Friday, March 4, 2011

Peano's Axioms for Non-Mathematicians (not dummies)

So when I looked at my previous post on Peano's axioms again, trying to see it as if I'd never seen the axioms before, I realised that maybe they aren't too accessible as they are...

Here is my attempt to explain in plain(er) English. I'll leave out some of the definitional fluff at the beginning and cut straight to the axioms themselves

  1. We assume that there is at least one natural number: 1 (some people prefer to start with 0, I'll just use the same convention as the one in the previous quotation). The set of natural numbers is not empty.
  2. For each number that is a member of our set (i.e. for each natural number) there exists a successor to that number. In other words, every natural number has a "next" number. The successor of 1 is 2, the successor of 2 is 3 and so forth. We'll label the successor of an unknown natural number x as x'.
  3. There is no natural number whose successor is 1. In other words, x'≠ 1, no matter what x is. In still other words, 1 is the first natural number. (It also works if you let 0 be this "first" natural number).
  4. Every natural number has one successor or no successor, i.e. the successor of a natural number is unique (if it has one). We write this symbolically as: x'=y' means that x=y.
  5. Let's make a special set of natural number that obeys the following rules. A: 1 is a member of this set. B: if a certain natural number is a member of the set, then its successor must also be a member of the set. THEN all the natural numbers have to be members of that set. This is the principle of induction.
I think perhaps I'll post about the principle of induction in more detail another certainly merits a post of its own!

In the meantime, I hope that helps you to understand Peano's axioms a bit better.

Oh, and by the way:

The coolest thing about the natural numbers is this: any set that obeys those five axioms (considered as rules) is, for all extent and purposes, essentially identical to the set of natural numbers.

This is deep. Think about it.

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