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.

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

Then the following are Peano's Axioms:

Then $\left({P, s, 0}\right)$ is called a Peano structure, $s$ is called the successor mapping, and $0$ is called the non-successor element.

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.

From the above we see that a Peano structure has the following features:

Successor Mapping
Let $\left({P, s, 0}\right)$ be a Peano structure.

Then the mapping $s: P \to P$ is called the successor mapping

Non-Successor Element
Let $\left({P, s, 0}\right)$ be a Peano structure.

Then the element $0 \in P$ is called the non-successor element.

This is justified by Axiom $(P3)$, which stipulates that $0$ is not in the image of the successor mapping $s$.

It would be nice if there were a name for this element more terse than non-successor element and more general than zero.

Westwood suggests primal element.

Principle of Induction
Axiom $(P4)$ is known as the Principle of Mathematical Induction.

Peano's Axioms define Natural Numbers Uniquely
One of the most important aspects of Peano's Axioms is that they uniquely define the set of natural numbers.

That is, not only do the natural numbers satisfy Peano's Axioms, but conversely, any set that satisfies Peano's Axioms also satisfies all the properties held by the set $\N$ of natural numbers.

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

Also see

 * Minimal Infinite Successor Set Fulfils Peano Axioms

They were formulated by Peano, and were later refined by Dedekind.

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.