completeness proofs

A completeness proof in logic shows that if a statement is true in all logical models or interpretations, then there’s a formal proof of that statement within a given logical system. Essentially, it confirms that the system is powerful enough to prove every statement that is universally valid. This assures us that if something is true in every possible scenario, the system can theoretically demonstrate its truth through a rigorous proof process. Completeness links the concepts of truth (semantic) and proof (syntactic), ensuring the logical system is both sound and capable of capturing all truths.