same size as the number of digits, which would be a googol. So enormous Kolmogorov complexity.
▶︎ 32:11 Now, the deep observation about Kolmogorov complexity is that actually we don't have any way of computing it. There's no computable procedure that will accept a given string or number as input and then tell you what is the Kolmogorov complexity of that number. It's in principle impossible to compute the Kolmogorov complexity of a number. And let me give you an argument for that.
▶︎ 32:36 Let's suppose toward contradiction that we could compute Kolmogorov complexity in general. Suppose we had a computable procedure that enabled us for any given number to compute the Kolmogorov complexity of that number. Now, of course, for a given level of complexity, there's only finitely many numbers of that complexity or less because there are only finitely many programs of that size or less. So therefore, the complexity of the numbers must eventually grow.
▶︎ 33:15 And so if I had a way of computing the complexity, I could go and search for a number that had a big complexity. So if we could compute complexity, then I could just try out the numbers one after the other and say, "Well, what's the complexity of this number? What's the complexity of the next number? What's the complexity of the number after that?" And so on. And I could simply iterate this process until I found a number that had a big complexity, and it's possible to design a program that would undertake this search, because it's just a simple looping. I'm just searching, try this number, try the next one, try the next one, until you find a number that has a complexity that's bigger than the very program that's undertaking that search, and then stop and produce that number as output.
▶︎ 33:59 So in other words, the algorithm is, if Kolmogorov complexity were computable, then we can design an algorithm that would produce a number as output that had a higher Kolmogorov complexity than the program itself. But that's contradictory, because Kolmogorov complexity, by definition, is the size of the smallest program that's able to produce the number. So we cannot produce with this searching program a number whose Kolmogorov complexity is bigger than the size of that program itself, because that would contradict the definition of Kolmogorov complexity.
▶︎ 34:31 So what this shows is that it's impossible in principle to know for certain what the Kolmogorov complexity of a given number is. There's no computable way to calculate it exactly. Let's return to the paradox of the largest tweetable number.
▶︎ 34:44 The paradox of the largest tweetable number. I claim that there's an absolutely winning entry to this contest, and it's the following. We gave an argument that there's only finitely many possible tweets, and some of those tweets may describe numbers, so therefore there's only finitely many tweetable numbers and therefore there is a largest tweetable number. And so my submission, my updated submission into the largest tweetable number contest is to submit the largest tweetable number. That fits in a tweet, and by definition, it is the largest number that you could possibly tweet. That's what the meaning of this phrase is.
▶︎ 35:49 No one can ever tweet a number that's bigger than this number, because if you could tweet a number, then this number would be at least that big because this number is the largest tweetable number. And so it would definitely win the contest. So now the question is someone might realize, "Hang on, what's going on?" Because maybe they want to submit the following tweet, the largest tweetable number plus one. Now, this number is bigger than the largest tweetable number, and yet it fits in a tweet.
▶︎ 36:28 That is the paradox of the largest tweetable number contest. Because it shows that the phrase the largest tweetable number can't really be meaningful maybe, because if it were meaningful, then this number would also be meaningful, but this number is tweetable, and yet it is bigger than any number that you could tweet, because it's bigger than the largest tweetable number. It's one more than that number. And so what the heck is going on?
▶︎ 36:52 Because it seemed like there was a rock solid argument. There's only finitely many tweets, some of those tweets describe numbers, so there's only finitely many tweetable numbers, so there must be a largest tweetable number, so therefore, I can talk about the largest tweetable number, that's a coherent concept, and therefore, I can tweet the largest tweetable number plus one. Paradox, because this number would have to be larger than any number that you can tweet and yet we tweeted it. What is going on with the largest tweetable number contest?
