Welcome to the Lectures on Infinity, my series of lectures exploring all my favorite paradoxes and conundrums.
In this instance, we shall consider the paradox of the largest tweetable number. What is the largest number that you can tweet? (to use the old terminology for making a post on X) The question will lead us on a cosmic journey to some truly astounding numbers and beyond, eventually into an evaporating mist of confusion, of logical complexity and paradox, ultimately revealing a fundamental underlying issue of mathematical determinacy—does every mathematical question have a determinate answer?
I shall be sharing the individual infinity lectures here on Infinitely More in the coming months.
Please enjoy!
Find the lectures here on Infinitely More in the lectures-on-infinity tag.
The lectures will appear on YouTube.
The whole lecture course is hosted at Ergo: Lectures on Infinity.
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This lecture is based on my essay The largest tweetable number.
The essay also appears in my new book, The Book of Infinity.
The Largest Tweetable Number
A lightly edited transcript. Timestamps link to the video on the Ergo website.
Can You Tweet the Biggest Possible Number?
I would like to tell you about the paradox of the largest tweetable number. The platform is called X now, of course, but I want to use the old terminology of tweeting and Twitter. There is a limitation when you make a tweet of 280 characters, and you could use that space to tweet numbers. For example, you could simply fill the tweet with digits, something like 28765 and so on, and you would be tweeting a certain number.
What is the largest possible number you could tweet in that way? Well, if we used nines instead of those other digits, we could fill the tweet completely with nines, and that would be an enormous number. It would be one less than 10 to the 280. 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, which is a number much bigger than what you get by filling the tweet with the digit nine. So what is the biggest possible number that you can tweet? This paradox, the paradox of the largest tweetable number, is really an exploration of how it is that we can describe enormous numbers with a very small description.
Why a Largest Tweetable Number Must Exist
We can describe much bigger numbers than this. Someone from the back might ask, “What about a googol?” A googol is the traditional term for 10 to the 100. But we have already tweeted bigger numbers than a googol simply by filling a tweet with digits: with 280 digits, the resulting number is already larger than a googol.
If we want to tweet even bigger numbers, we might think to use mathematical operations. For example, we could write a googol factorial. We could in fact fill an entire tweet with exclamation points after a googol, producing the factorial of that already enormous number. In this way, we can begin to tweet some truly staggering quantities.
There is a certain argument I want to make about the paradox of the largest tweetable number. First, we can observe that there are, in principle, only finitely many tweets one can make. Certain characters are allowed in a tweet, perhaps any Unicode symbol you could type on your computer, and there are 280 possible character positions under the normal limit. So if N is the number of characters you can type, the total number of possible tweets is N to the 280.
The reality is a little more complicated than that. If you look into it, some characters in the Unicode character set are control characters that cause accents to appear on preceding characters, and the actual Twitter algorithm allows two such characters to count as only one symbol. So the limit is not exactly N to the 280, but let us take that as a reasonable approximation.
The key point is that there are only finitely many possible tweets. Some of those tweets describe your breakfast, some describe your vacation in Athens, and some describe numbers. I could write a tweet saying “the number of grains of sand in the Sahara Desert,” and that picks out a certain number. I could write “the number of stars in the Milky Way galaxy as of this moment,” and that picks out another. Or I could describe some mathematical formalism that fits in a tweet, such as a googol factorial factorial factorial, where the factorial of a number is the product of that number with every positive integer below it down to one.
Since there are only finitely many possible tweets, and only some of those tweets describe numbers, there are only finitely many numbers that can possibly be tweeted. Therefore, there must be a largest number that we could tweet. That is the argument I want to examine. The question is whether it is a good argument, whether there really is a largest tweetable number, and whether we could ever hope to discover what it is.
The Million Dollar Largest Number Contest
A few years ago, I ran a contest on Twitter. I posted a tweet inviting people to submit the largest possible number, and I received many submissions almost immediately. I also offered a prize: in the tweet, I promised to award one million dollars to the winner. There was, however, a small asterisk, because the footnote condition specified that the prize amount would be one million dollars divided by the value of the winning number, the largest number submitted.
