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 these coincide for the natural numbers (when they are seen as elements of the minimal infinite successor set).

Also see

 * Definition:Minimal Infinite Successor Set