scary numbers

twenty-three! ah!-ah!-ah!

Linked by Truthshaper earlier today:

Some people are posting an ask meme where one of the questions is “a number that weirds you out?” and I’ve been having intrusive thoughts about it.

Almost all numbers really freak me out a lot, actually, and I just have to not think about them. Most of them I can’t write out here because they have an INFINITE FUCKING DESCRIPTION LENGTH. But for something short to write in a Tumblr post, an unsettling number is (pi + sqrt(2)) * a googolplex / 7. Yikes. Pretty much every part of that expression is horrifying? 7 is okay. 1/7 is pushing it. I’m embarrassed to have brought a googolplex into this, who even thinks of numbers that large? But most integers are much, much larger than that. And integers are the nice ones? Numbers were a mistake.

As an expert numerologist¹¹ Literally anyone can claim this title. My math degree and nerdiness helps, because it means I know about weirder numbers, but it’s not required.

…Or like, okay, maybe Celene somehow not knowing what’s special about 69 was enough of an accomplishment to actually make her meaningfully underqualified. Congratulations Celene., I know about a lot of scary numbers. For example, four and thirteen are famously scary. Obviously π is scary, because we should be using τ. According to the traditional mathematician’s folk tale, Hippasus of Metapontum was drowned by the Pythagoreans for proving the irrationality of √2²² There isn’t very much solid evidence anything like that actually happened.. And really, I think maybe Ms. McFly might be underrating seven—Doctor W.D. Gaster might be a little spooky, but really he’s just been a slightly ineffable pal so far. I’m kinda more spooked by the Angel³³ If you insist then I suppose you can think of six and seven in terms of Christianity instead of in terms of Deltarune. Deltarune did, of course, sorta inherit it from Christianity. On the other hand, Christianity won’t make nine spooky, but Undertale will..

Now, look. I’m not totally sure if my eight year old sister already knows what π or √2 are, and even if she’s heard of π I’m not sure quite how well she actually understands it. (Is it better thought of as the ratio of a circle’s circumference to its diameter, or as the area of a unit circle, or maybe as something wacky like half the period of one of those functions which are their own fourth derivative? Could she give any of those definitions?) But like, I could give something resembling a satisfying introductory explanation of what these numbers are. I can write down their decimal expansions. They’re comprehensible, even (to some extent) to a bright eight year old.

The point is, they’re way too flipping exoteric. Where are the real scary ones. I want the good stuff.

Set theory is scary

When I first saw the question of scary numbers posed, I quickly thought of Chaitin’s Constant⁴⁴ Arguably “constant” is a misnomer, because Ω is defined in terms of a specified prefix-free Universal Turing Machine. But it’s scary regardless of your choice of Universal Turing Machine. Ω. And, you know, it is kind of scary. Unlike the great majority of numbers you’ll ever see and just like almost every⁵⁵ In the technical sense, the one where “almost all” is kind of an understatement. You might as well think that it’s all of them, to the extent that quantity matters and not how likely you are to come across them. The set of computable real numbers isn't just of measure zero, it’s countable. real number, Ω is uncomputable—there is no terminating algorithm which can approximate it to a desired degree of precision. And you know, it’s kinda close to that Gödel/Turing cluster of ideas related to how it’s very difficult for anything to have a total and perfect understanding of itself⁶⁶ Unless you somehow handicap the thing’s ability to use that understanding to produce a contradiction..

But I’m pretty sure everyone knows about that one already? You, dear reader, have surely heard of Chaitin’s constant. There’s like, a Numberphile video on it featuring Matt Parker, whose most-viewed video (I will never stop being astounded) I am in some small way partially responsible for. You surely must already know about Chaitin’s constant. I can assert these things, because I am not a geologist.

So the actual answer I gave in Discord was 0#.

In the mathematical discipline of set theory, 0# (zero sharp) is the set of true formulae about indiscernibles and order-indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the natural numbers (using Gödel numbering), or as a subset of the hereditarily finite sets, or as a real number.

