Let $\Sigma$ be a maximal consistent set of formulas. Prove that |mod($\Sigma$)|= 1

Question

I know the definition of maximal consistent set of formulas, but until now I cannot see how to use the definitions to prove that there only exists one model of any maximal consistent set $\Sigma$. Maybe we have to use the fact that $\Sigma$ is the set of all its syntactic consequences or we just have to imagine two different models of $\Sigma$ and prove that they are the same one.

Answer 1

Hint: if $\Sigma$ is maximal consistent then for all $\varphi$ either $\Sigma\vDash\varphi$ or $\Sigma\vDash \neg\varphi$.

Now consider $\varphi$ of the form $p_i$ where $p_i$ is a propositional variable and remember that in propositional logic a model is just a truth valuation.