I think the original post is not specific enough to be useful.
I see two essential moot points:
1) Why should be there a system of continuous correspondences between the truth values of sentences that have to do anything with some intuitive notion of truth values?
2) Are the truth values (of the sentences) after taking the fix point actually useful? E.g. can’t it be that we end up truth values of 1⁄2 for almost every sentence we can come up?
Before these points are cleared, the original post is merely an extremely vague speculation.
A closely related analogue to the second issue: in NP-hard optimization problems with lot of {0,1} variables, it’s a most common problem that after a continuous relaxation the system is easily (polynomially) solvable but the solution is worthless as a large fraction of the variables end up to be half which basically says: “no information”.
An infinitely long proof is not a proof, since proofs are finite by definition.
The truth value of a statement does not depend on the existence of a proof anyways, the definition of truth is that it holds in any model. It is just a corollary of Goedel’s completeness theorem that syntactic truth (existence of a (finite) proof) coincides with semantic truth if the axiom system satisfies certain assumptions.