Are Maximal Consistent set of formulas unique?

Question

I know that for any consistent set of formulas $\Gamma$, there exists a maximal consistent set of formulas $\Theta$, such that $\Gamma\subseteq\Theta$. Given this, do we know if $\Theta$ is unique?

Answer 1

No, having a unique maximal consistent extension is equivalent to a theory being complete and consistent. So any incomplete consistent theory (e.g. PA) has more than one maximal consistent extension.

If $T$ is consistent and incomplete, and $\varphi$ is undecidable in $T,$ then $T\cup\{\varphi\}$ and $T\cup \{\lnot \varphi\}$ are both consistent, so have (necessarily different) maximal consistent extensions which are both maximal consistent extensions of $T$.