GIT (Goedel's Incompleteness Theorem 1) says that for every consistent formal system there is some statement about the natural numbers which can't be proved in

GIT (Goedel's Incompleteness Theorem 1) says that for every consistent formal system there is some statement about the natural numbers which can't be proved in it. This article made me think that maybe any statement about the naturals could be proved by F^n(M) where F is some sort of "strengthening" operation on the formal system M (probably Peano arithmetic), and n is an ordinal number. The problem of proving statements about integers then reduces to inventing notations for expressing ordinal numbers. It's then easy to believe that there might be diminishing returns on expressing larger ordinals, and therefore GIT doesn't matter much in practical reality. I haven't made the above into a formal conjecture. Is there any truth to the above? Have I missed something obvious? Is there a result in the literature to this effect? Cheers.