Definition:Successor Mapping/Also known as

Successor Mapping: Also known as
The successor mapping can also be seen referred to as the successor function.

Some sources call this the Halmos function, for who made extensive use of it in his $1960$ work.

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).

and, in their of $2010$, use a variant of $\sigma$ which looks like $o$ with $^\text {-}$ as a close superscript.