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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Is mathematics a posteriori?

The general consensus among mathematical philosophers (or philosophical mathematicians, for that matter) is that mathematics is a priori – independent of experience. However, a flourishing contemporary branch within the philosophy of mathematics is that of structuralism, which I will explain in detail in a little bit. Advocates of the structuralist approach are e.g. Shapiro [1] and Resnik [2], who argue how this approach answers a lot of important philosophical questions, including Resnik’s way of answering how mathematics can have applications in the physical world. There is a catch however: he assumes that mathematics is a posteriori – dependent of experience – contrary to popular belief.

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The unstable pillar

When reading about the foundations of mathematics, one is bound to stumble upon the notion of “the four pillars of the foundations of mathematics”; namely, proof theory, model theory, recursion theory and axiomatic set theory. But what exactly is it about these areas that makes them get this prestigious title? One can argue that these “pillars” have a metamathematical nature, in that they try to describe methods of existing mathematics instead of describing elements of the mathematical realm. Proof theory about the way of proofs, model theory about how we construct new structures and recursion theory deals with questions about which parts of mathematics can be computed. I would argue that the last “pillar”, axiomatic set theory, does not fall into this category of metamathematics.

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