Successor ordinal

Successor ordinal

When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation "S" to get the next higher one. Using von Neumann's ordinal numbers (the standard ordinals used in set theory), we have, for any ordinal number,

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

Since the ordering on the ordinal numbers α < β if and only if alpha in eta, it is immediate that there is no ordinal number between α and "S"(α) and it is also clear that α < "S"(α). An ordinal number which is "S"(β) for some ordinal β, or equivalently, an ordinal with a maximum element, is called a successor ordinal. Ordinals which are neither zero nor successors are called limit ordinals. We can use this operation to define ordinal addition rigorously via transfinite recursion as follows:

:alpha + 0 = alpha!:alpha + S(eta) = S(alpha + eta)!

and for a limit ordinal λ

:alpha + lambda = igcup_{eta < lambda} (alpha + eta)

In particular, "S"(α) = α + 1. Multiplication and exponentiation are defined similarly.

The successor points and zero are the isolated points of the class of ordinal numbers, with respect to the order topology.

ee also

*ordinal arithmetic
*limit ordinal
*successor cardinal

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