# Definition:Successor Mapping/Peano Structure

## Definition

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

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

### Successor Element

The image element $\map s x$ of an element $x$ is called the **successor element** or just **successor** of $x$.

## Also known as

The **successor mapping** can also be seen referred to as the **successor function**.

Some sources call this the **Halmos function**, for Paul R. Halmos who made extensive use of it in his $1960$ work *Naive Set Theory*.

Some sources use $x'$ rather than $x^+$.

Some sources use $x + 1$ rather than $x^+$, on the grounds that these coincide for the natural numbers (when they are seen as elements of the von Neumann construction of natural numbers).

Smullyan and Fitting, in their *Set Theory and the Continuum Problem, revised ed.* of $2010$, use a variant of $\sigma$ which looks like $o$ with $^\text {-}$ as a close superscript.

## Also see

- Results about
**the successor mapping**can be found**here**.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 4$: The natural numbers - 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 - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 1$: Introduction