Reflection principle and compactness theorem in ZFC

Question

I can't see where I'm wrong in the following reasoning.

For the Reflection principle, for every finite set $\Gamma$ of sentences of ZFC we can proof that $$ ZFC\vdash\exists M(M\models\Gamma). $$
So, this means that ZFC finitely consistent. Now by Compactness theorem, we can conclude that $$ ZFC\vdash\exists M(M\models ZFC). $$ I know that, by Gödel's second incompleteness theorem, this is imposible.

Then, where is my mistake?

Answer 1

The issue is that the reflection theorem is really a metatheorem - it is a scheme of theorems which says

For every finite $\Gamma \subseteq \text{ZFC}$, we have $\text{ZFC} \vdash \text{Con}(\Gamma)$.

The reflection theorem does not say

$ \text{ZFC} \vdash \text{For every finite } \Gamma \subseteq \text{ZFC}, \text{Con}(\Gamma)$.

If ZFC proved the latter, then because of compactness, as you wrote, ZFC would prove its own consistency. But the former does not contradict compactness, because the quantifier over finite subtheories is on the left side of the turnstyle.

The Wikipedia article is currently written in a confusing way, but one that is somewhat common in the way that many people write about these principles in set theory.

Also see: Montague's Reflection Principle and Compactness Theorem on MathOverflow