1
$\begingroup$

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?

1 Answers 1

4

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 $\Sigma_2$.

  • 0
    How do I prove this using formal proof and deduction? We use the Hilbert system.2011-08-12
  • 0
    not that they are equivalent. I want to prove these two questions separately using axioms from Hilbert system.2011-08-12
  • 0
    @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?2011-08-12
  • 0
    Well I would show that there is something it can't prove?2011-08-12
  • 0
    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?2011-08-12
  • 0
    Suppose $\gamma$ is provable from $\Sigma_1 \cup \Sigma_2$. Take each line in the formal proof where an axiom from $\Sigma_2$ is used, and replace that line of the proof with an entire proof of that axiom from $\Sigma_1$. The result is a proof of $\gamma$ in $\Sigma_1$. Note that the proof I just sketched is not itself in a Hilbert system, it is a proof in the metatheory.2011-08-12
  • 0
    The problem I am trying to prove is a bonus practice question: if $\Sigma_1 \cup \Sigma_2$ is not consistent, then there exists some $\beta$ such that $\Sigma_1 \vdash \beta$ and $\Sigma_2 \vdash \neg\beta$. I believe I can use the above fact to get a contradiction. However, I sure haven't heard of metatheory before. So is my approach correct, or is there some easier proof?2011-08-12
  • 0
    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