Would love to throw in some arguments (not all *for* logic):
1. I think we agree that "models of ZFC" should be considered to be structures in the sense of any other mathematical structures.
Set theory, as you note, is a kind of "study of structures" in the same vein as other sorts of structure. This I think qualifies it as a legitimate subfield of mathematics. Now where logic comes in is that to me, logic is indispensable as a tool for set theory. We would sort of be floundering around on level 0 without it.
I would go so far as to bat for the point you're making, in that I don't think logic subsumes set theory in any meaningful sense, nor does it subsume any other part of mathematics. It remains a tool and a pretty small part of the content of set theory.
1.a. I will say, however, that I disagree with the point you're making about the study of "realistic" axioms. Every branch of mathematics deals with in some sense separate families of axioms (the ring axioms are not realistic tools for group theory, to pick a silly example). If you are speaking instead of a background family of realistic axioms, then ZFC in that vein has taken a long time to crystallise and shouldn't really be given any special position of being more realistic. The underlying point being, the choice of appropriately realistic axioms is inescapable anywhere in mathematics.
It is certainly true that overexposure to "perverse" axiom systems warps what logicians consider worth studying, but imo there are certain families of axioms that make life simpler in a legitimately mathematical sense, mainly because they create interesting mathematical objects. More on this in point 3.
1.b I agree with you that community consensus is more important than the formal rigour of a proof.
2. You are completely correct about this, surprising results have slowly become extremely commonplace. This is a good thing, in some sense. We wouldn't have progressed unless generations of set theorists thought of incompleteness as inevitable and just moved on.
I'd like to raise, however, what I think are the surprising results that have come out of more recent work:
- the kunen inconsistency, which asserts that certain logical symmetry requirements on a model of set theory contradict the axiom of choice (more on the use of the word symmetry in point 3)
- the covering dichotomies, which assert very sharp order vs chaos dichotomies for the definability of sets (usually of the form, either all sets are approximated by highly definable sets, or the class of definable sets is extremely paltry, for various meanings of "definable")
- the KPT correspondence, which turns facts about continuous actions of certain groups into facts purely about model theory and vice versa
Naturally these are harder to give exact accounts of than incompleteness, but I expect a few of them to reach motivated undergrads within our lifetime.
3. I have the good fortune of working in a very "symmetry" heavy part of set theory. I do indeed spend a lot of my day drawing diagrams and developing visual intuition! The core reason for this is:
What can be thought of abstract logically as "failures of choice" can be thought of more mathematically as "failures of G-equivariant choice". In other words, one doesn't need to warp one's mind to study what happens when "choice doesn't hold" (since in the common consensus it does), one can leave choice untouched and study "G-equivariant" forms of it, and this gives consistency results for forms of choice anyway!
This is a family of methods (called permutation models) that forms a sort of parallel lineage to forcing, in a way that's made a bit clearer with some topos theory (forcing involves posets, which are categories with at most one arrow between any two objects, whereas group actions involve groups, which are categories with all arrows from the same object to itself; the synthesis of these two is the theory of Grothendieck topoi).
I completely share your frustration regarding this, I feel increasingly that forcing and most of its uses do not often have a sense of whimsy, and when they *do*, this is often because the poset being used has a very visual interpretation. It should however be stressed that while the results that pop out at the end are "consistency" results, they nevertheless have mathematical content (G-equivariant choice, for instance, isn't just about choice, but has bearing on G-equivariant cohomology).
The other sense in which symmetries and diagrams and so on crop up is the study of large cardinal axioms. The universe of set theory is in some sense deliberately constructed to avoid symmetries (ZFC typically "wants" sets to have unique descriptions), so when we assume that monoids of symmetries act on a model of set theory, things get very strange!
Here I think a crucial and unfortunate gap in communication takes place with other mathematicians. The study of large cardinal axioms is not that different from defining new forms of *structures* (as opposed to new axiom systems) in other areas of mathematics, and indeed this is the form a lot of "symmetry-based" axioms typically take. I personally wish they were communicated in this manner. The reason they're considered new *axioms* rather than simply definitions of new structures, however, is that their existence is unprovable. This is really the reason that a bit of logic becomes essential to their study.
A reasonable analogy for this is: often a ring will have, definable within it, another ring with different properties. This is a very common construction (think the ring of integers of a field). The only difference in set theory is that if this model of set theory definable inside another model is "too small", it directly bumps into Gödel's incompleteness theorem (more technically, no set sized structure that can be proved to exist from ZFC should be itself a model of ZFC). Large cardinal axioms often let one do this, and this makes them outside the purlieu of vanilla ZFC. This is really the thing (and I should stress the only thing) that sets the study of large cardinals apart from other studies of mathematical structure.
And even so, it bears some striking resemblance to the field of complexity theory, where one should not expect, for instance, for a ptime machine to solve a pspace complete problem (similarly a ZFC model shouldn't automatically have a large cardinal structure). This analogy is made exact in the literature, but this isn't the point of this post so I'll cite it another time.
I'd also argue that there is a sense in which these large cardinal axioms are quite realistic: they phrase precisely the intuition that "highly infinite objects are inscrutable". Furthermore, they often have the pleasing effect of making the underlying axiom system of set theory a lot simpler (in the sense that ZFC + new axiom can be expressed by a much simpler set of axioms). Though they study unrealistic "highly infinite" structures, they're rooted in notions of symmetry and "morphisms" for models of ZFC, and they formalise very realistic concepts.
To tie it all up, I think set theorists focus on the wrong things when talking to other mathematicians about their work. Defining new axiom systems is *fun*, but most of what we do is actually doing maths with them.
P.S. Most techniques in forcing and permutation models have a direct analogue in the theory of topoi, but topoi have received a much more mathematically palatable treatment (mathematicians don't pretend that any topos should be considered *the* universe of set theory, but that topoi are mathematical objects in their own right). Certainly it is hard to argue that Grothendieck topoi lack geometric intuition. I really hope set theory is eventually communicated well enough to receive this treatment (and brings with it the relevant parts of logic).
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