# Finite satisfiability and Zorn's lemma

Given a set $$\Sigma$$ of well-formed formulas(wffs), I want to get a set $$\Delta$$ with the following conditions: $$\Delta$$ is a set of wffs such that (i) $$\Sigma\subset\Delta$$ (ii) it is finitely satisfiable (iii) for every wff $$\alpha$$, either $$\alpha\in\Delta$$ or $$\neg\alpha\in\Delta$$.

To do that one needs to apply Zorn's lemma. But, I don't exactly know how to set up for the Zorn's lemma and to apply the Zorn's lemma. So far I know that we need to deal with a family of sets and the inclusion for the relation to apply the Zorn's lemma. But, how should I set up the sets?; of what form the sets should it be? And how should I apply the Zorn's lemma?

• Hint: What if you were looking for a set $\Delta$ with properties (i), (ii) and a property (iii') $\Delta$ is $\subseteq$-maximal with properties (i) and (ii)? Commented Oct 24, 2020 at 8:22
• This won't work for arbitrary $\Sigma$, though Commented Oct 24, 2020 at 8:24
• @JohannesKloos I need the (iii) condition..
– kkkk
Commented Oct 24, 2020 at 8:28

Let $$\mathcal Z$$ be the set of candidates for $$\Delta$$ if we ignore (iii), i.e., the elements of $$\mathcal Z$$ are all sets $$A$$ of wffs such that $$\Sigma \subseteq A$$ and $$A$$ is finitely satisfiable. Then $$\mathcal Z$$ is partially ordered by $$\subseteq$$. Show that it is inductively ordered and conclude.

However, this will work only if $$\Sigma$$ is nice.

• I need for arbitray set $\Sigma$ though...
– kkkk
Commented Oct 24, 2020 at 8:29
• And need (iii) condition..
– kkkk
Commented Oct 24, 2020 at 8:30
• @kkkk Condition (iii) will follow from maximality. Additionally, if $\Sigma$ is inconsistent, you will be in trouble... Commented Oct 24, 2020 at 8:50
• I see. By the way what do you mean by inconsistent?
– kkkk
Commented Oct 24, 2020 at 8:57
• To elaborate on the latter point, suppose $\Sigma = \{ p, \neg p \}$ for some proposition $p$. Every set $\Delta \subseteq \Sigma$ will not be finitely satisfiable, since it will contain both $p$ and $\neg p$, so the set you are looking for will not exist. Commented Oct 24, 2020 at 8:58