Gödel says that there are true statements that can't be proved, given a sound axiomatic system. Does anyone say anything about the provability of the provability of statements?
Is it still an open question that every logic statement that is provable can be proved to be provable? Or equivalently that every unprovable statement can be proved to be unprovable?
Or more specifically, is it still an open question if a subset of the provable statements (for example the true ones) can always be proved to be provable/unprovable so that one day we might come to the conclusion that all the true provable theorems can be proved to be provable?
I think my question may be a little outside of the box. I'll try to be more clear.
Gödel didn't specify the class of statements which are undecidable. What if, for example, all true theorems that can't be proved have some property P.
P being the property ‘useless for all practical purposes’.
Then we’d know that even though there are true theorems which cannot be proved, they are useless theorems. Wait a minute...is there such a thing as a useless theorem?