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.