Vollständigkeitssatz

Vollständigkeitssatz refers to a fundamental concept in mathematical logic, specifically within propositional logic and first-order logic. The term translates to "completeness theorem" in English. Essentially, a completeness theorem establishes that for a given logical system, the set of syntactically derivable statements (theorems) is precisely the same as the set of semantically true statements (logical consequences).

In simpler terms, if a statement can be proven to be true using the rules of inference of a logical system, then it must be true in all possible interpretations of that system. Conversely, if a statement is true in all interpretations, then there must exist a proof for it within the system. This ensures that the proof system accurately captures all logical truths.

The first significant completeness theorem was proven by Kurt Gödel in 1929 for first-order logic. His work demonstrated that the formal system of first-order logic is complete in the sense that any statement that is a logical consequence of a set of axioms can be derived as a theorem from those axioms using the rules of inference. This theorem is crucial for the foundations of mathematics and computer science, as it validates the power of formal proof methods. Variations and extensions of completeness theorems exist for different logical systems.