# Axiom:Peano's Axioms/Formulation 2

## 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 $0$.

Historically, these two presuppositions were considered the first two of **Peano's Axioms**. The other three can be formulated as follows:

\((P3)\) | $:$ | \(\displaystyle \forall m, n \in P:\) | \(\displaystyle \map s m = \map s n \implies m = n \) | $s$ is injective | ||||

\((P4)\) | $:$ | \(\displaystyle \Img s \ne P \) | $s$ is not surjective | |||||

\((P5)\) | $:$ | \(\displaystyle \forall A \subseteq P:\) | \(\displaystyle \paren {\paren {\exists x \in A: \neg \exists y \in P: x = \map s y} \land \paren {\forall z \in A: \map s z \in A} } \) | 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 $P$ |

## 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 see

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

## Source of Name

This entry was named for Giuseppe Peano and Richard Dedekind.

## 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.

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

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 4$: The natural numbers