nLab
successor

Given a natural number $n$, the successor $n^{+}$ of $n$ is simply $n+1$.

More generally, in any well-ordered set $S$, the successor $w^{+}$ or $w+1$ (if it exists) of an element $w$ is the smallest element of the subset of all elements which are (strictly) greater than $w$. If $S$ has no maximal element, then the successor map $w\mapsto w+1$ is always defined; it is sometimes used to make recursive definitions.

This notion is sometimes also used for some well-ordered proper classes, for example for the class $\mathrm{Ord}$ of ordinal numbers. For definitions by transfinite recursion?, one usually specifies the value at $0$, the rule for recursion along the successor map and a separate rule of recursion for the limiting ordinals (infinite ordinals which are not successors). (For example, the von Neumann hierarchy of well-founded pure sets is defined in that way.) One can (and in constructive mathematics must) also handle all three cases at once, and the successor function is used there as well.

In a topos, a natural numbers object $\mathbb{N}$ is equipped with a successor morphism $\mathbb{N}\to \mathbb{N}$; and it has an abstract property of recursion?.