Successor ordinal

Every ordinal other than 0 is either a successor ordinal or a limit ordinal. ==In Von Neumann's model==

Using von Neumann's ordinal numbers (the standard model of the ordinals used in set theory), the successor S(α) of an ordinal number α is given by the formula :S(\alpha) = \alpha \cup \{\alpha\}. Since the ordering on the ordinal numbers is given by α < β if and only if α ∈ β, it is immediate that there is no ordinal number between α and S(α), and it is also clear that α < S(α). ==Ordinal addition==

The successor operation can be used to define ordinal addition rigorously via transfinite recursion as follows: :\alpha + 0 = \alpha\! :\alpha + S(\beta) = S(\alpha + \beta) and for a limit ordinal λ :\alpha + \lambda = \bigcup_{\beta In particular, . Multiplication and exponentiation are defined similarly. ==Topology==

The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology. ==See also==