In a sound theory, $T$, for a Godel sentence $G_{T}$ is $\neg G_{T}$ being false a true or false statement Smith's "Intro to Godel's Theorems," defines $T$ as a sound theory as everything $T$ proves must be true.
He then mentions a Godel sentence $G_{T}$ which I would characterize as $T\vdash (G_{T}\leftrightarrow\neg\square G_{T})$.
"So $G_{T}$ is true if and only if $T$ can't prove it....Hence if $T$ is sound, $G_{T}$ is unprovable in $T$. Which makes $G_{T}$ true." 
I understand the above and that it follows from $T\vdash (G_{T}\leftrightarrow\neg\square G_{T})$.
Now comes my question: 
He goes on to say, "Hence $\neg G_{T}$ is false. And so that too can't be proved by $T$, since $T$ only proves truths." 
But in this instance, isn't $\neg G_{T}$ being false a truth?
Or, what probably is incorrect: $T\vdash(\neg G_{T}\leftrightarrow \square G_{T})$, and since I started with $G_{T}$ is true, then in this case $\neg G_{T}$ is true.
Thanks 
 A: $T$ does indeed prove $\neg G_T\iff\Box G_T$. However, I'm not sure how you get from this to the claim in your last sentence that $\neg G_T$ is true.
I think the key is the following: we're assuming that $T$ only proves true statements, but we're not assuming that $T$ proves all true statements. $G_T$ is true, and if $T$ proved $G_T$ then $G_T$ would be false, but we can't go from "$G_T$ is true" to "$T$ proves $G_T$" (in fact, that's the whole point of $G_T$).
So I think the issue is really linguistic here: that "$T$ only proves truths" is easy to misread as "the things $T$ proves are the truths."
A: You understood that $G_T$ is true. Well, then it follows that $\neg G_T$ is false. 
Of course, that also means that the statement "$\neg G_T$ is false" is a true statement .. but we're not really interested in that statement here.
Likewise, if we say that $A$ is true, then $\neg A$ is false. And again, it would then also be true that "$\neg A$ is false", but that does not make $\neg A$ true. In fact, if it is true that "$\neg A$ is false", then clearly $\neg A$ is false, not true. 
We would only have that $\neg A$ is true if "$\neg A$ is false" is false.
