Definition:Successor Set

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Definition

Let $S$ be a set.


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

$S^+ := S \cup \left\{{S}\right\}$


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 minimal infinite successor set).


Also see


Sources