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]
- [math]\displaystyle{ 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 addition
The successor operation can be used to define ordinal addition rigorously via transfinite recursion as follows:
- [math]\displaystyle{ \alpha + 0 = \alpha\! }[/math]
- [math]\displaystyle{ \alpha + S(\beta) = S(\alpha + \beta) }[/math]
and for a limit ordinal λ
- [math]\displaystyle{ \alpha + \lambda = \bigcup_{\beta \lt \lambda} (\alpha + \beta) }[/math]
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
References
- ↑ 1.0 1.1 Cameron, Peter J. (1999), Sets, Logic and Categories, Springer Undergraduate Mathematics Series, Springer, p. 46, ISBN 9781852330569, https://books.google.com/books?id=sDfdbBQ75MQC&pg=PA46.
- ↑ Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory, Undergraduate Texts in Mathematics, Springer, Exercise 3C, p. 100, ISBN 9780387940946, https://books.google.com/books?id=hCv-vFu4jskC&pg=PA100.
 |  |