It’s kind of a set, but sometimes set theorists have reason to encode it as a real number, so I can get away with calling it a number. But as I've given it more thought, I’m not actually sure 0# is very scary. Like, its existence is dependent on a large cardinal axiom, and I think when I was younger the idea of one of the good old real numbers simply not existing under certain conditions was very unsettling. But I am now older and wiser in the ways of mathematics and perhaps a tad more jaded, and I can observe that there's not like, some specific real number that doesn’t get added to the field of real numbers unless you have the right large cardinal axioms. It’s just that unless some technical criteria are met, you can't define a particular set, and to identify one of the preëxisting reals as encoding a set you do need to first be able to define that set. This isn’t that scary.

Wait, did someone say large cardinal?

Large Cardinals

Cardinal Johannes Baptist Franzelin was probably not very large, if I were to guess.

In 1876, despite his protests, he was raised to the cardinalate⁷⁷ Known among modern mathematicians as a cardinality. by Pope Pius IX, and participated in the papal conclave of 1878 which elected Pope Leo XIII⁸⁸ After whom our current Pope Leo XIV takes his name, at least in part, probably. I imagine he did some cool things but also he was mean to the Anglicans which I don’t like.. Though of delicate heath, the appointment made little change in his scrupulously simple lifestyle. As a cardinal, his sole departure from strict adherence to the Jesuit rule was to omit the daily recreation.

The daily recreation seems kind of important to me⁹⁹ For it is often good to follow in the footsteps of one’s creator.. I think one oughtn’t cut that out of their schedule. Though I suppose you could merely casually adhere to a daily recreation requirement, rather than strictly adhere to it, and that might not be so bad. The sabbath is made for man, after all.

The works of Cardinal Franzelin discuss cardinal numbers like Uno (one), as well as whatever sort of number(?) I’m supposed to call Trino (triune)¹⁰¹⁰ Maybe it's the sort of number that is an element of Z/2Z. This would clear up some confusing theological points..

Anyways, Cardinal Franzelin is relevant to this story because he acknowledged the validity of Georg Cantor’s arguments establishing the existence of multiple distinct cardinalities of infinite sets, and thus cardinal numbers were named after him¹¹¹¹ Or at least, I’m guessing that’s what happened. I have no plans to actually check, because then I would have to admit that this guess wasn’t very likely.. The Jesuits win yet again¹²¹² Other Catholics weren’t all so impressive, which is why they didn’t get a type of number named after them., outperforming well-known mathematicians like Henri Poincaré and Leopold Kronecker¹³, who obstructed Cantor’s work and deemed him a “corrupter of youth”.

Anyone who’s read Unsong knows how Cantor felt about his study of the infinite:

Cantor began talking about how his discoveries were direct and personal revelations from God, who wished him to preach the gospel of infinity so that an infinite Deity could be better understood. He posited an Absolute Infinite, beyond all the forms of infinity he had discovered, with which God might be identified. Finally, he declared:

“I have never proceeded from any Genus supremum of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no Genus supremum of the actual infinite. What surpasses all that is finite and transfinite is no Genus; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God.”¹⁴¹⁴ The Actus Purissimus (purest act)—or that which surpasses the finite and transfinite—is known to the modern mathematician as the proper class, a term which refers to any class (collection of sets) that is too large to itself be a set. And to be too large to be a set you really have to be just incredibly fucking large like jeez.

What I hope to convey before we start talking about the mathematics is that these numbers were like, a big deal. It’s like the 1800s version of when the Pythagoreans found out about the square root of two. Some of these mathematical objects are cursèd eldritch things. In that, they’re sort of like the quaternions, of which Lord Kelvin said:

Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.

Except, of course, that the quaternions stop being evil once you understand that their so-called vector part is in fact a bivector part.

So. A cardinal number is an equivalence class¹⁵¹⁵ If I was supposed to care when my equivalence classes turn out to be proper classes, they should’ve been called equivalence sets. It’s not like you can encode a function whose domain is the cardinal numbers as a set of ordered pairs anyways, the class of cardinal numbers will be proper any way you slice it. Also this is my blog and I didn’t sleep last night and I don’t need to worry about class propriety if I don’t want to. Choose a representative or something if you want to. of sets, where the equivalence relation is simply isomorphism. This definition will be very helpful if you somehow ended up learning category theory without ever learning about the cardinality of sets, which you definitely didn’t, how would that ever happen to literally anyone. But I claim that it is sort of the right way to think about cardinality if you’re the sort of person who is comfortable with the concept of an isomorphism. If you have a set, just a pure set with no additional structure, then identifying its cardinality is sort of the same thing as looking at a mug and noticing that it’s the same thing as a donut. Cardinality is the one and only meaningful property that a set has on its own—any other property a set has is a result of caring about what specific elements it has, or about something else that counts as “extra structure” according to category theorists.

