Proof of Lindenbaum's lemma for uncountable languages

Question

I am trying to understand this proof of Lindenbaum's lemma for uncountable languages. (First Proof, second proof is for countable lang)

Suppose Δ is consistent. Let P be the partially ordered set of all consistent supersets of Δ, ordered by inclusion ⊆

My questions are:

1. What P is it?

2. Why it is possible to partially order a set of consistent supersets and how would we do it?

Answer 1

$P$ is the set of all consistent supersets of $\Delta$, as it says. Simply look at all supersets of $\Delta$ and reject the non-consistent ones. There isn't really enough context in the web page for me to know where $\Delta$ comes from, but I assume there is some universal set $V$, and so we start with all subsets of $V$ that contain $\Delta$.

As for how we order $P$ it is partially ordered by $\subseteq$. This is true of any set of sets, as I'm sure you know.

Finally, the reason to do this is to apply Zorn's lemma. You must have seen proofs like this before. For example, the proof that every vector space has a basis.