Since I cannot easily afford a million dollars on a weekend, I immediately followed up my contest announcement by replying to my own tweet with a submission of my own: “One million.” I simply wrote out the words “one million.” This was, of course, a matter of some prudent urgency, because with one million as a submitted entry, the prize amount would be divided by a million, leaving me on the hook for roughly a dollar or less. That took care of my wallet. Nevertheless, I was alarmed when someone made the following submission to the contest.
Number vs. Numeral: A Giant Number One
What one contestant tweeted was an image of a gigantic number one. The question then arose whether this counted as a larger number than any of the numbers I had tweeted, given that I had submitted a million. If this entry were to be judged the winner, I would of course be dividing a million by one, which is quite worrisome for the prize.
But does it really count as a large number? It is just the number one, which is not very large on the numerical scale, even though it is very large as written. This is precisely the conflation between number and numeral. It is a large numeral, but it is not a large number. The numeral is the symbol we use to represent the number, whereas the number is the thing itself, the abstract quantity.
It reminds me of a wonderful children’s novel I read in my childhood, The Phantom Tollbooth by Norton Juster. In that novel there is the City of Digitopolis, and beneath it lies the number mine, where numbers are found by mining them out of stone. At one point they discover the largest number: an enormous number three, over four meters tall. That is precisely the same kind of mistake as the one made in this tweeting episode.
This distinction between number and numeral is an instance of what is sometimes called the syntax-semantics dichotomy.
The Use-Mention Distinction in Language
There is another way to illustrate this point. Once, at high table at Oxford, a formal college meal where scholars discuss various topics, my colleague Alex Moran was citing customary practice in the North of England. What he said was, “I generally use pants as trousers.” The question is whether he was talking about the words or the things. It is worth noting for any Americans in the audience that in British English, the word “pants” is sometimes used to refer to what in the United States we would call underwear or underpants, so his remark may carry a different meaning than one might first suppose. I took a discreet glance under the table and he did not appear to be wearing pants as trousers at the time, so I concluded he was talking about the words themselves, saying that he uses “pants” and “trousers” to mean the same thing as we generally do in the United States.
This is also known as the use-mention distinction, or the syntax-semantics dichotomy: the difference between using a word and mentioning it. My daughter Hypatia once asked me, “Does everything rhyme with itself?” And I said, “No, those two words don’t rhyme,” which was of course not what she was really asking. When we put a word in quotation marks we are mentioning the word rather than using it, so my answer was addressing the mention case: the word “everything” does not rhyme with the word “itself,” and indeed those two words do not rhyme.
But there are actually four combinations of use and mention to consider here. Take the question: does “everything” rhyme with itself? One might reasonably say yes, since perhaps every word rhymes with itself. It is a poor poet who rhymes a word with itself, but from a mathematical point of view it seems reasonable to treat rhyming as a reflexive relation, so that every word rhymes with itself and, in particular, “everything” rhymes with “everything.” Now consider the question: does everything rhyme with “itself”? The answer here is no, because the word “hippopotamus,” for instance, does not rhyme with the word “itself.” Finally, the fully unquoted version, does everything rhyme with itself, might again be answered yes, if we interpret “everything” as ranging over every word, since every word rhymes with itself. I think that is quite a reasonable answer.
Zero: The Largest Number Alphabetically?
Another entry to the largest tweetable number contest deserves attention. The person had simply tweeted the number zero. You might say, “Well, obviously that’s not the largest tweetable number.” But it depends on what order you are using.
In ordinary numerical order, zero is not a very large number. In fact, it is the smallest natural number. But what if you were thinking of the numbers in a different order, for example, in alphabetical order? In that case, zero would perhaps be the last number, the largest number in alphabetical order. It would come at the very end of the Book of Numbers, if such a book were arranged alphabetically. In that sense, zero is a very large number indeed.
Of course, we would then be dividing the million-dollar prize by zero, which might be considered infinite. I would really be on the hook in that case, except my lawyers would argue that a million divided by zero is undefined rather than infinite, and so I am safe.
