I already asked this but too many typos caused confusion and the editing was confusing me so I re-wrote it here:
If $\theta$ is a sentence in the language of number theory, and we know the following, with the corner braces being the Gödel number: $$N \vdash \theta \leftrightarrow Thm_N(\overline{\ulcorner \neg \theta \urcorner})$$ Note that $N$ is the acioms of Robinsons arithmetic and we have defined $Thm_N(f) \equiv (\exists c)Deduction(c,\,f)$ which basically says $f$ is a Gödel-number of a formula such that $c$ is a Gödel-number for a deduction of $f$.
The question is: is $\theta$ a true sentence (in $\mathfrak{N}$)?
My immediate thought is no, but Im worried I'm interpreting this wrong. Is $Thm_N(\overline{\ulcorner \neg \theta \urcorner})$ saying that there exists a deduction of $\neg \theta$? If so does that mean that $\theta$ must be false? I'm having trouble understanding.