What Does It Mean to Be a Number? (The Peano Axioms)

The natural numbers are pretty familiar.

But if I asked you to tell me what natural numbers are and how they work without using the notion of number in your answer, could you do it?

In the late 19th century, mathematicians were on a quest to put all of mathematics on firm logical footing.

Kelsey talked about this in a previous video, where she mentioned, among other things, that the number system can be constructed hierarchically starting from natural numbers and working your way up.

But what about the natural numbers themselves-- zero, one, two, three, et cetera?

Are they fundamental, or can they also be constructed from more basic ingredients?

Before we can tackle that question, we need a firmer handle on what exactly it means to be a natural number.

More specifically, we need a way to characterize the essential features of natural numbers without referencing directly or indirectly things that we already know about the natural numbers, because that would be circular.

Now it turns out that you can capture the essence of numberhood in a small set of axioms, analogous to Euclid's axioms in geometry, that will allow us to build a set N that behaves just like the natural numbers without ever explicitly mentioning numbers, or counting, or arithmetic as we do so.

These axioms were first published in 1889 more or less in their modern form by Giuseppe Peano building on and integrating earlier work by Peirce and Dedekind.

I'm going to show you how this is done.

But to appreciate what a massive achievement.

It is, I want you to try doing it yourself first.

Just casually try describing numbers without mentioning numbers or even inherently number-ish concepts, like one, or plus, or next.

You'll find it's pretty hard to do.

First, some preliminaries.

To make this somewhat self-contained, I need to assume your familiarity with some basic logical infrastructure that is not specific to numbers.

First of all, you should be familiar with some ideas surrounding sets, namely that the members of a set are called elements, that two sets with an identical roster of elements are really the same set, and with the idea of a subset of a set-- in particular, the fact that every set is a subset of itself.

Second, we need a concept of equals.

We'll be very bare bones about this, requiring only that whatever equality means the relation be reflexive, symmetric, and transitive.

And we'll also demand that the set N we're trying to build be closed under equality.

In other words, if x is a thing in N, and x equals y, then y is also a thing in N because, of course, x and y are the same entity.

Now that last part may seem self evident, but when you're laying out mathematical logic, it's best not to leave any assumptions unstated since that makes it harder to track which parts of an argument actually depend on which other parts.

And finally, we'll also rely on the concept of a function, or a map-- i.e.

a rule that associates an output with one or more inputs.

Armed with these basic tools, let's introduce Peano's axioms incrementally so that you can see exactly why each one needs to be there.

Now warning-- some parts of the argument are subtle, and you may need to watch this more than once to keep up with what's going on.

Let's begin.

Axiom one-- Zelda is an N. This is basically a declaration that set N isn't empty.

Zelda is in there.

Who or what is Zelda?

Don't worry about it.

It'll turn out to be the zero that you know and love.

But I'm trying to use nomenclature here that's not too suggestive so that you won't be tempted to bring in preexisting concepts.

Remember, the goal is to arrive at numbers without referencing numbers.

Now, to populate the rest of N, we can just keep including the next element.

But what is next?

The whole notion of next is loaded with conceptual baggage that can be hard to disentangle from your preexisting notions of counting and numbers.

So that's what the remaining axioms will do.

They will define next using only more basic and non-numerical ideas.

Let's see how that works starting with axiom two.

There exists a function S whose inputs and outputs are elements of N. In other words, if x is an N, then S of x is also in N. Now, on it's own, this doesn't bias anything because we haven't specified the mechanical rule underlying S. For instance, suppose that Zelda is in the set, and it turns out that S of Zelda equals Zelda.

This would still satisfy axioms one and two, but it looks nothing like the natural numbers.

So let's fix that.

Axiom three-- Zelda is never the output of S. More precisely, for any x in N, the statement S of x equals Zelda is always false.

Now, this is progress.

Axiom two says that S of Zelda is in N, and axiom three says that S of Zelda is different from Zelda itself.

But notice that having S of S of Zelda turn out to be S of Zelda would still be consistent with the axioms so far, and that would bring our building process to a halt.

So we need to prevent that possibility.

Enter axiom four.

If x and y are in N, and S of x equals S of y, then x equals y.

In other words, different inputs to S will always give different outputs.

Now, finally, we are guaranteed an infinite chain.

Watch.

Zelda is in our set by axiom one.

S of Zelda is in our set axiom two, and it's different from Zelda by axiom three.

S of S of Zelda is in our set by axiom two, and it's different from Zelda by axiom three, and different from S of Zelda by axiom four.

Now we keep using axioms two, three, and four over and over.

So s of s of s of Zelda will be in the set, and it's different from Zelda and from every element we've already mentioned and so on and so on.

This is starting to look like an infinite list of counting numbers.

If we introduce some shorthand names for these elements borrowing Arabic numerals, like 0 for Zelda, and one for S of Zelda, and two for S of S of Zelda, and so on, it will look exactly like the counting numbers.

Now, what about the function S?

Notice that we never specified its internal mechanics.

We just enforced some constraints on its behavior that apparently capture the intuitive notion of next, or successor, ergo the S. And we used as ingredients only the abstract concept of a function and the idea of equals.

So we're done, right?

Well, not so fast.

There's a subtle loophole in axioms one through four.

Suppose that we throw two other characters into N. Let's call them Mario and Luigi.

And suppose it turns out that S of Mario is Luigi and S of Luigi is Mario.

Tacking on these plumbers with this weird S behavior doesn't violate any of the axioms so far, but it does leave us with a set that looks somehow bigger than the familiar natural numbers.

So axioms two through four apparently do not quite capture the idea of next because they would allow Mario and Luigi to be in the set, but detached from everybody else so to speak.

To prevent this, we used Peano's fifth axiom also known as the axiom of induction, which cleans up this mess by fiat in a very subtle way.

Here's axiom five.

Suppose that T is any subset of N that satisfies two properties-- first that Zelda is in T and second that if x is in T, then S of x is also in T. Then by declaration, T is the set N. In other words, axiom five declares N to be the minimal set that will satisfy axioms one through four.

Now since the infinite chain we already have fits that bill by axiom five, that set is N. We are done.

And adjoining Mario and Luigi along with their weird S loop would give you something different from N. So what does it mean to be a natural number?

It means membership in a set that is endowed with the notion of successor, or next, and one of whose elements has the privileged status of not being anybody else's successor.

Now, the miracle of Peano's axioms, in my opinion, is the revelation that next is not a fundamental concept, that it can be reduced to more basic logical ideas.

And to me, the demonstration that this can even be done at all is one of the most beautiful arguments in all of mathematics.

Now, for some loose ends.

First, what about arithmetic?

It turns out that you can define both addition and multiplication solely in terms of the successor function S. Each of those operations, in fact, can be viewed as a function that takes two inputs from N and yields one output in N. And they can be defined recursively by the following rules.

Now, I don't have time to do it here, but it is a fun challenge to play around with these rules in order to derive simple facts like one plus two equals three, or that addition and multiplication are commutative and associative, or that they obey a distributive rule.

Super fun.

Second loose end.

We arrived at a notion of successorship, and we built a set that contains entities related to one another by iterated succession.

But we never specified what those entities actually were with Zelda, or zero, or any other members of N actually are.

We just asserted that here's a thing and here's some other things, and they're all in this set N. But we're leaving open the question, what are those things?

Are the irreducible fundamental ingredients in the mathematical pyramid?

Or can even these entities, the natural numbers, be built out of simpler entities in the same way that the integers get built from the naturals?

We'll answer that next time, and I'll see you then.