Usually, when explaining to the lay person, a mathematician will say something like “two sets have the same cardinality if they can be placed into one-to-one correspondence.” Personally, I find the mathematician’s terminology of bijection more natural and intuitive than “one-to-one correspondence”—whoever decided to also call injective functions “one-to-one functions” sort of ruined it—but your mileage may vary. If you imagine pairing off each element of set A with its own unique element of set B, and then not having any elements of B left over when you’re done going through the elements of A, that’s a bijection. Also, a function is bijective if and only if it has an inverse function.

So, all the natural numbers are basically cardinalities. Any two sets with four elements can be placed into bijection with each other. Any two sets with thirteen elements can be placed into bijection with each other. A set of four elements cannot be placed into bijection with a set of thirteen elements. It’s literally just the pigeonhole principle.

What Cantor figured out is that this gets really messed up for infinite sets. You see, you might at first think that there are fewer even integers than there are integers. Maybe there are half as many? But, insofar as we are talking about cardinality—I mean, maybe it’s not wrong to say that there are half as many, it’s not like ∞/2 is always less than ∞. But the even integers don’t have a smaller cardinality than the integers as a whole. Multiplying by two is a bijection between the integers and the even integers. Dividing by two is its inverse and takes you from the even integers to the integers as a whole. They’re in one-to-one correspondence.

So if you were wrong that there are fewer even integers than there are integers… you might start to get the idea that all infinite sets have the same cardinality. This is not, in fact, true. You’ve been double bamboozled. The usual line of reasoning people talk about is Cantor’s diagonal argument, which isn’t actually his first proof that the reals and naturals don’t have the same cardinality—he snuck that one past the censors by slipping it into a paper about putting the algebraic numbers (numbers which are roots of some polynomial) into bijection with the naturals. But it looks kind of complicated and topological and the diagonal argument is neat.

Basically, suppose you have some function f from N to R, putatively a bijection. Then consider a real which differs from f(0) in the zeroth digit past the decimal point, f(1) in the first digit, f(2) in the second digit, and so on ad infinitum¹⁶¹⁶ If you want to be careful, then you’re supposed to care about cases with infinite trailing nines.. The real you are currently considering differs from f(n) for all n, it is not in the image of f, and so f cannot be a bijection.

By showing that the reals have a larger cardinality than the naturals, Cantor pretty much invented set theory. But more importantly for our purposes, he discovered that cardinalities aren’t just like, boring finite numbers or straight up normal infinity. No, some transfinite cardinals are large compared to others. Eventually, Cantor would dub the cardinality of the naturals, known also as “countable infinity”, as א‎₁ (aleph-one), because countable infinity is the first infinite cardinality¹⁷. But then other mathematicians decided that they like zero indexing more, so it’s‎ א‎₀ (aleph-null) now. On their behalf, I announce that Kepe’s welcome. The aleph numbers form a sequence: א‎₁ is the smallest cardinal greater than א‎₀, and then comes א‎₂, and so on.

On the other hand: the cardinality of the reals, known also as the cardinality of the continuum, would come to be represented with the symbol 𝔠, though it wasn’t dubbed so by Cantor himself. And it is, I would venture, rather a lot larger than א‎₀. It’s a pretty large cardinal. But of course, “large cardinal axioms” as they’re called—which are very scary, why do they form such a nice hierarchy like that, what did the gods of mathematics do, I haven’t yet gathered even the slightest clue—usually concern themselves with, you know, cardinalities you can’t show to exist just with the axiom of infinity¹⁸¹⁸ The axiom of infinity is arguably itself a large cardinal axiom. ‎Like, really, א‎₀ is rather a lot larger than any of the naturals. And some of the naturals get scarily large already. And y’know, it is what they call a strong limit cardinal.

ALSO, if mathematicians weren’t COWARDS, then everyone would agree that the axiom of empty set is the first and most important large cardinal axiom. and the axiom of power set. So 𝔠 is merely the first cardinal proven to be larger than some other infinite cardinal, and you will confuse mathematicians if you call it a Large Cardinal.

