Henkin's Completeness Theorem

Henkin's Completeness Theorem states that in a formal logical system, if a statement is true in every logically consistent interpretation, then there is a formal proof of that statement within the system. Essentially, if something cannot be false under any consistent situation, then the system can verify this truth through its rules. This bridges the gap between semantic truth (truth in all models) and syntactic proof (derivable within the system), assuring that all logically valid statements can be proven within the logical framework, confirming the system's robustness and reliability.