@CoronaCoreanici I am aware of the philosophical concept of which you speak; I don't think the mathematicians have gotten around to arguing for its existence yet. Last I heard we were still trying to show that big things are really big and can be classified according to their bigness.

@CoronaCoreanici More recent than that; I'm over in https://en.wikipedia.org/wiki/Inaccessible_cardinal territory saying that ZFC basically says you can always find a way to reclassify 'the ultimate infinite' as a member of a larger set with a new set of properties."

@CoronaCoreanici There's a whole hierarchy of infinities that have been classified to date, but nothing yet that forces a paradox that's passed peer review of which I'm aware.

@CoronaCoreanici Here's the money shot: "Because ZFC + 'there is an inaccessible cardinal' does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + 'there is an inaccessible cardinal' then this latter theory would be able to prove its own consistency, which is impossible if it is consistent."

Basically, if you just straight up assume that they exist, you still end up with a consistent theory, but you can't derive it in advance.

@CoronaCoreanici In that fashion, the inaccessibles parallel the existence of the infinites, which more or less get waved into existence by delcaring "let there be an infinite set: the set of all natural numbers" which is basically Axiom VII of Zermelo Set Theory. (https://en.wikipedia.org/wiki/Zermelo_set_theory)

@CoronaCoreanici Sorry, I have no idea what people know -- I've never asked your field -- and I've done some armchair digging into this. I'm at best a gifted amateur.