The first version was developed by Ernst Zermelo in 1908; in
1922, Abraham Fraenkel and Thoralf Skolem? independently
proposed a precise first-order version with the axiom of
replacement; von Neumann added the axiom of foundation in 1925.
All of these versions included the axiom of choice, but this was
considered controversial for some time; one has merely if it is
taken out. is similar to the class theories (due to John von
Neumann, Paul Bernays?, and Kurt Gödel?) and (due to Anthony
Morse? and John Kelley, see Morse-Kelley set theory). The former
is a conservative, finitely axiomatisable extension of , while
the latter is stronger and cannot be finitely axiomatised
(without introducing meta-classes or something). Contemporary
set theorists often accept additional large cardinal axioms,
which can greatly increase the strength of , far beyond even .
Other additional axioms which are sometimes added are the axiom
of determinacy? (or various weaker versions of it) or the axiom
of constructibility?. There are al