Axiom:Peano's Axioms

Axioms
Peano's Axioms are a set of properties which can be used to serve as a basis for logical deduction of the properties of the natural numbers. Peano's Axioms are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: s(n) = n + 1$ and $0$ as an element of $\N$.

Let there be given a set $P$, a mapping $s: P \to P$, and a distinguished element $0$.

Historically, the existence of $s$ and the existence of $0$ were considered the first two of Peano's Axioms. The other three are as follows:

Peano Structure
Such a set $P$, together with the successor mapping $s$ and non-successor element $0$ as defined above, is known as a Peano structure.

Also defined as
There is nothing special about the symbol $0$ for the non-successor element. Another popular choice is $1$.

The distinction between these only comes into existence at the definition of addition.

Also known as
Peano's Axioms are also known as the Peano Axioms, the Dedekind-Peano Axioms and the Peano Postulates.

Also see

 * Equivalence of Formulations of Peano's Axioms


 * Minimal Infinite Successor Set forms Peano Structure

They were formulated by, and were later refined by.

Historical Note
According to :
 * [These] assertions ... are known as the Peano axioms; they used to be considered as the fountainhead of all mathematical knowledge.

However, as the Peano axioms can be deduced to hold for the minimal infinite successor set as defined by the Axiom of Infinity from the Zermelo-Fraenkel axioms, it has to be pointed out that they are now rarely considered as axiomatic as such. However, in their time they were groundbreaking.