Iterated Exponentials and Their Ambiguity
Let us get down to the details of describing very large numbers in tweets, which is really what the paradox of the largest tweetable number is about. It gives us a reason to think carefully about how we can describe enormous numbers in a very small space. One natural approach is to use exponentials: 2 to the 100 is a pretty big number, and 2 to the 1,000 is bigger still. But we can go further by using what are called iterated exponentials, such as a to the b to the c, where the exponentiation is applied repeatedly.
Iterated exponentials are, on the one hand, potentially ambiguous, because the expression a to the b to the c admits two different interpretations. We could read it as a raised to the quantity b to the c, associating upward, or we could read it as the quantity a to the b, all raised to the c. Both readings are grammatically natural renderings of a to the b to the c, yet they are not always equal. This inequality is simply another way of saying that exponentiation is not an associative operation: it matters which pair you exponentiate first.
We can see the difference clearly by noting that the second interpretation, a to the b raised to the c, equals a to the b times c, whereas the first interpretation gives a to the b to the c. In general, b to the c can be much larger than b times c, so the first interpretation typically yields the larger number. This observation also resolves the ambiguity: when someone writes a to the b to the c, we almost always mean the upward-associating interpretation, precisely because we already have a simpler way to express the other one using the rule for multiplying exponents.
With that convention in place, one strategy for describing an enormous number in a tweet is simply to fill the entire character limit with an iterated exponential: 10 to the 10 to the 10 to the 10, and so on, all the way across.
Googolplex and Incomprehensible Numbers
Let us talk about some other large numbers. We already mentioned a googol, which is 10 to the 100, and if you wrote that out in decimal it would be a one followed by 100 zeros. Incidentally, the company Google, as in the search engine, is named after this number, though spelled differently: the number is spelled with an O-L at the end, while the company ends in L-E. There is another number called a googolplex, which is 10 to the googol, or in other words, 10 to the 10 to the 100, an instance of iterated exponentials (I shall henceforth write it as googol plex). As a side note, the headquarters of Google is also called the Googleplex, which is rather fitting.
In decimal, a googol plex is a one followed by a googol number of zeros. Now, there is an interesting feature of the googol plex: it is very easy to describe. I just did it. The expression “10 to the 10 to the 100” fits comfortably in a tweet. But let us think carefully about a typical number less than a googol plex. Such a number would consist of approximately a googol many digits, and those digits would be essentially random. The only way to specify the particular value of such a number would be to recite its digits.
So could we actually do that? Suppose you were extraordinarily fast at reciting digits, say a million digits per second. Physicists tell us the age of the universe is approximately 13.8 billion years, which is less than 10 to the 18 seconds. If you recited a million, that is 10 to the 6, digits per second for the entire age of the universe, you would recite at most 10 to the 24 digits in total. But a typical number less than a googol plex requires reciting 10 to the 100 digits, so you would barely cover the tiniest fraction of them. Even reciting digits continuously since the Big Bang at that rate, you could not finish.
The point is this: we can easily describe a googol plex itself, since “10 to the 10 to the 100” is a short description. But a typical number smaller than a googol plex consists of essentially random digits of length a googol, and there is simply no way to hold such a number as an object of thought in our minds. We cannot recite its digits, and no short description captures it. The shortest description of such a number would itself be enormously long.
This phenomenon is directly related to the paradox of the largest tweetable number, which concerns describing enormous numbers with very short descriptions. The key insight is that just because you can describe one enormous number with a small description does not mean you can describe all the numbers smaller than it with equally small descriptions. There are simply not enough short descriptions to go around.
Googol Bang, Plex, and Stack Hierarchy
Let me describe another number: googol bang. A googol bang means you take a googol and compute its factorial, multiplying 1 times 2 times 3 times 4 and so on all the way up to a googol. Equivalently, starting from the top, it is googol times (googol minus 1) times (googol minus 2) and so on, all the way down to 1. In general, x-bang simply means x factorial, which is just a whimsical way of referring to the factorial function.
