In 1889 Giuseppe Peano published a set of five axioms which attempted to provide a rigorous basis for the natural numbers. In 1888 Richard Dedekind had published a set of axioms on the same topic, but Peano’s were more precisely stated. However, they are sometimes referred to as the Peano–Dedekind axioms in recognition of the foundation laid by Dedekind.

The axioms are:

1. (i) Zero is a natural number.

2. (ii) Every natural number has a successor in the natural numbers.

3. (iii) Zero is not the successor of any natural number.

4. (iv) If the successors of two natural numbers n, m are the same number, then n =  m.

5. (v) If a set S contains zero and the successor of every number in S is also in S, then the set S contains the natural numbers.

The last of these is known as the principle of mathematical induction.