What are Numbers Made of?

In the physical world, many seemingly basic things turn out to be built from even more basic things, like molecules being made of atoms, or atoms being made of protons, neutrons, and electrons.

But what about numbers?

What are they made of?

The natural numbers form the basis of the rest of the number system.

Last time, we asked whether you can describe the essence of the natural numbers without circularly evoking numerical concepts in that description.

Now, such a description might read as follows, specify a starting point, or a seed, that we arbitrarily call 0, and then include the next thing, and the next thing after that, and so on.

That sounds innocuous, but the tricky part is capturing what next means without inadvertently or indirectly referring to numbers.

Tricky, mind you, but not impossible via something called the Peano axioms.

Now, if you haven't seen that last episode, you should pause me now and go catch up on how that works, or today's episode may be hard to follow.

But here's the executive summary, just in case.

Imagine a function s, any function, whose inputs and outputs both come from some non-empty set, n, any set, as long as n and s are subject to the following conditions.

1, different inputs to s always yield different outputs, i.e.

the function s is 1 to 1.

Second, there's an element in n that is never the output of s. And third, n is the leanest, or minimal, non-empty set on which you can even define a function s that works like this.

Any set n and function s that meet these conditions together will behave, respectively, like the natural numbers and the operation next, or successor.

You can even define operations that fully mimic run of the mill addition and multiplication, in terms of any suitable function s, regardless of the details of how s works under the hood.

In this sense, the Peano axioms distill number hood down to its bare essentials.

But let's remember Kelsey's earlier episode about the 19th and 20th century efforts to describe all of math in terms only of sets and operations on sets, as codified by the axioms of Zermelo-Fraenkel set theory.

The implication there was that the natural numbers don't have to be viewed as indivisible entities, that even they can somehow be synthesized from sets.

Well today, we'll demonstrate a couple of different ways to do exactly that, both of which will follo wa very similar logical arc.

Remember, the Peano axioms are agnostic about both the nature of the objects that your calling numbers and about the mechanical details of the successor function s. Now, in set theory, the objects we have available to us to play with are sets.

So we're going to assemble a suitable mega set n, whose elements are themselves sets, along with the suitable function s that eats and spits out sets, such that n and s together satisfy all the Peano axioms.

In doing so, we will have built something that we can legitimately call natural numbers.

Now, before we start, some disclaimers.

I will not address nuances, like which aspects of these constructions rely on which specific axioms of ZF set theory.

And I won't even mention technical issues associated with so-called first order versus second order logical formulations of the Peano axioms or of these constructions.

That's not my point.

My objective today is simply to show, particularly for those who have never seen it before, that it's even possible to reduce something as seemingly basic as the counting numbers to even more basic ingredients.

Let's start with Zermelo's construction, or what I like to call the Russian matryoshka model of the numbers.

Here's the operation that Zermelo used as his function, s. You take a set x as input and then you output a set y that contains that set x as its only element.

Or if you want to visualize every set as a Russian doll that might have some stuff inside, then s is the operation, stick the input doll inside of a new hollow doll.

Not the contents of the input doll, mind you, just the input doll itself.

In other words, if the input were a doll that has three pieces of candy in it, then the output doll wouldn't have three pieces of candy inside of it.

It would only have one thing inside it, namely the original doll.

This is analogous to how nested arrays, or lists, work in most programming languages.

Asking what's inside a given container means peeling off only the outermost layer of its onion, without peeling any layers off any additional onions that you happen to find inside that onion.

Anyway, now let's unmask the elements of n. Zermelo starts with the empty set, or if you will, an empty Russian doll.

If you apply s to that doll, you get a new doll whose only element is the original empty doll.

If you apply s again, you get a doll that contains the immediately prior doll, and so forth.

OK, now let's aggregate all of these successively more nested dolls into a single collection, n. This collection n, together with the dollify function s, satisfy Peano's axioms together.

