Gödel's Paradox --- Every set of formulas is consistent

Question

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!

Answer 1

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 $\Gamma \not\models (\neg \psi)$". Namely, $\Gamma \not \models (\neg\psi)$ can only be valid if there is a model of $\Gamma$ in which $\neg \psi$ is false. In particular, we have tacitly assumed existence of a model of $\Gamma$ in the first place.

Thus your proof is circular.

Answer 2

If $\Gamma$ is inconsistent to begin with then it has no models. Therefore $\Gamma\models\psi$ holds vacuously, for every $\psi$.

In particular it follows that if $\Gamma$ is inconsistent then $\Gamma\models\lnot\psi$, contrary to your "proof".

That is, the step $(\Gamma\models\psi)\rightarrow(\Gamma\not\models\lnot\psi)$ requires the assumption that $\Gamma$ is consistent, and therefore resulting in circularity.

Answer 3

Check the second step. $\Gamma \not ⊨ \phi$ means there is a model of $\Gamma$, which cannot happen if $\Gamma$ is inconsistent.