Truth and undecidability 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.
 A: It is difficult to give a precise answer, since concepts you are working with are very vaguely defined. Nevertheless, I hope my answer would clarify things a little bit.
Given a theory $T$ and sentence $\phi$, the truth of $\phi$ in $T$ is not well defined (at least there is no universally accepted definition). However, given a structure ${\cal M}$, one can define the truth of $\phi$ in ${\cal M}$. Usually when discussing some theory, one has an intended structure in his/her mind. In this case one may casually call sentences true or false (every sentence is true or false with respect to a given structure). For example, for $PA$ (Peano Arithmetic) the intended structure is the set of natural numbers (with appropriately defined operations). In this example, Godel's sentence is undecidable in $PA$, but true nonetheless.
As you have correctly pointed out, $\phi$ is independent of $T$ if and only if $T + \phi$ and $T + \lnot \phi$ are consistent. A sufficiently strong theory can prove it himself. However, if $T$ is consistent, then for every $\phi$ either $T + \phi$ or $T + \lnot \phi$ is consistent. But the consistency of $T + \phi$ or $T + \lnot \phi$ implies the consistency of $T$. So a sufficiently strong theory $T$ cannot prove the consistency of $T + \phi$ for any $\phi$. 
