The axioms of Peano

The Peano axioms (also known as Peano's postulates) were a proposal by the Italian mathematician Geuseppe Peano in 1889, in order to axiomatically formalize the natural numbers, based on the set theory developed by Georg Cantor

Axioms, which is still used in the present

Natural numbers are defined as a set (called \mathbb{N}_0), an element (which assumes the role of zero and that we will denote as 0) and a “next” (or “successor”) element that is an application denoted by S so that it fulfills:

1. The zero is a natural number
2. The next natural number is also a natural number
3. There is No natural number whose next is zero
4. If the next of two natural numbers are equal, then the two numbers are equal
5. If S is a set of natural numbers such that zero is of S and whenever a natural number is of S also its next one is in S, then S is the set of natural numbers

Using a more formal or algebraic language, these five axioms can be stated like this:

1. 0\in \mathbb{N}_0
2. \exists s| \mathbb{N}_0\rightarrow \mathbb{N}_0 and in addition satisfies the axioms following
3. \not\exists n\in \mathbb{N}_0 |s(n)=0
4. s(n)=s(m)\Rightarrow n=m (s is injective)
5. S\in \mathbb{N}, 0\in S |\forall n\in S=s(n)\in S\Rightarrow S=\mathbb{N}_0