Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\Sigma_0 \models \tau$.

share|improve this question
    
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) –  Asaf Karagila Feb 18 '13 at 8:31
    
Its listed in the link to the other problem, but I'll include it to make this problem more clear –  Math_Illiterate Feb 18 '13 at 8:33
    
@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.) –  Brian M. Scott Feb 18 '13 at 8:33
    
Thank you, Ockham and @Brian. –  Asaf Karagila Feb 18 '13 at 8:34
add comment

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.