# Definition:Successor Mapping

## Definition

Let $V$ be a basic universe.

The **successor mapping** $s$ is the mapping on $V$ defined and denoted:

- $\forall x \in V: \map s x := x \cup \set x$

where $x$ is a set in $V$.

### Peano Structure

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

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

### Successor Mapping on Natural Numbers

Let $\N$ be the set of natural numbers.

Let $s: \N \to \N$ be the mapping defined as:

- $s = \set {\tuple {x, y}: x \in \N, y = x + 1}$

Considering $\N$ defined as a Peano structure, this is seen to be an instance of a successor mapping.

## Successor Set

For $x \in V$, the result of applying the **successor mapping** on $x$ is denoted $x^+$:

- $x^+ := \map s x = x \cup \set x$

$x^+$ is referred to as the **successor (set) 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

- 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: Definition $1.1$