I'm reading Fundamentals of Mathematical Logic by Peter G. Hinman, and I'm stuck with a corollary in section 1.4 about propositional theories:
Corollary 1.4.13 For any set $\Gamma$ of sentences and any propositional theory
$T$ , the following are equivalent:
(i) $T$ is a complete propositional theory with $\Gamma \in T$ (a complete extension
of $\Gamma$);
(ii) $T = \mathrm{Th}(V)$ for some model $V$ of $\Gamma$.
It is a corollary to the theorem that if $V$ is a truth assignment, then $\mathrm{Th}(V)$ is a complete and consistent propositional theory, and $V$ is the unique model of this theory. I'm having trouble obtaining a model $V$ of $T$ in the implication (i) => (ii). I tried to first prove that $T$ is consistent but was stuck. Any ideas?
Some definitions:
Definition 1.4.5 A set $T$ of sentences is a propositional theory iff it is closed
under tautological consequence — that is, for all sentences $\phi, T \vDash \phi \implies \phi \in T$.
Definition 1.4.8 For any set $\Gamma$ of sentences,
(i) $\Gamma$ is consistent or satisfiable iff Γ has at least one model; otherwise, Γ
is inconsistent or unsatisfiable;
(ii) $\Gamma$ is complete iff for every sentence $\phi$, either $\phi \in \Gamma$ or $\neg\phi \in \Gamma$, but not both — that is, $(\neg \phi) \in \Gamma \iff \phi \notin \Gamma$; otherwise Γ is incomplete.
Definition 1.4.10 For any truth assignment $V$ , the theory of $V$ is the set $\mathrm{Th}(V) = \{\psi : V (\psi) = T\}$.
Truth assignments are defined to respect connectives.
Best Answer
Since $T$ is complete, for each propositional variable $P$, exactly one of $P$ and $\neg P$ is in $T$. So this gives you a truth assignment: let $V$ be the truth assignment that says a propositional variable is true if it is in $T$ and false if its negation is in $T$. Now you can use the fact that $T$ is closed under tautological consequence to show that in fact $T=Th(V)$.