If you dollify any object in this collection, you always get something else that's in the collection.

If you dollify two different starting dolls, you get two different final dolls, and the empty doll isn't of dollifying anything.

So the empty doll, which is just a visualization of the empty set, plays the role for us of 0, and the doll containing 0 plays the role of 1.

And the doll containing 1 plays the role of 2, and so on.

Now for Von Neumann's construction, or as I like to think of it, the expanding suitcase model of the numbers.

The operation s here is a bit different.

The output set will be the union of the input set and a set that contains the input set.

In case you're not too fluent with set operations, here's another visual aid.

Picture a suitcase, x, that's full of items.

That will be the input to s. Now, take a bigger suitcase, y, and fill it as follows.

First, clone every item that's in x and put it into suitcase y.

Then take suitcase x itself along with all its contents and put that into suitcase y.

Suitcase y is the output of s, and it contains everything that suitcase x contains plus a copy of the entire suitcase x itself.

In other words, the output suitcase always has exactly one more element than the input suitcase.

Now to build our collection n, let's start again with the empty set, which this time we'll picture as an empty suitcase.

If you feed that into s, you get a new suitcase that contains clones of the contents of the input suitcase, except there aren't any in this case, because it's empty, plus the empty input suitcase itself.

Fine.

Now, let's feed that into s. You get a new suitcase that has two items in it, the original empty suitcase and a suitcase containing the empty suitcase.

Keep going like this.

Just as before, you'll end up with an infinite collection of items that satisfy the Peano axiom.

The empty suitcase plays the role of 0.

The suitcase containing the empty suitcase, i.e.

Containing 0, is 1.

The suitcase contains 0 and 1 plays the role of 2.

The suitcase containing 0 and 1 and 2 is 3, and so on.

But Neumann's model has another neat characteristic.

Every number in that construction is a set, each of whose elements is also a subset of that set.

Chew on that for a bit and verify for yourself that it's accurate.

And now the punch line.

Notice that both the Zermelo and Von Neumann constructions begin with the empty set, and using different set operations, they then synthesize every other number from the empty set, which means when we ask, what are numbers made of, at least in set theory, the answer seems to be nothing, provided, at least, that you stipulate nothing exists.

That's cute.

But now, which of these constructions is really the natural numbers?

That's a philosophically heavy question and I'll try to answer it in two ways.

First, the pragmatic answer.

The Von Neumann construction is the de facto standard set theory for a few reasons.

Relations like less than are less cumbersome to define with Von Neumann.

It's also somewhat simpler to define cardinal numbers, i.e.

answers to the question, how many, in terms of the Von Neumann naturals, which, if you think about it, have really so far been a model of ordinal numbers that only specify position along a sequence.

And finally, the Von Neumann ordinals lend themselves better to generalizing to transfinite arithmetic.

If you want a nice primer on what transfinite arithmetic means, Vsauce has a very popular video on how to count past infinity that you should go check out.

Of course, who's to say that someone tomorrow won't find a different set theoretic model of the naturals that's equally good for these purposes?

And that leads me to my second answer, which is just my personal view of the situation, namely, that asking what anything in math really is somewhat beside the point.

In the case of the naturals specifically, it can be shown, and I'm pretty sure Dedikam was the first to do so, that all realizations of the Peano axioms are isomorphic to one another, meaning that you can put any two of them into one-to-one correspondence, so that whatever the s function does in model a winds up with what the s function in model B does to corresponding elements in model B.

But more important, the details of one versus another construction matter a lot less than their actual existence.

I've always felt that the whole point of the reductionist enterprise of mathematics is less about uncovering some unique truth and more about minimizing the number of unavoidable assumptions.

In that sense, the existence of constructions, in terms of more basic ingredients, matters a lot more than the details.

There are, for instance, numerous ways to synthesize your way up to the real numbers starting from the natural.

And preferring one to another of those constructions is less a matter of correctness than a matter of taste.