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?


  • 5
    $\begingroup$ Take $\tau = \perp$. $\endgroup$ Commented Apr 27, 2011 at 20:56
  • 1
    $\begingroup$ Look at Enderton's proof of the corollary for the propositional case. That will give you some ideas. Hopefully that's in your edition; I'm looking at the old one. $\endgroup$
    – ShyPerson
    Commented Apr 28, 2011 at 3:32

1 Answer 1


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


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