The following definitions and claims appear on p. 31 of Fred Kröger's Temporal Logic of Programs (Springer, 1987). The formulas under discussions are those of propositional temporal logic, but I think it is safe for the purpose of my question to assume regular propositional logic.
Definition 1 A finite set $\mathcal{F} = \{A_1, \dots, A_n\}$ of formulas is called inconsistent if $\vdash\neg(A_1\wedge\dots\wedge A_n)$, otherwise it is called consistent.
Definition 2 Let $F$ be a formula. We define $\tau(F)$ to be the set of subformulas of $F$. If $\mathcal{F}$ is a set of formulas, then we let $\tau(\mathcal{F})=\{A|A\in\tau(F), F\in\mathcal{F}\}$.
Definition 3 A set $\mathcal{F}$ of formulas is called complete if for every formula $A\in\tau(\mathcal{F})$ either $A\in\mathcal{F}$ or $\neg A\in\mathcal{F}$ (but not both).
Lemma 4 Let $\mathcal{F}$ be a finite, consistent set of formulas. There is a finite, consistent and complete set $\mathcal{F}^*$ of formulas with $\mathcal{F}\subseteq\mathcal{F}^*$.
Definition 5 Let us call a set $\mathcal{F}^*$ according to Lemma 4 a completion of $\mathcal{F}$.
Claim 6 In general, there may exist different completions of $\mathcal{F}$, but obviously only finitely many.
Question
Re claim 6: why are there only finitely many completions of $\mathcal{F}$?