TITLE: Compactness theorem equivalent
QUESTION [9 upvotes]: I've been reading Enderton's Mathematical Introduction to Logic. One of the exercises on Compactness theorem requires the proof that the following corollary
[(Corollary 17A) Suppose $\Sigma \models \tau$, then there is a finite $\Sigma_0 \subseteq \Sigma$ such that $\Sigma_0 \models \tau$. ]
is equivalent to Compactness Theorem.
Can any one give me a hint on how to prove CT from this statement?
Thanks.

REPLY [4 votes]: HINT $\ (\Rightarrow)\ $ Apply CT to $\Sigma \cup \{\lnot \:\tau\}\:.\: $ $\rm\ (\Leftarrow)\ $ Let $\rm\:\tau\ =\ \exists x\ (x \ne x)$