# PEANO'S AXIOMS OF THE NATURAL NUMBERS

Peano's axioms (or postulates) are five rules, defined in set-theoretical terms, for the construction of natural numbers. The axioms are named after the Italian mathematician, Peano, who published them in 1889, although he himself attributed them to the German mathematician Dedekind. Given the set of natural numbers, Nat, whose initial elements are 0,1,2,3,4, ..., the axioms are defined as follows:

1. There is a constant 0:Nat. (This is also regarded as a nullary function to Nat).
2. There exists a total, unary function s, s:Nat->Nat, referred to as the successor function, i.e. for each element x in Nat, there is one and only one element y in Nat such that y = s(x).
3. There is one and only one element in Nat, the constant 0, which is not the successor of an element in Nat. Thus for all x in Nat, s(x) =|= 0, i.e. 0 is not a member of Ran(s).
4. For all x and y in Nat, s(x) = s(y) -> x = y. In other words for all x and y in Nat, x =|= y -> s(x) =|= s(y) (i.e. the successor function is an injection).
5. For all subsets A of Nat, if either 0 is a member of A or for all x belonging to A, s(x) is a member of A this then implies A = Nat. In other words given a collection A of elements in Nat such that 0 is in A and for each x in A s(x) is also in A, then A = Nat. This is referred to as the induction principle or the principle of mathematical induction. Consequently we can define addition of natural numbers in such a manner that s(x) = x+1, for all x in Nat.

The set Nat together with the constant 0:Nat and the function s:Nat->Nat are referred to as the Peano system (Nat,0,S).

The axioms (slightly refined) can also be applied to the set of positive numbers, Pos, whose initial elements are 1,2,3,4,5, ....

Created and maintained by Frans Coenen. Last updated 16 September 2000