Defining Infinity

[MUSIC PLAYING] Set theory is supposed to be a foundation of all of mathematics, but how does it handle infinity?

[MUSIC PLAYING] Set theory arose, at least in part, to get a grip on infinity.

Early naive versions were beset by apparent paradoxes and were superseded later by axiomatic versions that used formal rules to demarcate legal mathematical statements from gibberish.

Today, I want to explore how set theory defines different sorts of infinities and how it avoids paradoxes like the one I introduced in my previous episode.

I'll have to gloss over or omit many formal details, but I'll do my best to represent the overarching ideas faithfully.

Now, anything you hear that sounds like unfamiliar background is in these other videos that you may want to go watch.

I will rely on them and go somewhat quickly, so you may need to watch this episode a few times with pencil and paper before it clicks.

Here we go.

The most common axiomatization of set theory is Zermelo-Fraenkel, or ZFC, which, interestingly, doesn't actually define set.

Instead, its axioms just circumscribe what you're allowed to say exists.

And "set" is just ZFC speak for "thing that exists."

Now, characterizing infinities in set theory begins with the natural numbers, which in ZFC are just these sets.

This is von Neumann's definition of the naturals, which I covered earlier in this episode, in case you need a refresher.

Now, technically, numbers defined this way are ordinals, since they themselves are ordered-- i.e., a comes before b if a, as a set, is an element of b.

They can be used to index element positions in any other ordered set A.

Just map A's elements one-to-one, or bijectively, to a subset of successive naturals that starts at zero in a way that matches the ordering on A.

So in our example, the ordered set A maps to 0, 1, 2, which, you'll recall, is the ordinal number 3.

That's why we call 3 the order type of this ordering on set A.

So what about cardinal numbers that quantify how many?

Well, in ZFC, the cardinals piggyback on ordinals as follows-- the cardinality of set A is just the lowest ordinal, that, when thought of as a raw set with no order on it, has a bijection with A.

Why lowest?

Isn't there only one such ordinal?

Well if A is finite, yes, but if A is infinite, no.

More on this later.

For now, let's just note that two sets related by a bijection will have the same cardinality under this definition.

Also, cardinals are ordinals under the hood, and ordinals are sets, which means that the size of a set is itself a set.

And since it's a set, a size has a size, namely, itself.

For instance, the cardinality of the set 5 is the set 5.

Now let's see how these definitions accommodate infinite sets in ZFC.

The ZFC axioms let you assemble lots of infinite sets starting from the von Neumann naturals, and the axiom of infinity declares the naturals together to be a set.

We actually need that axiom.

Without it, you can derive that each individual natural exists, but not, interestingly, that the whole collection exists.

So infinity has to be introduced manually.

OK. Now imagine an ordered set x, with x0, x1, x2, et cetera, that you need all the naturals to index.

Just as the set 0, 1, 2 is 3, we can say that the set, 0, 1, 2, dot, dot, dot-- all the naturals-- is the ordinal after all finite ordinals.

Now, as a raw set, it's identical to the set n of all naturals, but when we interpret it as an ordinal or an order type, we label it omega.

Now, a w appended to the end of the finite ordered set xyz would end up having indexed 3, and the whole new set would have an order type 0, 1, 2, 3, i.e., 4.

Likewise, appending Mario to the end of the infinite ordered set of x's would give Mario a positional index omega, and the new resulting set would have order type 0,1,2, dot, dot, dot, omega.

We call this omega plus 1, since it's the next ordinal after omega.

Appending even more elements produces sets with order types like omega plus 2, omega plus 3, and so on.

In fact, ZFC lets you keep consolidating mutually-disconnected infinite sequences like this into a single set to produce something like a two-dimensional array that you're still indexing with a single counter that you allow to advance to the next row, even if the current row continues to the right forever.

That's actually how infinite ordinals end up indexing position, by encoding which subsequence an element is in and where it lies along that subsequence.

Now, writing out those ordinals can get cumbersome quickly, so we borrow familiar shorthands from ordinary multiplication and exponentiation just to make the notation more compact.

Now for infinite cardinals.

What's the cardinality of the naturals?

