i'm very familiar with all of this (i cut my teeth on baby rudin and i have dipped into papa rudin over the years). what i'm interested in is not the value (i understand the value of set theory like i understand the value of ZFC and Peano) but the "beauty". note i "know" point-set topology (at the level of munkres) and even a little cohomology so i'm looking for something a little more sophisticated than just hausdorff spaces or something like that.
edit: but that is a nice paper you linked to. thank you.
Given your background I'm surprised you'd think you might be called a hipster for not being a fan of axiomatic set theory :). Most math people I know learn just the naive set theory they need to move on to analysis and algebra (and later topology and algebraic geometry).
|\N| = |\Z| = |\Q^2457000| != |\R| isn't wondrous to you? Nor are space-filling curves? Or unmeasurable sets and paradoxes like Banach-Tarski? You don't see beauty in the proofs of Cantor's theorem (|S| < |P(S)| for all S) or the Bernstein-Schroeder theorem (injections X -> Y and Y -> X imply |X| = |Y|)? What about the plethora of mathematical claims that are equivalent to the axiom of choice -- some of them very sought-after properties, some of them unnerving and seemingly broken? Does the status of the continuum hypothesis (and the mathematical legitimacy of, to say nothing of the resolution to, a bunch of oddball "infinitary combinatorics" problems) interest you?
What about where set theory interplays with model theory? Would it surprise you to hear that the dizzying towers of infinities that can be spoken of in ZF can be modeled by a merely countable universe? What about a computer program (albeit an impractical one) that decides whether sentences about the natural numbers with successor are true or false, whose existence is discovered by proving that large uncountable models are isomorphic?
To be clear, it's okay to be bored by this stuff. I tend to get bored pretty quickly when people start talking about systems of DEs or eigenvectors, or really much of anything that looks too much like calculus, and I'm not about to apologize for my apathy for those things. But most of the above stuff tends catch the attention of even the humanities types, who are easily deterred by most mathematical topics.
Out of curiosity, I might ask whether you're similarly bored by computability theory or complexity theory, which leverage many of the same tools as those used in set theory (mappings as preorders, diagonal arguments, etc), but add computable enumerability/uniformity/syntacticity requirements to the mix.
yes Bernstein-schroeder wasn't interesting except in its "universality". Cantor diagonalization yes was very formative for me in my mathematical maturation and Godel too but no the complexity towers - recursively enumerable and nondenumerable don't interest me (or whatever the class that NP belongs to is called - I've forgotten most of the recursion theory I learned now). so yes computability theory was very boring to me. in the parlance I'm a geometer rather than an algebraist (so analysis, manifolds, relativity, etc.)
Personally (as someone who did several pure math courses as an undergraduate but is not by any means a pure mathematician) I think Cantor’s set theory and everything that flows from it is a bunch of hokum.
But if mathematicians’ convention is that they will refer to ZFC when writing their proofs about something else I am interested in, I am willing to humor them, and keep my caveats to myself.
I’m reasonably well convinced that most models mathematicians can make in the context of ZFC which have any real-world implications (e.g. accurately simulate some physical phenomenon) could be alternately proved under some more restrictive set of axioms, just sometimes with a lot more hassle.
In my opinion it is entirely speculative, built on largely unconvincing foundations (involving a lot of hand waving and circular reasoning when you start poking at them) whose implications are often absurd.
But it doesn’t really matter what I think. This is not a fight I care about. It’s like arguing with Buddhists about the nature of suffering or something.
Can a statement like Goodstein's theorem (which is about finite objects) be proven by finitist mathematics? It cannot be proven by Peano arithmetic, which is bi-interpretable with ZFC with the axiom of infinity replaced by its negation [1], or any weaker theory.
Could you expand on this a little? Which reasoning do you find to be circular and which implications to you find to be absurd? If you explain what you an issue with others may be able to help you understand it more clearly.
edit: but that is a nice paper you linked to. thank you.