Gödel’s Compactness Theorem

Abstract

So far we have considered only finite sets of clauses. But as we will see in the second part of this course, infinite sets play an important rôle. Therefore we extend the notion of satisfiability as follows: Satisfaction of an infinite set of clauses. Let S be a (finite or infinite) set of clauses. Let V ar (S) denote the set of variables that occur in the clauses of S. Then an assignment α is suitable for S if the domain of α contains V ar (S). We say that α satisfies S, and write

$$\alpha \vDash S,$$

if it satisfies each clause of S; S is unsatisfiable if no assignment satisfies it.