Completeness theorem for propositional logic

For a set of formulae $\Gamma$ and a formula $\phi$,Together with the soundness theorem for propositional logic, this says that semantic consequence and syntactic provability coincide. Equivalently, every consistent set of formulae has a satisfying Boolean valuation.

Assign a primitive proposition true exactly when it belongs to a maximal consistent set. An induction on formulas proves that truth agrees with membership: maximal consistency gives the negation dichotomy, and deductive closure gives the implication truth table. This builds a model of every consistent propositional theory.

Consistency is preserved under unions of chains because proofs use finitely many assumptions. A maximal consistent extension then gives a valuation through its truth lemma, proving propositional completeness.

When the primitive propositions are not countable, replace sequential enumeration of the formulae by Zorn lemma. The union of a chain of consistent extensions is consistent because every formal proof is finite, so every consistent theory extends to a maximal consistent set in propositional logic. Declaring an atom true exactly when it belongs to that maximal set and proving the truth lemma by structural induction produces a model.