Successor ordinal

In set theory, the successor of an ordinal number α is the smallest ordinal number greater than ÃŽÂ±. 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.

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

Since the ordering on the ordinal numbers is given by α&nbsp;<&nbsp;β if and only if α&nbsp;∈&nbsp;β, it is immediate that there is no ordinal number between α and S(α), and it is also clear that α&nbsp;<&nbsp;S(α).

Ordinal addition

The successor operation can be used to define ordinal addition rigorously via transfinite recursion as follows:

and for a limit ordinal λ

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

• Ordinal arithmetic
• Limit ordinal
• Successor cardinal

References