# 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**:

\((\text P 1)\) | $:$ | \(\displaystyle 0 \in P \) | $0$ is an element of $P$ | |||||

\((\text P 2)\) | $:$ | \(\displaystyle \forall n \in P:\) | \(\displaystyle \map s n \in P \) | For all $n \in P$, its successor $\map s n$ is also in $P$ |

The other three are as follows:

### Formulation 1

\((\text P 3)\) | $:$ | \(\displaystyle \forall m, n \in P:\) | \(\displaystyle \map s m = \map s n \implies m = n \) | $s$ is injective | ||||

\((\text P 4)\) | $:$ | \(\displaystyle \forall n \in P:\) | \(\displaystyle \map s n \ne 0 \) | $0$ is not in the image of $s$ | ||||

\((\text P 5)\) | $:$ | \(\displaystyle \forall A \subseteq P:\) | \(\displaystyle \paren {0 \in A \land \paren {\forall z \in A: \map s z \in A} } \implies A = P \) | Principle of Mathematical Induction: | ||||

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

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

### Formulation 2

\((\text P 3)\) | $:$ | \(\displaystyle \forall m, n \in P:\) | \(\displaystyle \map s m = \map s n \implies m = n \) | $s$ is injective | ||||

\((\text P 4)\) | $:$ | \(\displaystyle \Img s \ne P \) | $s$ is not surjective | |||||

\((\text P 5)\) | $:$ | \(\displaystyle \forall A \subseteq P:\) | \(\displaystyle \paren {\paren {\exists x \in A: \neg \exists y \in P: x = \map s y} \land \paren {\forall z \in A: \map s z \in A} } \) | 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 $\struct {P, s, 0}$ be a Peano structure.

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

### Non-Successor Element

Let $\struct {P, s, 0}$ be a Peano structure.

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

This is justified by Axiom $(\text P 4)$, 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 presented as

Some sources present the axioms in a different order.

For example, 1964: J. Hunter: *Number Theory* 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 defined as

Some treatments of Peano's axioms 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**(for Richard Dedekind)**the Peano postulates****Peano's postulates**

and so on.

## Also see

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

## Source of Name

This entry was named for Giuseppe Peano.

## Historical Note

A set of axioms on the same topic as **Peano's axioms** was initially formulated by Richard Dedekind in $1888$.

Giuseppe Peano published them in $1889$ according to his own formulation, in a more precisely stated form than Dedekind's.

Bertrand Russell pointed out that while **Peano's axioms** give the key properties of the natural numbers, they do not actually define what the natural numbers actually are.

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.

## Sources

- 1982: Alan G. Hamilton:
*Numbers, Sets and Axioms*... (previous) ... (next): $\S 1$: Numbers: $1.1$ Natural Numbers and Integers: Examples $1.1 \ \text {(e)}$