Gödel's Paradox --- Every set of formulas is consistent 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, $\Gamma\models\psi$. Then $\Gamma\not\models(\neg\psi)$. 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!

Two things.

First of all, $\Gamma \vdash \psi$ implying $\Gamma \models \psi$ is known as the _Soundness Theorem_ for the proof system $\vdash$ (i.e., "true premises do not prove false conclusions").

Now on your purported proof, the flaw occurs at "Then $\Gamma \
ot\models (\
eg \psi)$". Namely, $\Gamma \
ot \models (\
eg\psi)$ can only be valid if there is a model of $\Gamma$ in which $\
eg \psi$ is false. In particular, we have tacitly assumed _existence_ of a model of $\Gamma$ in the first place.

Thus your proof is circular.