Definition:Successor Mapping/Successor Set

Definition
Let $S$ be a set.

The successor (set) of $S$ is defined and denoted:
 * $S\,^+ := S \cup \left\{{S}\right\}$

This set is guaranteed to exist for any background set theory that includes an Axiom of Union (even a weaker finitary schema) and an Axiom of Pairing.

Also known as
Some sources call this the Halmos function, for Paul R. Halmos who made extensive use of it in his 1960 work.

Some sources use $S\,'$ rather than $S\,^+$.

Some sources use $S + 1$ rather than $S\,^+$, on the grounds that they mean the same thing when applied to the set of natural numbers.

Also see

 * Minimal Infinite Successor Set


 * Natural Numbers are Elements of Minimal Infinite Successor Set