Now consider a fun little puzzle: which is bigger, a googol bang or a googol plex? A googol plex is 10 to the googol, so written out as a product it is 10 times 10 times 10 and so on, with a googol number of factors. A googol bang, by contrast, is googol times (googol minus 1) times (googol minus 2) and so on, all the way down through 3, 2, 1. The number of terms is the same in both cases, namely a googol, except that in the factorial product most of the terms are much larger than 10. A few terms near the bottom are smaller than 10, but all the rest exceed 10, and many of them far exceed it. The factorial product therefore dominates, and we can see quite clearly that a googol bang is much bigger than a googol plex.
A slightly more challenging version of the puzzle allows iterated applications of these suffixes. One could write a tweet such as “googol bang plex bang bang plex plex bang,” and so on. Here, x-plex means 10 to the x, and x-bang means x factorial, and these operations can be chained in any order. One could simply fill an entire tweet with such suffixes. The difficulty is that if two contestants submit different strings of this kind, you need to determine which number is larger, and comparing them is not always immediately obvious. If you think carefully about it, however, there is in fact an algorithm for making that determination.
There is yet another suffix we might add: the google stack. A googol stack means 10 to the 10 to the 10 to the 10, an iterated exponential tower of 10s whose height is a googol. This number is far, far larger than both a googol bang and a googol plex, because iterated exponentials grow so rapidly. More generally, x-stack means a tower of 10s iterated x many times. One can then form expressions such as “googol bang plex stack” or “googol stack stack stack bang plex stack,” and so on. Comparing any two such expressions in order to judge which represents the largest tweetable number is the central challenge of the contest.
Knuth’s Up-Arrow Notation Explained
Donald Knuth introduced a notation that is extremely helpful and partakes of these ideas: his up-arrow notation. The basic single up-arrow, say two up-arrow four, refers to exponentiation, so it simply means two to the fourth, or two times two times two times two. In general, a up-arrow b is the base case of his recursion, meaning a multiplied by itself b times, which is the same as a to the b.
The next level of the recursion is the double up-arrow, where a double-up-arrow b means a up-arrow a up-arrow a up-arrow, and so on, with a appearing b times, associating to the right. This operation is also called tetration, or iterated exponentiation: it produces a tower a to the a to the a to the a, where the height of that iterated exponential is b. So the double up-arrow is the stack operation referred to earlier.
Going one level further, the triple up-arrow repeats the same idea using double up-arrows. It means a double-up-arrow a double-up-arrow a, and so on, with b terms. Since each double up-arrow is itself a stack, what this ultimately produces is a stack of as whose height is a stack of as whose height is a stack of as, and so on, with the total length of that iteration being b. In this way, each level of up-arrows is obtained by iterating the previous level b times.
These ideas are related to the Ackermann function, introduced in the early twentieth century. We can unify the definitions with a single recursion: a up-arrow-zero b means multiplication, and a up-arrow-(n+1) b means a up-arrow-n applied iteratively, with b terms. Numbers such as three quadruple-up-arrow three are mind-bogglingly large, and it is difficult even to describe them except by using this kind of Knuth notation. But the aim here is to go beyond Knuth even further.
Beyond Knuth: Strong Up-Arrow Notation
One way of climbing on top of what Knuth did is to define the strong double arrow, which is distinct from an ordinary Knuth double arrow. The strong double arrow a ⇑ b means that one performs the b-fold up-arrow of a with itself, thereby transcending what Knuth’s original notation achieves. From there, one can define the double strong double up-arrow, which carries out a similar kind of iteration in which the number of terms is b, and so on.
One then defines the n-fold iteration, the double strong up-arrow and beyond, continuing the recursion past each previous level. In this way we can describe some truly enormous numbers. For example, after defining the triple strong up-arrow and the quadruple strong up-arrow and so on, my entry in the largest number contest is 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 Knuth’s double up-arrows. This number is far, far larger than Graham’s number.
What Is Kolmogorov Complexity?
Let me speak more abstractly about the nature of these tweetable numbers by introducing the concept of Kolmogorov complexity. When you tweet a number, what you are really doing is tweeting a description of how to compute it, since we can view these descriptions as a kind of computer program. What I was doing earlier was giving recursive definitions of numbers, so what I tweeted was essentially a description of how to calculate them. The Kolmogorov complexity of a number, or of any finite string of symbols, is the size of the smallest program that will produce that string or number.
