### Peano axioms

The arithmetic of Peano defines the natural numbers.

##### Explanation

Peano's axioms contain the notion of a set. They define the (non-negative) integers as:

1. | 0 is a natural number |

2. | Every integer has a unique successor. |

3. | There is an integer which is not a successor of any integer. |

4. | Two distict integers cannot have the same successor. |

5. | If M is a set of integers such that 0 is in M and such that if an integer n is in M then its successor is in M, then every integer is in M. |

##### Additional information

The Italian mathematician Giuseppe Peano has contributed much to the number theory.