Let T be a true theory of arithmetic to which the incompleteness theorems apply. Consider two sentences in the language of T to be equivalent if they are provably equivalent over T. How many equivalence classes are there, under this relation, that contain a true-but-unprovable sentence?
Note: The top part answers an old version of the question, which is now irrelevant.
Now that I reread your question, I think you wanted to have infinitely many logically inequivalent statements each of which is independent of T. This is true when T has no axiomatizable complete extension, which is guaranteed Gödel's Theorem when T is a consistent axiomatizable theory that extends PA.
Indeed, if there were only finitely many statements φ1,...,φk independent of T, up to T-provable equivalence. Then we could get an axiomatizable extension of T by adding to T each such φi or its negation ¬φi while maintaining consistency. (For example, when the standard model satisfies T, we can pick whichever is true in the standard model.) Since we're only adding finitely many new axioms, the result would be an axiomatizable complete theory even if our finitely many decisions were very complex; this would contradict Gödel's Theorem.
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The original question can be read sensibly as follows: Let T be a true theory of arithmetic to which the incompleteness theorems apply. Consider two sentences in the language of T to be equivalent if they are provably equivalent over T. How many equivalence classes are there, under this relation, that contain a true-but-unprovable sentence? This avoids the issue of sentences like φ∧∃x(x=x), which I think is what the question means by "with decidable tautologies or decidable sentences disregarded".
The answer is trivial, though, assuming T is a true theory: there are still countably many such equivalence classes, which is as many as there possible could be. "True theory" means "satisfied by the standard model".
First, T + Con(T) is strictly stronger than to T. Also T + Con(T) is a true theory, and the incompleteness theorems apply to it, so it is strictly weaker than (T+ Con(T)) + Con(T +Con(T)). Continuing this way gives an ω-chain of stronger and stronger true theories extending T, each of which adds only a finite number of (true) axioms to T.
There is a more non-trivial fact that regardless whether T is a true theory, if T is essentially incomplete then the Lindenbaum algebra of sentences modulo provability over T is the countable atomless Boolean algebra, so it has all sorts of structure. This is because any coatom [φ] in this algebra would correspond to a complete, consistent, effective theory T + φ, which cannot exist.
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