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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.

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

1 Answer 1

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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 $T$.

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  • $\begingroup$ I know this works but I was trying to prove it just by using the compactness theorem. $\endgroup$
    – xzeo
    May 21, 2018 at 16:59

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