Successor ordinal

In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α. An ordinal number that is a successor is called a successor ordinal. The ordinals 1, 2, and 3 are the first three successor ordinals and the ordinals ω+1, ω+2 and ω+3 are the first three infinite successor ordinals.

Properties

Every ordinal other than 0 is either a successor ordinal or a limit ordinal.^[1]

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^[1]

${\displaystyle S(\alpha )=\alpha \cup \{\alpha \}.}$

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:

${\displaystyle \alpha +0=\alpha \!}$

${\displaystyle \alpha +S(\beta )=S(\alpha +\beta )}$

and for a limit ordinal λ

${\displaystyle \alpha +\lambda =\bigcup _{\beta <\lambda }(\alpha +\beta )}$

In particular, S(α) = α + 1. 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.^[2]

See also

• Ordinal arithmetic
• Limit ordinal
• Successor cardinal