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.
Contents

Properties 1

In Von Neumann's model 2

Ordinal addition 3

Topology 4

See also 5

References 6
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

^ ^{a} ^{b} Cameron, Peter J. (1999), Sets, Logic and Categories, Springer Undergraduate Mathematics Series, Springer, p. 46, .

^ Devlin, Keith (1993), The Joy of Sets: Fundamentals of Contemporary Set Theory, .
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