I'm going through Enderton's Mathematical Logic text and have encountered a problem that I'm having trouble solving. After searching this website I've found that another user had the same problem (you can check it out here), and even after looking at the hints listed I'm still really confused about the problem of proving the compactness theorem using its corollary.

I'd appreciate it if someone could clearly explain how to approach this problem and provide a little more insight. Thanks in advance!

(Corollary 17A) Suppose $\Sigma \models \tau$, then there is a finite $\Sigma_0 \subseteq \Sigma$ such that $\Sigma_0 \models \tau$.

  • $\begingroup$ And the corollary that the compactness theorem is proved from is? (for those of us who do not have the text in front of them) $\endgroup$ – Asaf Karagila Feb 18 '13 at 8:31
  • $\begingroup$ Its listed in the link to the other problem, but I'll include it to make this problem more clear $\endgroup$ – Math_Illiterate Feb 18 '13 at 8:33
  • $\begingroup$ @Asaf: At the link: If $\Sigma\models\tau$, then there is a finite $\Sigma_0\subseteq\Sigma$ such that $\Sigma_0\models\tau$. (But it would be good to have this in the question.) $\endgroup$ – Brian M. Scott Feb 18 '13 at 8:33
  • $\begingroup$ Thank you, Ockham and @Brian. $\endgroup$ – Asaf Karagila Feb 18 '13 at 8:34

Suppose that every finite subset of $\Sigma$ has a model. If $\Sigma$ has no model, then it’s vacuously true that $\Sigma\models\tau$ for any $\tau$ whatsoever. Take $\tau$ to be $\exists x(x\ne x)$. By the corollary there is a finite $\Sigma_0\subseteq\Sigma$ such that $\Sigma_0\models\tau$, which is absurd, since $\Sigma_0$ by hypothesis does have a model.

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