Set theory as formalised by Frege was a "naive" set theory in which an unqualified principle of abstraction was available.
This means that the extension of any property expressible in the formal language corresponded to that of some set, and this
liberal ontological principle was shown by Russell's paradox to render the logical system inconsistent.

Two earliest response to this problem, published by Russell and Zermelo in 1908, both effectively realised consistency by
constraining the ontology to be well-founded (though Russell's solution was more elaborate, its central principle was the
avoidance of "vicious" circularity).

The dominant set theory since those early times is a derivative of Zermelo's system called Zermelo-Frankel set theory, particularly
the version with a choice principle (ZFC).