Nonetheless, 𝔠 might well be the largest cardinal you’ve encountered, unless you’ve had reason to work with something like the set of functions from R to R, which has cardinality 2^𝔠. And really for the cardinality of that set to actually be relevant you need to be trying to like, do measure theory on it or something. Which is certainly a thing people do—but if you’re just working with like, a Hilbert space or something, then no, that (usually) just has the same cardinality as R.

Okay but like how many real numbers are there

It turns out that the reals can be placed in bijection with the set of subsets of the natural numbers¹⁹¹⁹ Okay, so, consider just the unit interval. If you have some set containing naturals {p,q,r}, then consider the real number in the unit interval written in binary that has a 1 at the pᵗʰ, qᵗʰ, and rᵗʰ positions to the right of the decimal point. This will cover the closed unit interval. Now pick your favorite bijection between the closed unit interval and R.

…the logit function is a bijection between the open unit interval and R? I don’t feel like fixing this. I am trusting you to not run away and conclude that 𝔠 and 2^א‎₀ must not be equal on the basis that actually 2^א‎₀ - 2 = 𝔠. I believe in you. You can avoid forming that belief.

I’m sorry that there are no good bijections between the closed unit interval and R, blame topology, the problem is that they aren’t homeomorphic, you’ll need some garbage discontinuous thing. If you don’t have a favorite bijection it’s fine to just pick one you hate slightly less than the others instead, no one will judge you., which is also known as the power set of the naturals. It’s called a “power set” because it’s the set you get when you raise 2 to the Nᵗʰ power:

which works because X^Y is the set of functions from Y to X²⁰²⁰ The order has to be reversed because we would like 2³ to be 8 and not 9. For example., and a subset of N is pretty much just a function from N to 2²¹²¹ As in, a subset of a set X can be determined by a function which tells you, for each element xᵢ of X, whether xᵢ is in the set or not in the set. And {in the subset, not in the subset} has cardinality 2 and is thus isomorphic to 2. Do you follow? You follow.

Really, this is just a simple generalization of the notation that gives us R², which is obviously the set of functions from 2 to R.

…What do you mean that wasn’t what you thought R² meant? “Cartesian nᵗʰ power”? Well, Wikipedia is telling me that the Cartesian nᵗʰ power of a set X is just the set of functions from n to X. So who’s laughing now..

So you know, in a sense it’s pretty straightforward. I think the real numbers are like, a set people basically understand, to some extent. It’s the uncountable set people work with the most. Lots of other sets are of the same cardinality: the vector spaces people are most likely to work with, the set of… I don’t know, analytic functions? The set of strings in any alphabet with countably many characters? If someone talks about “uncountable infinity” as if it’s a specific number rather than “all cardinalities greater than א‎₀”, they probably mean 𝔠.

So, our prophet of infinite cardinals²²²² Well he also gave us the infinite ordinals, like ω, but let’s not get sidetracked now. Even though they are also scary. There are a deeply ludicrous number of ordinals between ω and the first uncountable ordinal, ω₁.

Or, you know, in a sense there’s at most 𝔠, and far far less than that if the continuum hypothesis is false! But something about the way that the ordinals are arranged all in a row, with a whole + 1 added between each one—it just makes them feel much vaster.

So like, ω isn’t very scary. Even ε₀ isn’t all that bad. But the Church-Kleene ordinal? Now that one’s terrifying. It’ll give you vertigo if you think about it too hard.

But like I said, let’s not get sidetracked. had a conjecture:

which is now known as the “continuum hypothesis”. Cantor did not have access to the ZFC axioms yet, so the tools of modern set theory were not available and the ultimate premises a proof either way would have to rely on were not crisply laid out. He thought 𝔠 was א‎₁, that it was the next smallest cardinal after א‎₀. He could never prove it. Nor could he ever disprove it. How many reals even are there?

Or is it more like, how many different weird screwed up sorts of sets that fail to have bijections to either N or R can we smoosh in between א‎₀ and 𝔠? I think that’s maybe closer to what's going on. But I’m not totally sure, set theory gets weird when you start stepping between lots of different models and I cannot follow all of it.

