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.

If $\Sigma_1$ and $\Sigma_2$ are consistent sets and if $\Sigma_1 \vdash \alpha$ for every $\alpha \in \Sigma_2$, is $\Sigma_1 \cup \Sigma_2$ consistent? Intuitively I think it is consistent, but I am not sure how to prove it.

I would also like to know if $\Sigma_1 \vdash \alpha$ for every $\alpha$ such that $\Sigma_2 \vdash \alpha$ is $\Sigma_1 \cup \Sigma_2$ consistent?

Finally, are any difference(s) between the first and second question?

share|improve this question

1 Answer 1

up vote 4 down vote accepted

Under the conditions in the question, $\Sigma_1$ has a model because it is consistent, and that model is a model of $\Sigma_1 \cup \Sigma_2$, because $\Sigma_1$ proves each axiom of $\Sigma_2$. So $\Sigma_1 \cup \Sigma_2$ is consistent.

There is no difference between the two questions; you can show directly that $\Sigma_1$ proves every $\alpha$ in $\Sigma_2$ if and only if $\Sigma_1$ proves every $\alpha$ that is provable from $\Sigma_2$.

share|improve this answer
    
How do I prove this using formal proof and deduction? We use the Hilbert system. –  Mark Aug 12 '11 at 1:52
    
not that they are equivalent. I want to prove these two questions separately using axioms from Hilbert system. –  Mark Aug 12 '11 at 1:59
    
@Mark: The questions don't require proving things in $\Sigma_1 \cup \Sigma_2$, they require proving things about $\Sigma_1 \cup \Sigma_2$. It is conceivably possible that you could do that in a formal system, but that system would not be $\Sigma_1$, it would be some metatheory. It appears more likely that you might be confused about the distinction in the first sentence of this comment. How exactly would you express "$\Sigma_1$ is consistent" in your formal system? –  Carl Mummert Aug 12 '11 at 2:07
    
Well I would show that there is something it can't prove? –  Mark Aug 12 '11 at 2:09
    
I should note that the course teaches very little semantics. Imagine if you didn't know any semantics, what would the problem look like? Right now I only know formal proofs. So could you elaborate a little more about your last point? –  Mark Aug 12 '11 at 2:17

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.