User blog comment:Rgetar/Help to name function/@comment-30754445-20180423200016

"Cardinality" is the proper official term for this function.

And yes, Ωβ is both a cardinal and an ordinal. A "cardinal" is nothing more than an ordinal α which obeys: for all β<α, there's no bijection from α to β. Alternatively, the "cardinality" of a given set A is the minimal ordinal α for which there exists a bijection from A to α.

(if the above confuses you, remember that every ordinal is also a set: an "ordinal" is defined as the set of all ordinals smaller then itself. Thus the ordinal 5 is the set {0,1,2,3,4} and the ordinal ω is the set of all non-negative integers)