# Proof for subtheories using compactness theorem

I'm trying to understand the use of the Compactness Theorem to proof certain properties for theories in languages. I've tried to prove the following:

If $\phi$ holds in every model of the theory $T$, then there is a finite subtheory $T'$ such that $\phi$ holds in every model of $T'$.

My proof is:

Because $\phi$ holds in $T$, $T \cup \{\neg \phi\}$ is an inconsistent theory. Compactness theorem then says that there is a finite subtheory of this theory that is inconsistent. This theory will be of the from $T' \cup \{\neg \phi\}$ with $T'$ a finite subtheory of $T$. Because $T' \cup \{\neg \phi\}$ is inconsistent, $\phi$ holds in every model of $T'$.

I'm not too sure if this proof is true, especially the part where I take the inconsistent finite subtheory.

• Your inconsistent finite subtheory could be of the form $T'$, without the $\{\neg \phi\}$ - you need to rule out this case. May 21, 2018 at 16:08
• @B.Mehta: No, you don't. May 21, 2018 at 16:11
• If T is consistent, this is ruled out becaue of compactness theorem. What if T is inconsistent?
– xzeo
May 21, 2018 at 16:12
• @AsafKaragila Could you explain where I'm mistaken? May 21, 2018 at 16:12
• @B.Mehta: If $T'$ is a subset of $T$, then $T$ is inconsistent and has no models. The statement holds vacuously. May 21, 2018 at 16:13

Try using Godel's completeness theorem. If $\phi$ is a statement true in every structure, then $\phi$ is a theorem of $T$ and so $\phi$ has a finite proof in $T$.