Full Idea
For any set x, we define the 'successor' of x to be the set S(x) = x U {x}.
Gist of Idea
A successor is the union of a set with its singleton
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
Book Reference
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.58
A Reaction
This is the Fregean approach to successor, where the Dedekind approach takes 'successor' to be a primitive. Frege 1884:§76.