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Comment by Dellort | Hacker News Reader

Then you start looking at foundational mathematics. So you learn ZFC set theory, but wait axioms in ZFC are in FOL. Right so FOL is a foundation that comes before ZFC, but wait sets are used in FOL... Back and forth like this for a while and then you're tired of looking for where the beginning is.

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I'm not sure what this has to do with the original post, but I'll go with it...

You also have category theory, proof theory, model theory, recursion theory, type theory... They all fail as "foundations" because they all emit paradoxes. In some cases these can effectively be ignored because they don't play into anything (ZFC, for example), and in others they can be ignored because we can work around them (e.g. category theory).

In fact, few working mathematicians actually care about mathematical "foundations" because there is no way to know if they are "correct".

In what way is ZFC paradoxical? One could certainly accuse it of being arbitrary (given the independence proofs) and incomplete (given Gödel's 1931), but I wasn't aware that one could derive a contradiction in ZFC. Russell's paradox, for example, is not a paradox in ZFC but a proof that there is no set of all sets.

Good catch. I'm using "paradox" very loosely here. What I was trying to say is that we ignore the shortcomings of ZFC: undecidable statements, unnecessarily strong axioms added (regularity and replacement). In some ways avoiding Russell's paradox have made ZFC a weaker theory.

Obviously paradoxes don't have to be actual contradictions, merely results which fail to conform to our intuitions (e.g. the Banach-Tarski paradox), but in my experience when the term is used in the context of the foundations of mathematics, it does mean that a contradiction is derivable within a foundational theory (the class paradoxes being the obvious case in point). This is not a complaint, I'm just explaining why I took your remark in slightly the wrong way.

Working mathematicians ignore the shortcomings of ZFC because foundational issues just aren't what they concern themselves with day-to-day; I'm sure you're aware of the remark that mathematicians are platonists during the week and formalists at the weekend. It's generally left to logicians and philosophers of mathematics (two tribes which, while not coextensional, have a rather large intersection) to worry about these things. Certainly according to structuralists like Shapiro, this is not actually a problem. [1]

[1] Shapiro 1997, http://bit.ly/3om0CO

Well that's not entirely true. There has been a vast effort to make a foundation for mathematics with the various set theories. Naive set theory is usually used as a language for creating the mathematical primitives most working mathematicians deal with.

I agree, there was a vast effort, but that was decades ago. As I said, most working mathematicians don't concern themselves with these issues and just stick with ZFC.

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