The peano axioms
or the peano postulates, are a set of axioms for the natural numbers
Within these axioms, there are three kinds of statement
The first four axioms are statements about equality
The second four axioms are about the succession operator, S()
The ninth, and last axiom, is a second order statement, defining the principle of mathematical induction
They are, in order:
1. Equality is reflexive. For every natural number x, x = x.
2. Equality is symmetric. For all natural numbers x and y, if x = y, then y = x.
3. Equality is transitive. For all natural numbers x, y and z, if x = y and y = z, then x = z.
4. The natural numbers are closed under equality. For all a and b, if a is a natural number and a = b, then b is also a natural number.
5. 0 is a natural number.
6. For every natural number n, S(n) is a natural number.
7. For every natural number n, S(n) = 0 is False.
8. For all natural numbers m and n, if S(m) = S(n), then m = n.
9. If K is a set such that:
* 0 is in K, and
* for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number.