The **Peano axioms** were proposed by Giuseppe Peano to derive the theory of arithmetic. Together, these axioms describe the set of natural numbers, $ \N $ , including zero.

## The axioms

- There exists a natural number zero (0).
- For each $ n\in\N $ , there exists a natural number that is the
*successor*of $ n $ , denoted by $ n' $ . - 0 is not the successor of any natural number. (For each $ n\in\N,n'\ne0 $)
- For all natural numbers $ n,m\in\N $ , if $ n'=m' $ , then $ n=m $ .
- Given any predicate $ P $ on the natural numbers, if $ P(0) $ is true and $ P(k) $ implies $ P(k') $ for any $ k\in\N $ , then $ P(n) $ is true for all $ n\in\N $ . (This is also known as the principle of mathematical induction.)

A variation of these axioms substitute the natural number one (1) in place of zero in axioms 1, 3 and 5 above. It is possible, therefore, to regard the natural numbers as excluding zero while the whole numbers include zero. Usage varies on this point, however.

## Description

The Peano axioms set up the natural numbers by introducing a particular element (zero) with the first axiom, and then describing a particular function on the natural numbers. This function, namely the successor function, sets up a chain that begins at zero (axiom 3), and continues indefinitely (axiom 2). Axiom 5 ensures that the chain will eventually cover every natural number, and axiom 4 ensures that every link in the chain is preceded by at most one other link.

## Properties

Intuitively, there are more properties of the natural numbers than are listed in the axioms. However, these properties can be proven from the axioms. Here are some such properties:

- No natural number is its own successor
- Every natural number except for zero is a successor to some other natural number

## Important Functions

The natural numbers, as described by the Peano axioms have addition and multiplication operations defined.