Full Question: Show that if $\Delta(\Gamma)$ is the deductive closure of a consistent $\Gamma$ then $\Delta(\Gamma)$ is the intersection of all maximal consistent sets of formulas $\Lambda$ with $\Gamma \subseteq \Lambda$.
I'm really stuck with this question, I honestly don't know how to even start. Can someone please provide some assistance?
You have two things to prove: (1) that $\Delta(\Gamma)$ is a subset of any maximal consistent superset of $\Gamma$; (2) that if $\phi \not\in \Delta(\Gamma)$ then there is a maximal consistent superset of $\Gamma$ that does not contain $\phi$.
For (1), if $\chi \in \Delta(\Gamma)$ and $\Lambda$ is a consistent superset of $\Gamma$, then so is $\Lambda \cup \{\chi\}$. So if $\Lambda$ is a maximal consistent superset, then $\chi \in \Lambda$.
For (2), use Zorn's lemma: consider the set $\cal C$ of all consistent supersets of $\Gamma$ that do not contain $\phi$. There is at least one member of $\cal C$, namely $\Gamma$ itself. The union of a chain of elements of $\cal C$ is also a member of $\cal C$, so by Zorn's lemma $\cal C$ has a maximal element.