[throwaway81523]

Imho, software people should study foundations (particularly proof theory) much more than they do. That is how you ensure correct code, after all. The people deeply involved in software verification are basically logicians. Also, the test suite for the HOL Light proof assistant (used in software verification) uses a large cardinal axiom, sort of. It uses a version of itself with the large cardinal added, to prove the consistency of the normal version without the large cardinal. Neither version can prove itself consistent, because of Gödel's theorem, but the one with the large cardinal can prove the consistency of the one without it.

One can say that if either is inconsistent then they can prove everything, but that makes it even sharper: the large cardinal is used purely to give engineering assurance of software correctness and not real mathematical rigor. So it's a pure engineering use of one of the most "out there" mathematical objects. It doesn't seem worse than using IEEE floating point arithmetic to design airplanes....