▶︎ 37:29 This brings us to another paradox called Berry's paradox. Berry was a librarian in Oxford, at the beginning of the 20th century, and Bertrand Russell described him as the only one in Oxford who could understand logic. What Berry talked about is, of course, he didn't have Twitter or tweets or anything like that, but he introduced a paradox about numbers that you can describe, and so he described a certain number like this. This is Berry's number. Whoops. Berry's number B is the smallest number not definable in fewer than a dozen words.
▶︎ 38:53 Of course, we're talking about English words here. There's only finitely many English words, and so there's only finitely many phrases with a dozen words in them, and some of those phrases describe numbers, and so there must be some numbers that are not definable in fewer than a dozen words. Now if I've done this right, let's see. We've described this number B as the smallest number not definable in fewer than a dozen words. One, two, three, four, five, six, seven, eight, nine, ten, eleven. I've described a number B in fewer than a dozen words, and it, by definition, is the smallest number that is not definable in fewer than a dozen words. So I seem to have defined B in fewer than a dozen words, even though it's the smallest number that's not definable in fewer than a dozen words.
▶︎ 39:47 That's a kind of contradiction. If you think we have a valid notion of what it means to define a number in a certain number of words, then this should be a definition of a certain number, and yet, it can't be, because it would have to be smaller than itself. It's exactly the same paradox as the paradox of the largest tweetable number. Well, it's not exactly the same, because it's really a version of, instead of tweeting the largest tweetable number plus one, it's like tweeting the smallest untweetable number.
▶︎ 40:21 Are those the same? You might ask, the largest tweetable number plus one is bigger than every tweetable number, and so it's not tweetable. If we take a naive view about it being sensible to talk about whether numbers are tweetable, then you might think that the smallest non-tweetable number and the largest tweetable number plus one are the same, but that's wrong actually, because the smallest untweetable number, I claim, is in fact much less than a googolplex, because there aren't enough tweets. There aren't a googolplex many tweets. There's only N^280 many possible tweets, where N is the size of the Unicode alphabet, and this is much less than a googolplex.
▶︎ 41:06 So therefore, not all the numbers up to a googolplex can be tweetable, because there aren't that many tweets. So therefore, the smallest untweetable number is gonna be less than a googolplex even. So we cannot, there's no way that number's gonna win the contest, because I can beat the smallest untweetable number by tweeting a googolplex, which is easily tweetable. Berry's paradox is more analogous to the smallest untweetable number than it is to the largest tweetable number plus one. But I can have a version of Berry's paradox that talked about the largest number definable in fewer than 20 words plus one, and that's fewer than 20 words, and so there would be an analog in that direction.
▶︎ 41:53 What is going on with the tweetable number paradox and Berry's paradox and so on? Let's try to have a more sophisticated perspective. Maybe we have the view that when you submit a number, what you're doing is submitting a computer program that's going to compute the number. And imagine that you're the judge of this contest, so you have a bunch of submissions which are computer programs. Of course, if someone submits a program that doesn't actually produce a number, maybe it produces some gibberish string or maybe it doesn't halt at all. It never stops. It doesn't give you an answer, so you would need, in order to be the judge, you have to compare these programs, the numerical answers.
▶︎ 42:38 That requires you, first of all, that even in the case when the programs do halt, they're giving you these numbers, you need to be able to compare the sizes of these numbers. So, for example, this is related to the problem of deciding whether a Googolplex bang is bigger than a Googolbang plex or with longer iterates. But maybe if you insist that the program is actually producing the digits of the number in decimals, then the comparison process maybe will be a bit easier, but still you have to know if the program is going to stop or not, if it's going to halt. And so it seems that in order to be a judge of the tweetable number contest, you would need to solve these instances of the halting problem.
▶︎ 43:19 But this is a famously undecidable problem. Alan Turing, in 1936, introduced the concept of Turing machines and talked about the possibility of some problems being undecidable, and we know now that the halting problem, the question of whether a given computer program ever halts and gives output, is computably undecidable. It's similar to the Kolmogorov complexity problem. In fact, we can prove that the halting problem is undecidable on the basis of Kolmogorov complexity being undecidable, because if you could solve the halting problem, then you could compute Kolmogorov complexity of a number.
