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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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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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    The metatheory is just the theory in which you prove all the results about formal theories. Usually these proofs are just written in English, unlike the formal proofs of statements in the formal theories being studied. It seems like like the method you just sketched will also work; to prove the statement in the first line you need to use the fact that $\Sigma_1$ and $\Sigma_2$ are consistent individually (you might already know this). All the proofs will be about the same level of difficulty, I think, so it's not worth chasing for an even easier one once you have a proof you are happy with.2011-08-12