How to count past infinity

Hey Vsauce! Michael here.

What is the biggest number you can think of? A googol? A googolplex? A milli-million-oplex? Well in reality, the biggest number is 40.

Covering more than 12,000 square metres of Earth, this 40, made out of strategically-planted trees in Russia, is larger than the battalion markers on Signal Hill in Calgary, the 6 found on the Fovant Badges in England—even the mile of pi Brady unrolled on Numberphile. 40 is the biggest number … on Earth in terms of surface area.

But in terms of amount of things, which is normally what we mean by a number being ‘big’, 40 probably isn’t the biggest. For example, there’s 41. Oh and then there’s 42, and 43 … a billion, a trillion; you know, no matter how big of a number you can think of, you could always go higher.

So there is no biggest, last number… except infinity? No. Infinity is not a number. Instead, it’s a kind of number. You need infinite numbers to talk about and compare amounts that are unending, but some unending amounts—some infinities—are literally bigger than others. Let’s visit some of them and count past them.

First things first. When a number refers to how many things there are, it is called a ‘cardinal number’. For example, 4 bananas. 12 flags. 20 dots. 20 is the ‘cardinality’ of this set of dots. Now, two sets have the same cardinality when they contain the same number of things. We can demonstrate this equality by pairing each member of one set one-to-one with each member of the other. Same cardinality, pretty simple.

We use the natural numbers—that is, 0, 1, 2, 3, 4, 5 and so on—as cardinals whenever we talk about how many things there are, but how many natural numbers are there? It can’t be some number in the naturals, because there’ll always be 1 plus that number after it. Instead, there is a unique name for this amount: ‘aleph-null’ (ℵ₀). Aleph is the first letter of the Hebrew alphabet, and aleph-null is the first smallest infinity. It’s how many natural numbers there are. It’s also how many even numbers there are, how many odd numbers there are; it’s also how many rational numbers—that is, fractions—there are. That may sound surprising, since fractions appear more numerous on the number line, but as Cantor showed, there’s a way to arrange every single possible rational such that the naturals can be put into a one-to-one correspondence with them. They have the same cardinality.

The point is, aleph-null is a big amount; bigger than any finite amount. A googol, a googolplex, a googolplex factorial to the power of a googolplex to a googolplex squared times Graham’s number? Aleph-null is bigger. But we can count past it. How? Well, let’s use our old friend, the super-task. If we draw a bunch of lines and make each next line a fraction of the size and a fraction of the distance from each last line, well we can fit an unending number of lines into a finite space. The number of lines here is equal to the number of natural numbers that there are. The two can be matched one-to-one. There’s always a next natural, but there’s also always a next line. Both sets have the cardinality aleph-null.

But what happens when I do this? Now how many lines are there? Aleph-null plus one? No. Unending amounts aren’t like finite amounts. There are still only aleph-null lines here because I can match the naturals one-to-one just like before. I just start here and then continue from the beginning. Clearly the amount of lines hasn’t changed. I can even add two more lines, three more, four more—I always end up with only aleph-null things. I can even add another infinite aleph-null of lines and still not change the quantity. Every even number can pair with these and every odd number with these. There is still a line for every natural.

Another cool way to see these lines don’t add to the total is to show that you can make this same sequence without drawing new lines at all. Just take every other line and move them all together to the end. It’s the same thing.

But hold on a second. This and this may have the same number of things in them, but clearly there’s something different about them, right? I mean, if it’s not how many things they’re made of, what is it? Well, let’s go back to having just one line after an aleph-null-sized collection. What if, instead of matching the naturals one-to-one, we insist on numbering each line according to the order it was drawn in? So we have to start here and number left-to-right. Now, what number does this line get? In the realm of the infinite , labelling things in order is pretty different than counting them. You see, this line doesn’t contribute to the total, but in order to label it according to the order it appeared in, well we need a set of labels of numbers that extends past the naturals. We need ordinal numbers.

The first trans-finite ordinal is omega (ω), the lower-case Greek letter omega. This isn’t a joke or a trick, it’s literally just the next label you’ll need after using up the infinite collection of every single counting number first. If you got ω-th place in a race, that would mean that an infinite number of people finished the race, and then you did. After ω comes ω+1, which doesn’t really look like a number, but it is, just like 2 or 12 or 800. Then comes ω+2, ω+3 … ordinal numbers label things in order. Ordinals aren’t about how many things there are, instead they tell us how those things are arranged—their order type.

The order type of a set is just the first ordinal number not needed to label everything in the set in order. So for finite numbers, cardinality and order type are the same. The order type of all the naturals is ω. The order type of this sequence is ω+1, and now it’s ω+2. No matter how long an arrangement becomes, as long as it’s well-ordered, as long as every part of it contains a beginning element, the whole thing describes a new ordinal number. Always. This will be very important later on.