At the turn of the 20th Century, David Hilbert listed as the first of his famous twenty-three²³²³ I propose the theory that the Illuminati convinced David Hilbert not to include his 24th problem. problems:

Cantor’s problem of the cardinal number of the continuum.

Probably most people who have heard of the continuum hypothesis know roughly how this goes. In 1938, Kurt Gödel proved the (relative) consistency of the continuum hypothesis and the axiom of choice with ZF, by showing that they hold in the constructible universe L, which is a model of ZF—that is, if you only consider sets that are elements of L to exist, then the collection of sets you have access to satisfies the ZF axioms.

And then, the modern set theoretic technique of forcing, which I don’t really understand on any level deeper than “it’s a way to make ‘bigger’ models of set theory from littler ones”, is first introduced by Paul Cohen in 1963 to show that the negation of the axiom of choice and the continuum hypothesis are also both (relatively) consistent with the ZF axioms. Combined with Gödel’s result, we can conclude that the continuum hypothesis is independent of ZFC, which is sort of like a resolution to the question except for the part where it means we don’t get to actually settle on an answer. Cohen probably did deserve his Fields medal nonetheless.

And of course it turns out that it’s compatible with ZFC for 𝔠 to be pretty much any old cardinal you want, with some small restrictions. There could be just an absolutely monstrous quantity of cardinals sitting between א‎₀ and 𝔠. And yet, every single specific subset of R you ever actually meet is going to have a cardinality of either א‎₀ or 𝔠. Where are the in-between cardinalities supposed to fit? In hyperabstract conceptual set theory land, I have to suppose. You could suppose the continuum hypothesis is false and then use the axiom of choice to conjure up an injective function from ω₁ to R, and then tell me about how it has an image of cardinality א‎₁ < 𝔠, but… well, I don’t know, I suppose that example doesn’t speak to me any less ambiguously than the idea of well-ordering R by applying the axiom of choice does.

There is far from a consensus, but it seems to me that many of the set theorists with opinions on the matter are skeptical of the continuum hypothesis. It’s actually not too uncommon a stance that the best candidate for 𝔠 is specifically א‎₂, though I can hardly wrap my head around why. And I would be remiss not to point out the set-theoretic multiverse view from the wonderful Joel David Hamkins of infinite chess fame—or at least, he’s famous to me in particular for his infinite chess work!

This all is why 𝔠 is scary. Every mathematician has to encounter it, sets of that size are pretty inescapable unless you only ever work with finite algebraic structures or something. Or I guess you could be a computer scientist and limit yourself to the 2^64ish floats. But you know, it’s like, the reals. Yes, there are unfathomably many of them, oodles of numbers squeezed betwixt each and every rational, but like, eh.

And the set theorists CANNOT FIGURE OUT HOW LARGE A NUMBER IT IS. They just don’t know! It’s independent of ZFC! How many real numbers there are shouldn’t be something you can just change. They’re the reals. The complete Archimedean ordered field. You can’t just make more of them. But sometimes set theorists, at the very least, say things that sound a lot like they’re forcing new never before seen real numbers out of their hats. This is not how counting things is supposed to work, and I am scared—I suppose this is what happens when one stares down the uncountable.

If Cantor’s work and its elaboration by his heirs are meant to be a gospel of infinity such that an infinite Divine might be better understood, as was perhaps Cantor’s belief… what is one ever supposed to make of this? Even with the axiom of choice, even with whatever large cardinal axiom you like, we cannot figure out definitively how many real numbers there are. It’s almost comical.

The only way this could improve someone’s grasp of the Divine is if they think they understand the infinite and desperately need to get humbled. Or maybe trying to understand a multiversal view of set theory could teach someone to cope with uncertainty. Or maybe if you stare at the set theory long enough, you’ll see whatever Gödel and Cohen saw and fill in enough of the gaps to convince the rest of us that in the platonic ideal version of set theory there are precisely א‎₂ real numbers. I might become a little less scared if that happened.

Also, I think it’s a little spooky that 196883 + 1 = 196884. It’s a remarkably weird coincidence and I don’t really have any clue what’s up with it. It probably doesn’t help that I hardly know a thing about elliptic curves or modular functions or whatever the j thing is. It just gets spooky when mathematicians start calling things “the voice of God,” with, as far as I can guess, no more irony than Cantor had in anything he said.