# For the Compactness Theorem for Propositional Logic, show that the extension is not unique.

During the proof of the compactness theorem, from an arbitrary finitely satisfiable set $\Sigma$ of WFFs, we construct a finitely satisfiable set $\Delta\supseteq \sigma$ such that for every WFF $\alpha$, either $\alpha\in\Delta$ or $\lnot\alpha \in\Delta$. Show that $\Delta$ need not be unique by describing an infinite, finitely satisable set $\Sigma$ of WFFs such that there is more than one possible extension $\Delta$.

Could someone please give me some guidance in answering this question? Much appreciated. Thanks.

• If $\beta\notin \Sigma$ then $\Sigma\cup\{\beta\}$ and $\Sigma\cup\{\lnot\beta\}$ can be extended maximal finitely satisfiable set and they are not same. Commented Oct 6, 2013 at 1:08
• @tetori: Not so fast. How about $\Sigma=\{\alpha, \alpha\to\beta\}$? Then $\Sigma\cup\{\neg\beta\}$ will not be satisfiable. Commented Oct 6, 2013 at 2:12
• @HenningMakholm Oh, it is my mistake :( Commented Oct 6, 2013 at 2:40

Hint. Suppose the language contains some unary predicate that is not mentioned in $\Sigma$ at all ...