Does Every Ordinal Have a Following Cardinal?

[ibc]

(not assuming any kind of continuum hypothesis of course) does every cardinal have a following one, i.e a minimal cardinal that is strictly larger?edit: ops, the title should be "following cardinal"

 

[Preno]

Yes (assuming the definition of cardinals as a special type of ordinals), it's the minimum of the class of ordinals with greater cardinality than your cardinal (this works because any initial segment of the class of ordinals is a set, and because the ordinals are well-ordered). If you don't want to assume the axiom of choice, you'll have to replace "cardinal" with "well-ordered cardinal".