Wednesday, February 20, 2013

Natural Numbers

The first kind of numbers humanity discovered were the Natural Numbers. Now, this statement needs to be explained.

First, what does “kind of numbers” mean? Well, there are many of them! Like, for instance, the numbers you use to count (one apple, two apples, three apples) aren’t the same as the numbers you use to talk about quantum mechanics (such-and-such has an amplitude of √2(1+i)…)

Then there’s “discovered.” So that implies that the numbers are, somehow, “out there”? Well, that’s a philosophical issue, but I’d say that yes, yes they are. I mean, 2 apples plus 2 apples equals 4 apples, but it seems to me that the statement 2 + 2 = 4 is somehow deeper than that.

Anyway. There are the Natural Numbers. But what are they? How do we define them?

"Well, there’s 0, 1, 2, 3… you know, numbers."

Yes, but how would you explain these numbers to an alien species that doesn’t already know them?

"Uh… well…"

Of course you don’t need to do that, because in your head you already know what the Natural Numbers are. Still, it’d be nice if we had a set of rules that define them. So let’s do that.

1. 0 is a natural number.

Okay? That sounds like a good starting point. Now, we have an equality relation.

2. For every natural number n, n = n.

3. For every natural numbers m and n, if m = n then n = m.

4. For every natural numbers m, n and p, if m = n and n = p then m = p.

5. If m is a natural number and m = n, then n is a natural number.

Further, we have an operation called successorship, S(n).

6. For every natural number n, S(n) is also a natural number. (This statement is usually referred to as “the natural numbers are closed under S.”)

"Okay, so we have that 0 is a natural number, and S0 is a natural number. But since S0 is a natural number, SS0 is also a natural number. Okay! We’re done, aren’t we?"

Not quite.

7. For every natural numbers m and n, if S(m) = S(n) then m = n.

8. There is no natural number n such that S(n) = 0.

"Okay, now we’ve defined natural numbers, haven’t we?”

Indeed. But there’s a problem with the above. There are sequences that obey these rules and are not Natural Numbers.

"Huh? How is that?"

Well, the sequence …-2*, -1*, 0*, 1*, 2*,… obeys these rules nicely.

"Hold on… that doesn’t…"

Just look at it. 0 isn’t in it, but no one said it had to be. The numbers are also unique, and closed under S. And in fact, there’s an infinite number of infinite-in-both-directions sequences that obey all these rules.

"Alright! So we can just say that… that any sequence that’s the natural numbers must have 0 or something…”

Yes, yes indeed.

9. If K is a set that contains 0 and is closed under S, then K contains all natural numbers.

There. Now we’re done. We’ve just pinpointed all of the Natural Numbers.

These rules are called the Peano Axioms, and they uniquely define the Natural Numbers. That is to say, if anyone ever asks you what the Natural Numbers are, you just tell them to follow these strict rules, and they will create a sequence that is isomorphic to them.

Notes

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