# 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.