Well, remember our definition-- the lowest ordinal that's in bijection with n when you ignore ordering is omega, since it's identical to n as a raw set.

So the answer is omega.

Except as a cardinal, we don't call it that.

We call it aleph0.

Remember, as raw sets, omega and aleph0-- and n for that matter-- are all the same object.

The different monikers just reflect different usages-- position, size, or raw set.

Now check out these two ordered sets that have order types 2 omega and 3 omega.

I can actually resequence their elements like this to give each set an order type omega.

So unlike finite sets, it appears infinite sets can support multiple order types.

The lowest ordinal in bijection with these particular sets looks like omega, which means both of them have cardinality aleph0.

Now, you can resequence lots of these sets to put them in bijection with omega, which means all of them will have cardinality aleph0.

To get larger cardinalities, you would have to lose bijection with omega, and the resulting monstrosity of a set has to have a higher order type, which we denote omega1.

And it has a corresponding higher cardinality that we denote aleph1.

Again, for the record, omega1 and aleph1 are the same raw set.

The different names just reflect whether we're viewing it as an ordinal or as a cardinal.

And you can keep on adding ordinals and cardinals like this, building a sequence.

Vsauce has covered the process in this other video, so I'll move on.

Now, what about that cascade of cardinalities of powersets that I discussed in my previous episode?

Where do they fall in this sequence?

That's undetermined, because that cascade forms a logically-separate sequence of cardinalities called the Beth numbers.

Now, Beth0 and aleph0 are clearly the same because of how they're constructed, and Cantor proved that the reals have cardinality Beth1.

How does that compare to aleph1?

If you stipulate that Beth1 and aleph1 are equal, that's called the continuum hypothesis.

If you stipulate that all the Beths and alephs match up, that's the generalized continuum hypothesis.

And shockingly, ZFC is consistent both with accepting and rejecting either hypothesis.

You can see Kelsey's hierarchy episode for details.

So to summarize, ordinals are sets whose elements are all previous ordinals.

Every ordering of a set A has an associated ordinal, whose elements index A's elements.

Every set A also has a cardinality.

That's the lowest ordinal that, when taken as a raw set, has a bijection with A.

And finally, since ordinals and cardinals are sets, they themselves have both order types and sizes.

Recently, I posed an infinity puzzle.

Namely, if you take all the cardinals and put them in a bag, what's the cardinality of that bag?

We ended up with a paradoxical answer, that the size of that bag is bigger than itself.

Why did that happen?

Because at least in ZFC, there is no such bag.

Remember, in ZFC, only sets have sizes, because only sets are things.

But the proposition, there exists a bag of all cardinals, turns out not to be derivable from the ZFC axioms.

So that bag is not a set and its size is undefined.

So in ZFC, there isn't actually an infinity of infinities, at least not if you mean, how many?

All infinities taken together simply doesn't have a size, and asking for one is like asking the blood type of a donut.

It doesn't sound like gibberish, but it is.

A similar thing happens with the bag of all ordinals.

Loosely speaking, if that were a set, you could show that its order type would have to be greater than itself.

This is called the Burali-Forti paradox.

You can look it up.

Now, you can still refer to all cardinals or ordinals.

I mean, we just did.

You just can't do it from within ZFC, which is the problem.

Now, there is an extension of ZFC called NBG that actually allows for two different types of entities-- sets, which can both have and be members of things, and proper classes, which can only have members.

Now, the bags of all infinities turn out to be proper classes in NBG, so there are legitimate things that you can refer to.

But even in NBG, it only sets that end up having cardinalities.

Thus, an infinity of infinities is still gobbledegook.

Notice a strange self-limiting consequence of the original axiom of infinity.

We needed it to talk formally about infinite collections in the first place and then ended up spawning a bounty of different varieties of infinities.

But it also rendered that bounty in its entirety somehow indescribable, even though we can describe every individual infinite item in it.

Something along these lines seems to afflict axiomatic systems quite generally, and maybe we'll talk about it another time.

For now, look around for the Subscribe button, hit it, and turn on notifications so you can see when we do.

Until then, I'll see you.