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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. Every ordinal other than 0 is either a successor ordinal or a limit ordinal.
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 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).
and for a limit ordinal λ
In particular, S(α) = α + 1. Multiplication and exponentiation are defined similarly.