To give some examples: the Kolmogorov complexity of a googol plex is very small, because I can describe it so easily. The program that computes 10 to the 10 to the 100 is very short, fits easily in a tweet, yet the number itself is enormous. Similarly, a googol plex plex plex tower has a very small program that computes it, even though the number itself is vast.
This brings out a general phenomenon worth dwelling on. A googol plex is very large, yet it has very small Kolmogorov complexity. By contrast, the typical numbers smaller than a googol plex cannot really be held as objects of thought at all, because it would take longer than the time since the Big Bang even to recite their digits. Another way to describe that situation is to say that those numbers have very high Kolmogorov complexity. If their digits were truly random, the shortest program that could produce them would simply be one that hard-codes the digits directly into the program. You would not be able to compress that information substantially, and so the program would be approximately the same size as the number of digits, which would be a googol. That is an enormous Kolmogorov complexity.
Why Kolmogorov Complexity Is Uncomputable
The deep observation about Kolmogorov complexity is that there is no computable procedure that will accept a given string or number as input and tell you what its Kolmogorov complexity is. It is in principle impossible to compute the Kolmogorov complexity of a number. Let me give you an argument for that.
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, for a given level of complexity, there are only finitely many numbers of that complexity or less, because there are only finitely many programs of that size or less. Therefore, the complexity of numbers must eventually grow.
If we could compute complexity, we could simply go and search for a number with a large complexity. We could try out the numbers one after another and ask what the complexity of each one is, iterating this process until we found a number whose complexity exceeded the size of the very program undertaking the search. It is possible to design such a program, since it involves only a simple loop: try this number, try the next one, try the next one, until you find a number whose complexity is bigger than the program itself, then stop and produce that number as output.
In other words, if Kolmogorov complexity were computable, we could design an algorithm that produces as output a number whose Kolmogorov complexity is higher than the complexity of the program itself. But that is contradictory, because Kolmogorov complexity is by definition the size of the smallest program able to produce the number. We cannot produce, with this searching program, a number whose Kolmogorov complexity exceeds the size of that program, since that would contradict the definition directly.
What this shows is that it is impossible in principle to know with certainty what the Kolmogorov complexity of a given number is. There is no computable way to calculate it exactly.
The Winning Entry That Breaks Everything
Let us return to the paradox of the largest tweetable number. I claim there is an absolutely winning entry to this contest, and it is the following. We gave an argument that there are only finitely many possible tweets, and some of those tweets may describe numbers, so therefore there are only finitely many tweetable numbers, and therefore there is a largest tweetable number. My submission into the largest tweetable number contest is simply to submit: the largest tweetable number.
That phrase fits in a tweet, and by definition it is the largest number one could possibly tweet. No one can ever tweet a number bigger than this, because if you could tweet a number, then this number would be at least that big, since it is the largest tweetable number. It would therefore definitively win the contest.
Now someone might realize that something has gone wrong, because one could also submit the following tweet: the largest tweetable number plus one. This number is bigger than the largest tweetable number, and yet it fits in a tweet. That is the paradox of the largest tweetable number contest.
The paradox shows that the phrase “the largest tweetable number” cannot really be meaningful, because if it were meaningful, then this new number would also be meaningful. But this new number is tweetable, and yet it is bigger than any number you could tweet, since it is one more than the largest tweetable number. So what is going on?
The argument seemed rock solid: there are only finitely many tweets, some of those tweets describe numbers, so there are only finitely many tweetable numbers, so there must be a largest tweetable number, so I can coherently refer to the largest tweetable number, and therefore I can tweet the largest tweetable number plus one. Paradox: this number would have to be larger than any number you can tweet, and yet we just tweeted it. What exactly is going on here?
