[FOM] Quantifier free sentences in set theory

[Studtmann, Paul]

Robinson's Arithmetic is complete with respect to quantifier free sentences.
I am wondering whether anyone can tell me if an analog of this holds in set
theory.  Suppose, for instance, that the language contains two constants --
one for the  empty set and one for the set of finite ordinals -- as well as
function symbols for the basic set theoretic operations like set union, set
difference, power set, pairing, etc.  Is ZF (or a fragment thereof) or some
other theory complete with respect to all the quantifier free sentences in
the language?


Sincerely,

Paul Studtmann