Completeness Theorem (Gödel)

Gödel’s Completeness Theorem states that if a statement is logically true—meaning it is true in every possible situation—then there is a formal proof of that statement within a logical system. In other words, all universally valid logical truths can be proven using the rules of the system. This bridges the gap between semantic truth (truth in all models) and syntactic proof (derivable through formal rules), ensuring that if something must be true, there is a systematic way to verify it with mathematical rigor.