I am trying to hastily answer the question in the title because I need it for another problem. I have been consulting multiple sources and am confused about the standard meanings of the following terms and the relations among them: successor cardinal, weak/strong limit cardinal, weakly/strongly inaccessible cardinal, large cardinal.
On the terminology front, is there a term analogous to successor cardinal but for the powerset operation instead, e.g., something like powerset cardinal? When repeated applications of an operation are being considered, does this always mean countably many applications or something more general?
From what I understand so far, assuming GCH should not affect the number of weak limit cardinals because it says only that every weak limit cardinal is also a strong limit cardinal and vice versa. Is this correct?
Finally, how many weak limit cardinals are there assuming and not assuming GCH, if this makes a difference? Does GCH change the number of strong limit cardinals?
If anyone wants to point out some cool questions and results surrounding this question, that would also be welcome.