Full Idea
The main alternative to ZF is two-sorted theories, with some variables ranging over classes. Classes have more generous existence assumptions: there is a universal class, containing all sets, and a class containing all ordinals. Classes are not members.
Gist of Idea
The main alternative to ZF is one which includes looser classes as well as sets
Source
Michael Dummett (The Philosophy of Mathematics [1998], 7.1.1)
Book Reference
'Philosophy 2: further through the subject', ed/tr. Grayling,A.C. [OUP 1998], p.168
A Reaction
My intuition is to prefer strict systems when it comes to logical theories. The whole point is precision. Otherwise we could just think about things, and skip all this difficult symbolic stuff.