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 x
- 1 is not the successor of any natural number
- From x+ = y+ follows 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 M.
Then M contains all natural numbers.
In the language of these axioms, addition is defined by setting
- x + 1 = x+
- x + 2 = x++
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.