Ever couldn't sleep or eat thinking what happens when truth is replaced with provability in the omega liar? Now you can put your qualms to rest, this issue (needless to say, of ultimate relevance to Great Philosophical Questions) has been handled! Cezary Cieslinski and I have finished drafting a paper about this. (Of course, the entailment of corollaries about The Meaning of Life is so obvious that we didn't bother stating them.)
The paper is available here.
Abstract. We investigate what happens when `truth' is replaced with `provability' in Yablo's paradox. By diagonalization, appropriate sequences of sentences can be constructed. Such sequences contain no sentence decided by the background consistent and sufficiently strong arithmetical theory. If the provability predicate satisfies the derivability conditions, each such sentence is provably equivalent to the consistency statement and to the Godel sentence. Thus each two such sentences are provably equivalent to each other. The same holds for the arithmetization of the existential Yablo paradox. We also look at a formulation which employs Rosser's provability predicate.