It should be noted at this point that if you are ever playing a game of who can name the biggest number, and you’re considering saying ‘omega plus one’, you should be careful. Your opponents might require the number you name to be a cardinal that refers to an amount. These numbers refer to the same amount of stuff, just arranged differently. ω+1 isn’t bigger than ω, it just comes after ω.

But aleph-null isn’t the end. Why? Well, because it can be shown that there are infinities bigger than aleph-null that literally contain more things. One of the best ways to do this is with Cantor’s diagonal argument. In my episode on the Banach-Tarski paradox, I used it to show that the number of real numbers is larger than the number of natural numbers. But for the purposes of this video, let’s focus on another thing bigger than aleph-null: the power set of aleph-null.

The power set of a set is the set of all the different subsets you can make from it. For example, from the set of 1 and 2, I can make a set of nothing, or 1, or 2, or 1 and 2. The power set of 1,2,3 is: the empty set, 1, and 2, and 3, and 1 and 2, and 1 and 3, and 2 and 3, and 1,2,3. As you can see, a power set contains many more members than the original set. Two to the power of however many members the original set had, to be exact. So what’s the power set of all the natural?

Well let’s see. Imagine a list of every natural number. Cool. Now the subset of all, say, even numbers would look like this: yes, no , yes, no, yes, no, and so on. The subset of all odd numbers would look like this. Here’s the subset of just 3, 7 and 12. And how about every number—except 5. Or, no number—except 5. Obviously this list of subsets is going to be, well, infinite. But imagine matching them all one-to-one with a natural. If, even then, there’s a way to keep producing new subsets that are clearly not listed anywhere here, we will know that we’ve got a set with more members than there are natural numbers—a bigger infinity than aleph-null.

The way to do this is to start up here in the first subset and just do the opposite of what we see. 0 is a member of this one, so our new set will not contain 0. Next, move diagonally-down to 1’s membership in the second subset. 1 is a member of it so it will not be in our new one. 2 is not in the third subset so it will be in ours, and so on. As you can see, we are describing a subset that will be, by definition, different in at least one way from every single other subset on this aleph-null-sized list. Even if we put this new subset back in, diagonalization can still be done.

The power set of the naturals will always resist a one-to-one correspondence with the naturals. It’s an infinity bigger than aleph-null. Repeated applications of power set will produce sets that can’t be put into one-to-one correspondence with the last, so it’s a great way to quickly produce bigger and bigger infinities. The point is, there are more cardinals after aleph-null. Let’s try to reach them.

Now remember that after ω, ordinals split, and these numbers are no longer cardinals. They don’t refer to a greater amount than the last cardinal we reached—but maybe they can take us to one. Wait … what are we doing? Aleph-null? Omega? Come on, we’ve been using these numbers like there’s no problem, but if at any point down here you can always add one—always—can we really talk about it, this endless process, as a totality, and then follow it with something?

Of course we can. This is math, not science! The things we assume to be true in math are called axioms, and an axiom we come up with isn’t more likely to be true if it better explains or predicts what we observe . Instead, it’s true because we say it is. Its consequences just become what we observe. We are not fitting our theories to some physical universe, whose behaviour and underlying laws would be the same whether we were here or not; we are creating this universe ourselves. If the axioms we declare to be true lead us to contradictions or paradoxes, we can go back and tweak them, or just abandon them altogether, or we can just refuse to allow ourselves to do the things that cause the paradoxes. That’s it. What’s fascinating though is that in making sure the axioms we accept don’t lead to problems, we’ve made math into something that is, as the saying goes, ‘unreasonably effective in the Natural Sciences’. So to what extent we’re inventing all of this, or discovering it—it’s hard to say . All we have to do to get ω is say ‘let there be omega’, and it will be good.

That’s what Ernest Zermelo did in 1908 when he included the Axiom of Infinity in his list of axioms for doing stuff in math. The axiom of infinity is simply the declaration that one infinite set exists—the set of all natural numbers. If you refuse to accept it, that’s fine—that makes you a finitist, one who believes only finite things exist. But if you accept it, as most mathematicians do, you can go pretty far—past these, and through these … eventually we get to ω+ω, except we’ve reached another ceiling. Going all the way out to ω+ ω would be to create another infinite set, and the axiom of infinity only guarantees that this one exists.

