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

## The axioms

1. There exists a natural number zero (0).
2. For each ${\displaystyle n \in \N}$ , there exists a natural number that is the successor of ${\displaystyle n}$ , denoted by ${\displaystyle n'}$ .
3. 0 is not the successor of any natural number. (For each ${\displaystyle n\in\N,n'\ne0}$)
4. For all natural numbers ${\displaystyle n,m\in\N}$ , if ${\displaystyle n'=m'}$ , then ${\displaystyle n = m}$ .
5. Given any predicate ${\displaystyle P}$ on the natural numbers, if ${\displaystyle P(0)}$ is true and ${\displaystyle P(k)}$ implies ${\displaystyle P(k')}$ for any ${\displaystyle k\in\mathbb{N}}$ , then ${\displaystyle P(n)}$ is true for all ${\displaystyle n \in \N}$ . (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.