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 $s: \N \to \N: \map s n = n + 1$ 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 presented as
Some sources present the axioms in a different order.

For example, places the induction axiom $(\text P 5)$ as axiom $(3)$ and moves $(\text P 3)$ and $(\text P 4)$ down to be $(4)$ and $(5)$ respectively.

Also see

 * Equivalence of Formulations of Peano's Axioms


 * Minimally Inductive Set forms Peano Structure


 * Principle of Mathematical Induction for Peano Structure