Peano's Axioms

1. Zero is a number.

2. If  is a number, the successor of  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  of numbers contains zero and also the successor of every number in , then every number is in .

Peano's axioms are the basis for the version of number theory known as Peano arithmetic.

REFERENCES

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1152, 2002.