The Completeness Theorem

The Completeness Theorem states that if a statement is logically valid—meaning it’s true in every possible situation—then there is a formal proof of it using the rules of the logical system. In other words, if something must always be true, then it can be proven through logical reasoning from the system’s axioms. Conversely, if a statement can be proven within the system, it is necessarily true in all situations. This linking of "truth" and "provability" ensures that the logical system is capable of capturing all universally valid truths.