[Doctor Stochastic]

Any logical system up to the level of specificity to make it worthwhile to use will have truths that are incapable of proof within it.

Not really. Euclidean geometry is catagorical. All theorems are provable. The same is true for Pressberger arithmetic (ordinary arithmetic without multiplication; repeated addition up to any number is allowed.)

I don't know what your term "level of specificity" is supposed to mean, it does not appear in any of the works I have read on Gödel's theorem. I do know what the hypotheses of Gödel's theorem actually is though.

[bvw]

Pressburger arithmetic is a new one to me. And I hadn't known that eucildean geometry is catagorical -- a usage of "catagorical" that also new to me.

I remember blowing through proofs in a few seconds in Geometry class, so perhaps it might be so. Yet I can still hang my hat on what "useful" means, and thereby not eat it.

Arithmetic without X and / is not too useful. And pythagoreus was stumped by irrational numbers -- a suggestion that euclidean geometry -- in that clasical grade school "proof" context -- stops being useful at some point.