# Definition:Successor Set

## Definition

Let $S$ be a set.

The successor (set) of $S$ is defined and denoted:

$S^+ := S \cup \set S$

## Also known as

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 $S'$ rather than $S^+$.

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