# If $\Sigma_1 \vdash \alpha$ for every $\alpha \in \Sigma_2$, is $\Sigma_1 \cup \Sigma_2$ consistent?

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?

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## 1 Answer

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 $\Sigma_2$.

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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