Prove that there exists a sentence $\varphi$

Question

I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah.

I have to do this and I don't know how to start.

Let $\Sigma_1 $ and $\Sigma_2$ be sets of sentences such that there is no model $M$ such that both $M\vDash \Sigma_1$ and $M\vDash \Sigma_2$.

Prove that there exists a sentence $\varphi$ such that:

Every model of $\Sigma_1$ satisfies $\varphi$ and every model of $\Sigma_2$ satisfies $\neg \varphi$.

I will be very grateful if you could give me a hint :) Thank you.

Answer 1

Hint/Sketch: Show that if any of $\Sigma_1$ or $\Sigma_2$ are inconsistent then the "theorem" hold. Thus we may assume both $\Sigma_1$ and $\Sigma_2$ are consistent. What can we now say about $\Sigma_1\cup\Sigma_2$? Use the compactness theorem on this larger set to now deduce the theorem.

Edit Further solution: By compactness we have an inconsistent finite subset $A\subseteq \Sigma_1\cup\Sigma_2$. As both $\Sigma_1$ and $\Sigma_2$ are consistent, $A\not\subseteq \Sigma_1$ and $A\not\subseteq \Sigma_2$. Let $\varphi$ be a conjunction of all sentences from $\Sigma_1$ which are in $A$ and let $\psi$ be a conjunction of all sentences from $\Sigma_2$ which are in $A$.

If $N\models \Sigma_1$ then clearly $N\models \varphi$. If $M\models \Sigma_2$ then $M\models \psi$ but then if $M\models \varphi $ too $M\models A$ which it can't since $A$ is inconsistent. Thus $M\not\models \varphi$ i.e. $M\models \neg \varphi$.

Answer 2

By the completeness theorem, "there is no model $M$…" means "$\Sigma_1 \cup \Sigma_2$ is inconsistent". So there are $\alpha$ provable from $\Sigma_1$, $\beta$ provable from $\Sigma_2$ such that $\{ \alpha, \beta \} \vdash \bot$, or else one of $\Sigma_1, \Sigma_2$ is inconsistent.

Now consider $\alpha \wedge \neg \beta$.