# Axiom:Peano's Axioms

## Contents

## 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 $s: \N \to \N: s \left({n}\right) = n + 1$ 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:

### Formulation 1

\((P3)\) | $:$ | \(\displaystyle \forall m, n \in P:\) | \(\displaystyle s \left({m}\right) = s \left({n}\right) \implies m = n \) | $s$ is injective | ||||

\((P4)\) | $:$ | \(\displaystyle \forall n \in P:\) | \(\displaystyle s \left({n}\right) \ne 0 \) | $0$ is not in the image of $s$ | ||||

\((P5)\) | $:$ | \(\displaystyle \forall A \subseteq P:\) | \(\displaystyle \left({0 \in A \land \left({\forall z \in A: s \left({z}\right) \in A}\right)}\right) \implies A = P \) | Principle of Mathematical Induction: | ||||

Any subset $A$ of $P$, containing $0$ and | ||||||||

closed under $s$, is equal to $P$ |

### Formulation 2

\((P3)\) | $:$ | \(\displaystyle \forall m, n \in P:\) | \(\displaystyle s \left({m}\right) = s \left({n}\right) \implies m = n \) | $s$ is injective | ||||

\((P4)\) | $:$ | \(\displaystyle \operatorname{Im} \left({s}\right) \ne P \) | $s$ is not surjective | |||||

\((P5)\) | $:$ | \(\displaystyle \forall A \subseteq P:\) | \(\displaystyle \left({\left({\exists x \in A: \neg \exists y \in P: x = s \left({y}\right)}\right) \land \left({\forall z \in A: s \left({z}\right) \in A}\right)}\right) \) | Principle of Mathematical Induction: | ||||

\(\displaystyle \implies A = P \) | Any subset $A$ of $P$, containing an element not | |||||||

in the image of $s$ and closed under $s$, is equal to $P$ |

## Terminology

### Successor Mapping

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

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

### 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 $(P4)$, which stipulates that $0$ is not in the image of the successor mapping $s$.

### 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

Some treatments of this topic define the non-successor element (or **primal element**) to be $1$ and not $0$.

The treatments are similar, but the $1$-based system results in an algebraic structure which has no identity element for addition, and so no zero for multiplication.

## Also known as

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

## Also see

- Results about
**Peano's axioms**can be found here.

## Source of Name

This entry was named for Giuseppe Peano.

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

## Historical Note

According to 1960: Paul R. Halmos: *Naive Set Theory*:

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

It is worth pointing out that 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. Thus they are now rarely considered as axiomatic as such. However, in their time they were groundbreaking.