Successor ordinal
{{Short description|Operation on ordinal numbers}}
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.{{citation|title=Sets, Logic and Categories|series=Springer Undergraduate Mathematics Series|first=Peter J.|last=Cameron|publisher=Springer|year=1999|isbn=9781852330569|page=46|url=https://books.google.com/books?id=sDfdbBQ75MQC&pg=PA46}}.
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 < 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:
:
:
and for a limit ordinal λ
:
In particular, {{nowrap|1=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.{{citation|title=The Joy of Sets: Fundamentals of Contemporary Set Theory|series=Undergraduate Texts in Mathematics|first=Keith|last=Devlin|publisher=Springer|year=1993|isbn=9780387940946|at=Exercise 3C, p. 100|url=https://books.google.com/books?id=hCv-vFu4jskC&pg=PA100}}.