[lmm]

> I don't get it. There is a lot of variants of set theory. ZFC is just one of them, albeit the most common. But some proofs are founded on an extension of ZFC. For example, the Mizar system, which is another proof assistant, has adopted Tarski–Grothendieck set theory, an extension of ZFC.

Some mathematicians accept extensions of ZFC. But I would definitely expect a proof that relied on such an extension to have in big red letters "we use these additional axioms", and the mathematical community would hope for and expect a proof without it. (Indeed proofs that make use of the axiom of choice still often flag this up and we look for proofs without it where possible). I am quite surprised that Mizar would do this, and certainly hope that it provides a way to distinguish between proofs that are valid in ZFC and those that aren't.

> According to "The QED Manifesto Revisited" [1], it can be difficult to use Coq as-is for some kind of proofs:

The statement about aleph-0 and aleph-2 (the one that Coq has problems with) isn't really mathematics so much as metamathematics/proof theory, at least if we're viewing it as a statement about the consequences of axioms. Which sure is a valuable field and one we'd like to be able to reason about, but it's circumscribed and doesn't affect normal working mathematics.

If we're talking about accepting that axiom and actually using it in a proof, then it's certainly not "totally non-problematic". The paper admits that most mathematicians wouldn't even know the statement of the proper forcing axiom. They certainly wouldn't write proofs that depended on it.

[ngrilly]

Thanks a lot @lmm. Now I get it :)

About Mizar, I don't know if you can use it without the additional axioms of Tarski-Grothendieck set theory.