Compactness theorem equivalent

Question

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.

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. — ShyPerson, Apr 28 '11 at 03:32

score 4 · Answer 1 · edited Apr 27 '11 at 21:46

4

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

edited Apr 27 '11 at 21:46

answered Apr 27 '11 at 21:40

Bill Dubuque

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Compactness theorem equivalent

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