I am sure I have made a gross misunderstanding of Gödel's completeness theorems, as to me, it seems to follow that all sets of formulas are consistent.
Let $\Gamma$ be a set of formulas. If $\Gamma\vdash\psi$, then by Gödel's completeness theorem, $\Gamma\models\psi$.
By Gödel again, $\Gamma\not\vdash(\neg\psi)$.
Hence, $\Gamma$ is consistent.
I am pretty sure there is a flaw in the above argument, but I can't quite pinpoint it.
Any help is sincerely appreciated!