I think it's fairer to say that ZFC was a response to paradoxes in naïve set theory of the sort discovered by Russell than in response to Cantor; see, e.g.,

> In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox.

(https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory). Cantor discovered a rigorous basis for the intuitively paradoxical situation of two sets where it seems that one is manifestly bigger, but they are actually the same size; but I think that this work is best viewed as a resolution of the paradox, not a paradox itself.

He practically invented the field.