▶︎ 43:55 Because given a number, you look at all the programs up to a certain size, and you ask the halting problem which ones of those programs halt, and then you run them, the ones that do halt, you run them and see if they produce the number and so on. And in this way, you would be able to compute Kolmogorov complexity. But we already argued that you cannot compute Kolmogorov complexity, therefore you cannot solve the halting problem.
▶︎ 44:18 So in general, there's no standard by which there should be no expectation in the most general case about how to serve as a judge in the largest number contest. One objection you might make to this kind of argument is that, actually, there's only finitely many programs we're talking about in the largest tweetable number contests because there's only finitely many tweets, and therefore the halting problem, in that finite instance, is computably decidable because we could hard code the answer for all possible tweets as to whether they halt or not.
▶︎ 44:56 But now, this opens up the door to a much more profound attitude about the sort of problem in the largest tweetable number contests and contests of that form, which is the question of whether there's a fact of the matter about whether the programs halt. In fact, you can show that the halting problem is computably undecidable, and it follows as a consequence of that that actually, for any foundational theory that you might have, so if you take, say, as an axiomatic theory for all of mathematics, say Peano arithmetic or Zermelo-Fraenkel set theory, or any of the standard axiomatizations of mathematics, then there must be programs, and in fact relatively small programs, that don't halt, but the theory cannot prove this. And so this calls into question whether there's a fact of the matter about whether those programs halt or not.
▶︎ 45:55 And so when you're talking about judging the largest number contest, maybe you have in mind that we should be looking for proofs from a formal system that this program halts with a bigger number than that one does. And the point is that those kind of questions can be independent of our foundational axioms, and so to talk about the tweetable numbers, whether a number is tweetable or not, can be a fact that's independent of the axioms of mathematics.
▶︎ 46:36 And then it makes you wonder, really, then, in order to be talking about a certain description of a number in a tweet, I should be talking about the axiomatic framework in which I can prove that that description in fact makes that number halt, and those kind of supplementary descriptions don't fit in the tweet, and we can't just take them for granted precisely because the axiomatic systems that we're working in admit independence and don't prove all the instances of this. And so it only makes sense to talk about the question of which programs halt with a given output, with a given definite output, in the context of an axiomatic framework that itself does not fit in the tweet.
▶︎ 47:18 And so from that point of view, you realize that the phrase "the largest tweetable number" is a phrase that's dependent on the underlying axiomatic framework of the system and that's not part of the tweet and it doesn't fit in the tweet, and to specify it more fully is always going to be just beyond what's tweetable. And this is one way of coming to an understanding of the paradox of the largest tweetable number.
▶︎ 47:44 This point about logical undecidability in the largest number of contexts, maybe our most fundamental axioms of mathematics don't determine the answer to the question of whether a given description defines a number even, or whether one description defines a number that's larger than the number defined by another description, and maybe this relates to Berry's paradox, where we talk about the smallest number that's not definable in fewer than a dozen words.
▶︎ 48:11 One way of resolving this paradox is to inquire, well, what is this notion of definability, and is it actually definable? Tarski thought about that quite a bit and proved an absolutely wonderful theorem. It's called Tarski's theorem on the non-definability of truth. In fact, say if we have a formal language of arithmetic, then the question of whether a given formula defines a number is not actually expressible in that language. This is exactly the content of Tarski's theorem on the non-definability of truth.
▶︎ 48:48 In a quite formal way, definability is not definable, and maybe it's easy to take Berry's paradox in a naive way as meaningful that he's using this word "definable," but when you drill into it and you apply the tools of mathematical logic and Tarski's theorem, you realize that actually it's pulling a fast one there. Definability is not definable, and similarly, tweetability is not actually definable in a tweet. So this is one way of resolving the paradox of the largest tweetable number.