Sets of "Isolated" Cardinals

Question

Let $C\neq\emptyset$ be a set of infinite cardinals with the property that NO member of $C$ occurs as the supremum of strictly smaller members of $C$. So the cardinals in $C$ are sort of "isolated".

I am wondering if there is a name for such sets somewhere in literature, and where in mathematics they occur "naturally", if at all.

Answer 1

Maybe you are looking for successor cardinals. They are all regular, so cannot be written as the sup of any smaller set. By contrast limit cardinals like $\aleph_\omega$ can be written as the sup of smaller cardinals. It doesn't seem natural to consider sets like your $C$, as $\aleph_\omega$ could be in it as long as there are only finitely many smaller cardinals in the set.

Answer 2

It is consistent with ZFC that the regular cardinals - those which cannot be written as a "small" union of "small" cardinals - form such a class. Under this condition, of course, "regular" is the same as "successor," but regularity might seem a more natural property.