Are we going to have to add a new axiom every time we describe aleph-null-more numbers? No. The Axiom of Replacement can help us here. This assumption states that if you take a set—like, say, the set of all natural numbers—and replace each element with something else—like, say, bananas—what you’re left with is also a set. That sounds simple, but it’s incredibly useful. Try this: take every ordinal up to ω and then, instead of bananas, put ‘ω+’ in front of each. Now we’ve reached ω+ω, or ω×2. Using replacement we can make jumps of any size we want, so long as we only use numbers we’ve already achieved. We can replace every ordinal up to ω with omega times it, to reach ω×ω , ω². We’re cooking now! The axiom of replacement allows us to construct new ordinals without end. Eventually we get to ω to the ω to the ω to the ω to the ω… and we run out of standard mathematical notation. No problem! This is just called ‘epsilon-naught’ (ε₀), and we continue from here.

But now think about all of these ordinals. All the different ways to arrange aleph-null things. Well these are well-ordered, so they have an order type—some ordinal that comes after all of them. In this case, that ordinal is called ‘omega-one’ (ω1). Now because, by definition, ω₁ comes after every single order type or aleph-null things, it must describe an arrangement of literally more stuff than the last aleph. I mean, if it didn’t, it would be somewhere in here—but it’s not. The cardinal number describing the amount of things used to make an arrangement with order type ω₁ is ‘aleph-one’ (ℵ₁).

It’s not known where the power set of the naturals falls on this line. It can’t be between these cardinals because, well, there aren’t cardinals between them. It could be equal to aleph-one—that belief is called the Continuum Hypothesis. But it could also be larger; we just don’t know. The Continuum Hypothesis, by the way, is probably the greatest unanswered question in this entire subject, and today, in this video, I will not be solving it—but I will be going higher and higher, to bigger and bigger infinities.

Now using the replacement axiom, we can take any ordinal we already reached—like, say, ω—and jump from aleph to aleph all the way out to aleph-omega. Or heck, why not use a bigger ordinal like ω² to construct aleph-omega-squared? Aleph-omega-omega-omega-omega-omega-omega-o… Our notation only allows me to add countably-many omegas here, but replacement doesn’t care about whether or not I have a way to write the numbers it reaches. Wherever I land will be a place of even bigger numbers, allowing me to make even bigger and more numerous jumps than before. The whole thing is a wildly-accelerating feedback loop of embiggening. We can keep going like this, reaching bigger and bigger infinities from below.

Replacement, and repeated power sets which may or may not line up with the alephs, can keep our climb going forever. So clearly there’s nothing beyond them, right? Not so fast. That’s what we said about getting past the finite to omega. Why not accept as an axiom that there exists some next number so big, no amount of replacement or power-setting on anything smaller could ever get you there. Such a number is called an ‘inaccessible cardinal’ because you can’t reach it from below.

Now interestingly, within the numbers we’ve already reached, a shadow of such a number can be found: aleph-null. You can’t reach this number from below either. All numbers less than it are finite, and a finite number of finite numbers can’t be added, multiplied, exponentiated, replaced with finite jumps a finite number of times or even power set a finite number of times to give you anything but another finite amount. Sure, the power set of a milli-million to a googolplex to a googolplex to a googolplex is really big—but it’s still just finite. Not even close to aleph-null, the first smallest infinity. For this reason, aleph-null is often considered an inaccessible number. Some authors don’t do this though, saying an inaccessible must also be uncountable, which, okay, makes sense—I mean, we’ve already accessed aleph-null, but remember the only way we could is by straight-up declaring its existence axiomatically. We will have to do the same for inaccessible cardinals.

It’s really hard to get across just how unfathomable the size of an inaccessible cardinal is. I’ll just leave it at this: the conceptual jump from nothing to the first infinity is like the jump from the first infinity to an inaccessible. Set theorists have described numbers bigger than inaccessibles, each one requiring a new large-cardinal axiom asserting its existence, expanding the height of our universe of numbers. Will there ever come a point where we devise an axiom implying the existence of so many things that it implies contradictory things? Will we someday answer the Continuum Hypothesis? Maybe not, but there are promising directions, and until then, the amazing fact remains that many of these infinities—perhaps all of them—are so big, it’s not exactly clear whether they even truly exist, or could be shown to, in the physical universe. If they do, if one day physics finds a use for them, that’s great—but if not, that’s great too. That would mean that we have, with this brain, a tiny thing, a septillion times smaller than the tiny planet it lives on, discovered something true outside of the physical realm. Something that applies to the real world, but is also strong enough to go further, past what even the universe itself can contain, or show us, or be.

And as always, thanks for watching.

Another interesting fact about trans-finite ordinals is that arithmetic with them is a little bit different. Normally 2+1 is the same as 1+2, but ω+1 is not the same as 1+ω. One plus omega is actually just omega. Think about them as order types: one thing placed before omega just uses up all the naturals and leaves us with order type omega. One thing placed after omega requires every natural number and then omega, leaving us with omega plus one as the order type.