The worst thing is when someone who thinks that "math is 100% infallible and all about rigor, you gotta show your work and include all the steps" yet they think that set theory is good enough and it doesnt have problems
they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different object that exists in some unexplainable sidecar of set theory"
They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff
My favorite was when a manager tried to get me to agree with the statement that "math was just for the numbers right?". Meaning not character strings nor dates. I was dumbstruck by the question.
It's 100% possible to base logic and proof theory off of set theory. For example, you can treat proofs as natural numbers via Gödel encoding (or any other reasonable encoding) and we know that natural numbers can be represented by sets in multiple different ways.
You may prefer type theory or other foundations, but set theory is definitely rigorous enough and about as "infallible" (or not) as other approaches.
Yeah, they should have heard about ZFC and have a notion what a formal proof is. On the other hand, I'm not sure your last sentence is really that relevant.
> They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff
I'm doing a PhD in algebraic geometry, and that stuff isn't relevant at all. To me "everything is a set" pretty much applies. Hell, even the stacks-project[1] contains that phrase!
My claim is that mathematics as practiced by mathematicians is not as rigorous as they think it is, and infact they're not even aware of the advances in rigorous mathematics that they're not using. Even though those advances are super important.
Well, you've been claiming that these advances are "super important" and that set theory is not rigorous, but you have provided no evidence for either claim.
I never said "set theory is not rigorous." Euclid wrote Elements without knowing anything about set theory. Math was done for thousands of years without modern set theory or any modern notion of logical foundations. Set theory is more rigorous than what came before it.
It's not as rigorous as type theory (yes, this is an umbrella term) because type theory can be verified by a computer. Homotopy type theory is an example of the type of math that set theory can't handle
There are so many layers of ignorance to unpack here and I don't care to be your unpaid tutor
they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different object that exists in some unexplainable sidecar of set theory"
They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff