# Peano postulates

The arithmetic of the integers, like the geometry of the plane, can be made to depend on a few axioms, in the sense that everything else follows from them by accepted logical rules. One such set of axioms was given by G. Peano in 1889; it characterises the set (class, condition) of natural numbers 1, 2, 3, etc., and consists of the following **Peano postulates** (also called **Peano axioms**):

*1*is a natural number- To each natural number
*x*there corresponds a second natural number*x*, called the successor of_{1}*x* *1*is not the successor of any natural number- From
*x*follows_{1}= y_{1}*x = y* - Let
*M*be a set of natural numbers with the following properties:*1*belongs to*M*- If
*x*belongs to*M*, then*x*also belongs to_{1}*M*.

Then *M* contains all natural numbers.

In the language of these axioms, addition is defined by setting

*x*+ 1 =*x*_{1}*x*+ 2 =*x*_{11}

etc., and multiplication is defined in terms of addition:

*ab*=*a*+*a*+ ... +*a*

where there are *b* terms on the right hand side.

The usual rules of algebra can then be deduced, as they apply to the natural numbers, and the inequality symbol *<* can be introduced as required. Finally, zero and the negative integers can be defined in terms of the natural numbers.