Definition:Successor Mapping

Definition

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 $V$.

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

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

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.

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 $x$.

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.