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.
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
See also
- Ordinal arithmetic
- Limit ordinal
- Successor cardinal