kompaktaessatsen

The compactness theorem, also known as the completeness theorem for propositional logic, is a fundamental result in mathematical logic. It states that a set of propositional formulas is satisfiable if and only if it is consistent. In simpler terms, it means that if a set of statements does not lead to a contradiction, then there exists a way to assign truth values to the individual statements such that all statements in the set are true simultaneously.

This theorem has several important implications. Firstly, it establishes a strong connection between semantic notions (satisfiability) and syntactic notions (consistency, often expressed in terms of derivability from a set of axioms or lack of a proof of contradiction). Secondly, it serves as a cornerstone for many other results in logic, including the completeness of various proof systems for propositional logic. The proof of the compactness theorem often involves constructing a model or truth assignment from a consistent set of formulas, demonstrating the existence of such a model. It is a crucial tool in model theory and provides a foundation for understanding the properties of logical theories.