[lanstin]

I can’t tell if this is crazy or brilliant. Math has been working diligently for a long time to reduce the axioms. Most of the obvious Gödel sentences are stupid things like there is a number that is the proof of itself. The whole project is to derive all of the structure of mathematics, with a high information complexity, from basic axioms but also from comp,ex definitions. I think the definitions (natural numbers as sets, integers as equivalence sets of pairs of natural numbers, etc.) pump up the information complexity from the axioms. Like the initial state of Life allowing arbitrary computation from the simple Life rules.

The idea that there might be more axioms that would let one deduce more about computable complexity classes or the like seems pretty unlikely.

The number of provable statements and unprovable statements is countably infinite and we aren’t lacking the ability to prove things due to obviously true missing axioms.