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by jtc1983
3259 days ago
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Looking at TAPL, it seems like a good text coming at things from the theoretical computer science angle. I'm unsure how exactly to branch out from there, since my own background is more in math logic. So prob best to take my recommendations with that in mind (I think there's a ton of work in CS in the last 20 years that I'm just totally unfamiliar with). You could check out Lambek and Scott (1970s), which is excellent. Also Girard (citation for this one was in TAPL). I also think Martin Lof himself is pretty good for the feel of things, but he has always seemed kind of cryptic and mystical to me. There are also some good videos on YouTube that Steve Awodey (CMU) recorded for more intro/motivation. I'd also add Vovodsky's three Bernays lectures on YouTube for the general idea of using typed lambda calculus vs set theory as foundations. Sociologically, my impression is that most mathematicians trust that the usual axioms of set theory are consistent and safely ignore them. So, yes -- they choose axioms in their own field that are known to be derivable in set theory. Even in intro Analysis textbooks, where the role of set theoretic axioms themselves can be pretty easily extracted from the construction of the reals, I don't think you'd see set theoretic axioms explicitly appealed to (might be wrong about this). In constructive mathematics, it ends up mattering where Choice is used I guess. And it matters how these uses can be "constructivized." But constructive analysis is primarily motivated by foundational considerations. So it is not really an example of ordinary mathematics. Some axioms, like Regularity, are almost never even implicitly used outside of set theory itself except in very specific cases. Most every case where there is a question of whether a given bit of mathematics is derivable from the usual axioms of set theory arise only within set theory itself (CH, measurability of projective sets of reals, etc), and so such questions are easy to ignore for the vast majority of mathematicians. They simply don't care that much about logic and set theory aside from "just working" under the hood, as it were. |
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If one is a mathematician, I think it's entirely reasonable to just "trust" your basic axioms of your "sub-field", but OTOH I can really understand the worry that the rug is going to get pulled out from under you... which is a pretty remote worry given the complexity of math nowadays.
Anyway, thank you for taking the time for that response.