It's more of a theoretical corpus for the sake of this discussion. Just think of it as "all math humans have discovered/invented".

> Are all theorems or axioms useful?

Definitively no! Check out the first incompleteness theorem proof by diagonalization. The whole proof is based around generating new "facts" that are completely useless and uninteresting, outside of their utility in the proof itself.

It's actually a great question: to what extent do the incompleteness theorems actually apply to stuff that isn't silly and useless! The same goes for a couple similar proofs in other domains. An important one in Computer Science is Turing's proof that the halting problem can't be solved. There's at least one other important proof that is similar but I can't remember it at the moment.

To massively oversimplify, they can be boiled down to variations of:

> This statement is false.

Which proves that allowing self-referential statements can create a statement that isn't true or false. But they all raise the question: how often does this actually happen with statements we actually care about?