Berry’s Paradox and Definability
This brings us to another paradox called Berry’s paradox. Berry was a librarian in Oxford at the beginning of the twentieth century, and Bertrand Russell described him as the only person in Oxford who could understand logic. Berry introduced a paradox about numbers that can be described in a certain number of words. Berry’s number b is defined as the smallest number not definable in fewer than a dozen words.
We are talking about English words here. There are only finitely many English words, and so there are only finitely many phrases consisting of a dozen words. Some of those phrases describe numbers, and so there must be some numbers that are not definable in fewer than a dozen words. Now, we have just described b as “the smallest number not definable in fewer than a dozen words,” which is, by count, eleven words. So we have defined b in fewer than a dozen words, even though b is, by definition, the smallest number not definable in fewer than a dozen words.
That is a contradiction. If we accept that there is a valid notion of what it means to define a number in a certain number of words, then this phrase should succeed as a definition of some number, and yet it cannot, because the number it defines would have to be smaller than itself. This is essentially the same paradox as the paradox of the largest tweetable number, though not exactly the same. Rather than tweeting the largest tweetable number plus one, it is more like tweeting the smallest untweetable number.
One might ask whether those two things amount to the same thing. The largest tweetable number plus one is bigger than every tweetable number, so it is not tweetable. Taking a naive view in which it is sensible to speak of numbers being tweetable, one might think the smallest untweetable number and the largest tweetable number plus one coincide. But that is actually wrong, because the smallest untweetable number is in fact much less than a googol plex. There are not a googol plex many possible tweets; there are only N^280 possible tweets, where N is the size of the Unicode alphabet, and this is much less than a googol plex.
Since there are not enough distinct tweets to cover all numbers up to a googol plex, the smallest untweetable number must be less than a googol plex. That number could never win a largest-tweetable-number contest, because one could simply beat it by tweeting “a googol plex,” which is easily tweetable. So Berry’s paradox is more analogous to the smallest untweetable number than to the largest tweetable number plus one. There is, however, a version of Berry’s paradox that runs in the other direction: consider “the largest number definable in fewer than twenty words, plus one,” which is itself fewer than twenty words, and that gives an analog of the largest-tweetable-number paradox.
The Halting Problem Blocks the Judge
What is really going on with the tweetable number paradox, Berry’s paradox, and related puzzles? Let us try to adopt a more sophisticated perspective. Suppose that when you submit a number to the contest, what you are actually submitting is a computer program that will compute the number. Imagine that you are the judge of this contest, and you have before you a collection of submissions, each of which is a computer program. Of course, if someone submits a program that does not actually produce a number, perhaps it outputs some gibberish string, or perhaps it never halts at all and gives no answer, you would need to handle those cases. In order to serve as judge, you must be able to compare the numerical answers these programs produce.
That comparison task is already non-trivial. Even in the cases where the programs do halt and yield numbers, you need to be able to compare the sizes of those numbers. This is related, for instance, to the problem of deciding whether a googol plex bang is larger than a googol bang plex, or similar expressions with longer iterates. Perhaps if you insist that every program produce the digits of its number in decimal, the comparison process becomes somewhat easier, but you still face a prior question: you need to know whether a given program is going to halt at all. It therefore seems that in order to be a judge of the tweetable number contest, you would need to solve particular instances of the halting problem.
The halting problem, however, is famously undecidable. Alan Turing, in 1936, introduced the concept of Turing machines and raised the possibility of certain problems being undecidable. We now know that the question of whether a given computer program ever halts and produces output is computably undecidable. The halting problem is closely related to the undecidability of Kolmogorov complexity, and in fact we can prove the former from the latter. If you could solve the halting problem, then you could compute the Kolmogorov complexity of any number: given a number, you would examine all programs up to a certain size, use the halting oracle to determine which of those programs halt, run the ones that do, and check whether they produce the number in question. In this way you would be able to compute Kolmogorov complexity. But we have already argued that Kolmogorov complexity is not computable, and therefore the halting problem cannot be solved either.
Logical Independence and Axiomatic Limits
One objection you might raise to this kind of argument is that, in the largest tweetable number contest, we are only ever dealing with finitely many programs, since there are only finitely many possible tweets. Therefore, the halting problem in that finite instance is computably decidable, because we could simply hard-code the answer for every possible tweet as to whether it halts or not.
