Peano's Axioms for natural numbers

The modern formalism of numbers generally begins with the "Natural Numbers" ℕ, which are fairly easy to model, and then builds from there. From natural numbers, one can create a set of definitions for integers, rational numbers, real numbers, and complex numbers, all the way making appropriate definitions for the operators and relations we want to hold for all these different numbers and verifying that the operators and relations hold the way we want. But it all begins with ℕ.

Wikipedia lists 9 axioms in for Peano's Axioms of ℕ, but most formulations I've seen generally list 4 -- the other 5 are either assumed or subsumed by the 4. Peano stated, for instance, that equality of natural numbers was reflexive (a=a), symmetric (a=b implies b=a), transitive (a=b and b=c implies a=c) and closed (if a is a natural number, and a=b, then b is a natural number). In general, people tend to assume these 4 properties of equality, but they are reasonable to state in a full formalization. When we get to defining other numbers in terms of natural numbers, we are going to have to define equality as well, and show that these 4 properties hold for our definitions, so it's good to be explicit about them. But generally, most mathematicians today hold those four properties as being part of the concept of equality.

As for the rest, Peano assumed that the system of natural numbers consisted of two things at it's base, a set ℕ, and a function S:ℕ→ℕ (i.e. a function which takes natural numbers to natural numbers), with the following properties:

A. 1 is a natural number.
B. If n is a natural number, then S(n) is a natural number.
C. For all natural numbers n, S(n) ≠ 1.
D. For all natural numbers n, m, if S(n) = S(m), then n=m
and
E. If P(n) is a proposition on natural numbers such that P(1) is true and if P(n) is true then P(S(n)) is true for all natural numbers n, then P(n) is true for all natural numbers.

The last one is a bit complex, but it allows us to prove things for all natural numbers rather easily. It's simple once you get the hang of it.

S is called the "successor" function, and the axioms can be stated as "B. Every natural number has a unique successor", "C. 1 is not the successor of any number", "D. Every natural number is the successor of at most one number", and "E. If P is true for 1, and true for the successor of any number it is true for, then it is true for all numbers". While this restatement is somewhat less formal, it provides a more fluid way of speaking of the axioms.

Here's an example of a theorem that can be proved using the axioms (and a demonstration of axiom E)

Theorem: Every natural number n other than 1 is the successor of a natural number.
Proof: Let P(n) be the proposition that n is 1 or the successor of a natural number. If P(n) is true for all n, then that proves our theorem. P(1) is obviously true. Assume P(n) is true, then P(S(n)) is true since S(n), by definition, is the successor of a natural number. So P(1) is true, and P(n) implies P(S(n)). By Axiom E, P(n) is therefor true for all natural numbers. That is what we were to demonstrate.

This technique of proving P(1) and proving P(n)→P(S(n)) to prove P(ℕ) is called "mathematical induction", and is a powerful technique. Needless to say, we will be using it alot.

Tomorrow, I'll discuss order.