A version of set theory which is a formal system expressed in first-order predicate logic. Zermelo-Fraenkel set theory
is based on the Zermelo-Fraenkel axioms.

Zermelo-Fraenkel set theory is not finitely axiomatized. For example, the axiom of replacement is not really a single axiom, but an infinite family of axioms,
since it is preceded by the stipulation that it is true "For any set-theoretic
formula ."
Montague (1961) proved that Zermelo-Fraenkel set theory is not finitely axiomatizable,
i.e., there is no finite set of axioms which is logically equivalent to the infinite
set of Zermelo-Fraenkel axioms. von
Neumann-Bernays-Gödel set theory provides an equivalent finitely axiomized
system.

Montague, R. "Semantic Closure and Non-Finite Axiomatizability. I." In Infinitistic Methods, Proceedings of the Symposium on Foundations
of Mathematics, (Warsaw, 2-9 September 1959). Oxford, England: Pergamon, pp. 45-69,
1961.Zermelo, E. "Über Grenzzahlen und Mengenbereiche."
Fund. Math.16, 29-47, 1930.