Compactness Theorem (Propositional Logic)

Let and .
If then there is a finite subset of s.t. .

Proof

Trivial if is replaced with by the Completeness Theorem.

Corollary

Let .
If every finite subset of has a model, then has a model.

Proof

If doesn’t have a model, then .
By previous, there is a finite subset of s.t.