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.

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]

S(\alpha) = \alpha \cup \{\alpha\}.

Since the ordering on the ordinal numbers 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:

\alpha + 0 = \alpha\!
\alpha + S(\beta) = S(\alpha + \beta)\!

and for a limit ordinal λ

\alpha + \lambda = \bigcup_{\beta < \lambda} (\alpha + \beta)

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. ↑ 1.0 1.1 Cameron, Peter J. (1999), Sets, Logic and Categories, Springer Undergraduate Mathematics Series, Springer, p. 46, ISBN 9781852330569.
  2. ↑ Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory, Undergraduate Texts in Mathematics, Springer, Exercise 3C, p. 100, ISBN 9780387940946.