Peano's Axioms

1. Zero is a number.

2. If $a$ is a number, the successor of $a$ is a number.

3. Zero is not the successor of a number.

4. Two numbers of which the successors are equal are themselves equal.

5. (Induction Axiom.) If a set $S$ of numbers contains Zero and also the successor of every number in $S$, then every number is in $S$.

Peano's axioms are the basis for the version of Number Theory known as Peano Arithmetic.

See also Induction Axiom, Peano Arithmetic