Successor ordinal
Template:Short description 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.
PropertiesEdit
Every ordinal other than 0 is either a successor ordinal or a limit ordinal.<ref name="cameron">Template:Citation.</ref>
In Von Neumann's modelEdit
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<ref name="cameron"/>
- <math>S(\alpha) = \alpha \cup \{\alpha\}.</math>
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 additionEdit
The successor operation can be used to define ordinal addition rigorously via transfinite recursion as follows:
- <math>\alpha + 0 = \alpha\!</math>
- <math>\alpha + S(\beta) = S(\alpha + \beta)</math>
and for a limit ordinal λ
- <math>\alpha + \lambda = \bigcup_{\beta < \lambda} (\alpha + \beta)</math>
In particular, Template:Nowrap. Multiplication and exponentiation are defined similarly.
TopologyEdit
The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology.<ref>Template:Citation.</ref>