I believe this is more of a philosophical question.
Given a consistent theory T and a statement S independent of T. Can S be true or false in T? (I don't see any contradiction with that)
I read that Godel thinks the Continuum Hypothesis is false (in ZFC ?) even though independent of ZFC. (so a statement could be undecidable and false according to him)
But at the same time his first theorem shows that if statement S is independent of T then T+S AND T+(not S) are both consistent.
If S was true or false in T then we would have T+S OR T+(not S) consistent. (but you couldn't prove the consistency of T+S or T+(not S) otherwise S would be decidable in T)
So to conclude from the above statement I get: if S is true or false in the consistent theory T and independent of T then the consistency of T+S is undecidable (in T) ?
So my questions are, Do my above statements make sense ? Can their exist undecidable statements that are true ? Is it possible to build one ? is this question undecidable ? if yes in what axiomatic system ? or is it just philosophy ? ... is decidability always decidable (in T) ?
Note: Each time I wrote undecidable I tried to write the theory i'm working with, but I am not sure I am working in the right one.