But this observation opens the door to a far more profound concern about the largest tweetable number contest and competitions of that form: the question of whether there is a fact of the matter about whether the relevant programs halt at all. One can show that the halting problem is computably undecidable, and it follows as a consequence that for any foundational theory one might adopt as an axiomatic basis for mathematics, whether Peano arithmetic, Zermelo-Fraenkel set theory, or any of the standard axiomatizations, there must exist programs, in fact relatively small programs, that do not halt but whose non-halting the theory cannot prove.
This calls into question whether there is a determinate fact of the matter about whether those programs halt. When thinking about how to judge a largest number contest, one natural idea is to look for proofs, within some formal system, that a given program halts and produces a number larger than its competitors. The difficulty is that questions of this kind can be independent of our foundational axioms, so whether a number counts as tweetable can itself be a fact that is independent of the axioms of mathematics.
This forces us to recognize that, in order to meaningfully speak of a description in a tweet as picking out a definite number, one must specify the axiomatic framework within which one can prove that the described program halts with that output. Those supplementary specifications do not fit inside the tweet, and we cannot simply take them for granted, precisely because the axiomatic systems we work in admit independence phenomena and do not settle all instances of the halting question.
It therefore only makes sense to ask which programs halt with a given definite output relative to an axiomatic framework that itself does not fit in the tweet. From this point of view, the phrase “the largest tweetable number” is a phrase whose meaning depends on an underlying axiomatic framework that is no part of the tweet and cannot be made part of it. Any attempt to specify that framework more fully will always exceed what is tweetable. This is one way of arriving at a genuine understanding of the paradox of the largest tweetable number.
Tarski’s Theorem Resolves the Paradox
The point about logical undecidability in the largest-number contexts is that our most fundamental axioms of mathematics may not determine the answer to the question of whether a given description defines a number at all, or whether one description defines a number larger than the number defined by another description. This may relate to Berry’s paradox, where we speak of the smallest number not definable in fewer than a dozen words. One way of resolving that paradox is to ask what this notion of definability actually is, and whether it is itself definable.
Tarski thought about that question carefully and proved an absolutely wonderful theorem: Tarski’s theorem on the non-definability of truth. If we have a formal language of arithmetic, then the question of whether a given formula defines a number is not expressible in that language. This is exactly the content of Tarski’s theorem on the non-definability of truth.
So in a quite formal sense, definability is not definable. It may be tempting to take Berry’s paradox naively as meaningful, since it uses the word “definable,” but when you drill into it and apply the tools of mathematical logic and Tarski’s theorem, you realize that it is pulling a fast one. Definability is not definable, and similarly, tweetability is not actually definable in a tweet. This is one way of resolving the paradox of the largest tweetable number.
I hope you enjoyed this account of the paradox of the largest tweetable number, exploring various mind-boggling numbers, which open the door to philosophical aspects of the nature of having small descriptions of huge numbers. I hope to see you again next time!
The Lectures on Infinity are appearing in the lectures-on-infinity tag. The full collection of essays is available on Infinitely More at The Book of Infinity, now available in printed form:
Read more about the paradox of the largest tweetable number in my essay The largest tweetable number, which also appears as chapter 8 of The Book of Infinity.
One way forward is to replace "tweetable" with something we can define, like: strings of length at most 280 on the alphabet {0,1,2,3,4,5,6,7,8,9,+,-,*,/} that are well-formed expressions with the usual meanings attached to the symbols. I think I could work out the largest natural number that can be so expressed, but finding the smallest number that cannot be so expressed feels rather difficult.
It strikes me that there is some point of contact between this paradox and the finitists and/or some intuitionists and/or Reeb. On the one hand, a googolplex - arguably we can all hold the concept of that - the program that emits it - in our heads. So perhaps everyone in the tent agrees that it 'exists'. On the other hand, some arbitrary number 'a bit smaller' for which we lack any concise description: we can't exhibit a program that produces it, and we arguably can't be said to hold a concept of it. So perhaps it doesn't 'exist'?