Page "Morse–Kelley set theory" Paragraph 39

MK can be confused with second-order ZFC, ZFC with second-order logic ( representing second-order objects in set rather than predicate language ) as its background logic.

The language of second-order ZFC is similar to that of MK ( although a set and a class having the same extension can no longer be identified ), and their syntactical resources for practical proof are almost identical ( and are identical if MK includes the strong form of Limitation of Size ).

But the semantics of second-order ZFC are quite different from those of MK.

For example, if MK is consistent then it has a countable first-order model, while second-order ZFC has no countable models.