# Peano axioms

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The **Peano axioms** are a set of axioms that formally describes the natural numbers (0, 1, 2, 3 ...).
They were proposed by the Italian mathematician Giuseppe Peano in 1889.
They consist of a few basic — and intuitively obvious — properties that, however,
are sufficient to define the natural numbers:

- There is a smallest natural number (either 0 or 1), starting from which all natural numbers can be reached by moving finitely often to the "next" number (obtained by adding 1).

## The axioms

Today the Peano axioms are usually formulated as follows:

- Zero is a natural number.
- Every natural number has a unique successor that also is a natural number.
- Zero is not the successor of any natural number.
- Different natural numbers have different successors.
- If it is true that

- (a) Zero has property
*P*, and - (b) if any given natural number has property
*P*then its successor also has property*P*

- (a) Zero has property
- then all natural numbers have property
*P*.

The last axiom is called the axiom (or rule) of induction.