Which pattern does set theory study?

This post originates from a question I read on stackexchange. If we assume that the structuralist thesis is true, saying that mathematics is really the study of patterns and nothing else, where does this leave set theory? We have some intuition about number theory studying the pattern of natural numbers, real analysis about the reals and so on, but prima facie it seems that set theory doesn’t fit patterns in the same way. One might start off with the naive answer that set theory studies the pattern of sets – but then we encounter the classical paradoxes. If we instead fix ourselves on ZFC, how come set theory studies (almost exclusively) objects independent of ZFC?

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