# Axiom:Peano's Axioms/Formulation 1

## Axioms

**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**:

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

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

The other three are as follows:

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

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

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

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

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

## 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 minimally inductive 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

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 12$: The Peano Axioms - 1964: J. Hunter:
*Number Theory*... (previous) ... (next): Chapter $\text {I}$: Number Systems and Algebraic Structures: $2$. The positive integers - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 4$: Number systems $\text{I}$: Peano's Axioms - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 1$: Introduction - 1982: Alan G. Hamilton:
*Numbers, Sets and Axioms*... (previous) ... (next): $\S 1$: Numbers: $1.1$ Natural Numbers and Integers - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Peano's postulates** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Peano's postulates** - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 1$ Preliminaries - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Peano's axioms**