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?
First of all I would split set theory up into parts; the part that deals with ZFC-provable theorems and the part independent of ZFC. If we start off with ZFC set theory, the pattern would be the ZFC pattern – but what is that? We have for instance both V and L being models of ZFC, so would both exemplify the ZFC set theory scheme? Would it require an arbitrary choice of such a pattern? Intuition tells us that such an arbitrary choice should only be permissible if the choice doesn’t matter; which in this case would be equivalent to every such pattern being isomorphic. In our special case of V and L, this would be equivalent to the axiom of constructibility, which is independent of ZFC.
On the independent side of set theory, we have multiple patterns arising for every choice of axioms different from ZFC when adding specific axioms as in the study of large cardinals, e.g. by assuming the existence of a measurable cardinal, we study the pattern of ZFC+”there exist a measurable cardinal”. We arrive at the same problem as before though, the question of which models exemplify this structure. Clearly, if the exemplification of ZFC set theory is decided, the resulting exemplification of this pattern would just be that exemplification along with a measurable cardinal.
To bring some perspective to the question, we can look at our intuition regarding number theory studying the natural number pattern. Which model exemplifies this structure? We clearly have the natural numbers, but we also have the finite ordinals (or, the reduct of that to get rid of the membership relation). These structures are isomorphic though, so it doesn’t cause the same problem as the “V = L” question. If we look at the finite ordinals without getting rid of the membership relation, we would end up with a richer structure which isn’t isomorphic to the naturals. Intuition tells us that number theory studies the naturals, and not the naturals along with a membership relation, since such a membership relation isn’t a part of number theory. This leads us to wanting our exemplification of our patterns to be minimal and every such be isomorphic, just as the former reduct of finite ordinals and the naturals was.
This treatment is intuitively clear, but leads to several conjectures independent of ZFC. If we for instance take analysis studying the pattern of the reals with the exemplification being the reals as well, we run into troubles with both the minimal and the isomorphism requirements. Regarding the minimal requirement, we’re basically asking if there exists a proper subset of the reals, satisfying the same axioms as the reals. This requires an answer to the Continuum Hypothesis, which is independent of ZFC. For the isomorphism, we’re asking that if there exist another structure satisfying the axioms of the reals, would this structure be isomorphic to the reals – this is known as the Suslin Hypothesis, which is independent of both ZFC and the Continuum Hypothesis.
If we, despite the apparent complications, stick with these two requirements, we can try to apply this to the two parts of set theory. For ZFC set theory, a minimal structure has been found to be L. In fact, it has been found to be the least structure satisfying ZFC, making it the only structure satisfying this condition; thus trivializing the isomorphism requirement. So, under the assumption of our previously stated requirements, ZFC set theory studies the pattern L. For the other part of set theory concerning extra existence axioms, this would then be L along with such objects.
A natural question though, which I would guess is independent of ZFC as well, would be the question if the isomorphism requirement is necessary. In other words, given two such minimal exemplifications, would they necessarily be isomorphic? I would suppose this could be viewed as sort of a generalized Suslin Hypothesis. Nevertheless, this would help classifying these “representations” of the patterns under investigation, and thus give further arguments for structuralism.