P.S. I solved the problem, my instructor admitted that there is a typo in the definition of completeness. The set $P$ should be $Fml_P$.
The following is an exercise about compactness in propositional logic. (This is from the book A First Journey through Logic.)
Let $Fml_P$ be the set of propositional formulas over an infinite set $P$ of propositional variables. A set $\Sigma \subseteq Fml_P$ is called satisfiable if there is an assignment $\delta : P \rightarrow > \{0,1\}$ such that $\delta^*(F)=1$ for all $F\in \Sigma$; it is called finitely satisfiable if every finite subset $\Sigma_0 \subseteq \Sigma$ is satisfiable.
The aim of this exercise is to prove the following result: A set $\Sigma \subseteq Fml_P$ of propositional formulas is satisfiable if and only if it is finitely satisfiable.
(1) Prove the theorem in the special case when $\Sigma$ is complete, that is, when for every $p\in P$ one has $p\in \Sigma$ or $\neg p > \in \Sigma$.
(2) Prove that every finitely satisfiable set of propositional formulas is contained in a finitely satisfiable set which is complete.
(3) Conclude.
Since this is my homework, I don't want to answer, of course. However, I'm not sure about the definition of completeness. I can prove completely if I take the definition of completenes as follows: when for every $p\in Fml_P$ one has $p\in \Sigma$ or $\neg p \in \Sigma$. The book assumed this for only propositional variables. I wonder if this definition is right, or not. Also, I wonder if it is possible to extend this to all propositional formulas (maybe inductively). If someone share an idea about that, it would be really helpful.
Let me share the related definitions, too.
One fixes a set $P = \{p_i | i ∈ \mathbb{N}\}$, where the $p_i$ are called propositional variables (they will take only “true” or “false” as values). Propositional calculus formulas are defined as words over the alphabet $P \cup \{\neg, \land ,( ,)\}$, formed according to the following rules:
• $p_i$ is a formula for any $i$;
• if $F$ and $G$ are formulas, then $\neg F$ and $(F\land G)$ are formulas, too.
Let $Fml_P$ denote the set of propositional calculus formulas. As before we introduce the abbreviations $\lor , \rightarrow, > \leftrightarrow$.
One identifies $\mathbb{Z}/2 = \{0, 1\}$ with { “false”, “true”}, by assigning 0 to “false” and 1 to “true”. An assignment is a function $\delta : P \rightarrow \{0,1\}$. Any assignment $\delta$ induces a function $\delta^*: Fml_P \rightarrow \{0, 1\}$ defined by $\delta^*(p_i) = \delta (p_i), \delta^*(\neg F) = 1−\delta^*(F)$ and $\delta^*(F \land G) =\delta^*(F)·\delta^*(G)$ , inductively.