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.