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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 formula²

S ( α ) = α ∪ { α } . {\displaystyle S(\alpha )=\alpha \cup \{\alpha \}.}

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:

α + 0 = α {\displaystyle \alpha +0=\alpha \!} α + S ( β ) = S ( α + β ) {\displaystyle \alpha +S(\beta )=S(\alpha +\beta )}

and for a limit ordinal λ

α + λ = ⋃ β < λ ( α + β ) {\displaystyle \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.³

See also

• Ordinal arithmetic
• Limit ordinal
• Successor cardinal

References

1. Cameron, Peter J. (1999), Sets, Logic and Categories, Springer Undergraduate Mathematics Series, Springer, p. 46, ISBN 9781852330569. 9781852330569 ↩

2. Cameron, Peter J. (1999), Sets, Logic and Categories, Springer Undergraduate Mathematics Series, Springer, p. 46, ISBN 9781852330569. 9781852330569 ↩

3. Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory, Undergraduate Texts in Mathematics, Springer, Exercise 3C, p. 100, ISBN 9780387940946. 9780387940946 ↩