Fullständighetssatsen

Fullständighetssatsen, or the Completeness Theorem, is a fundamental result in mathematical logic, most notably established by Kurt Gödel in 1929. It pertains to first-order logic, also known as predicate logic. The theorem states that a formula in first-order logic is logically valid if and only if it is provable. In simpler terms, it establishes an equivalence between semantic consequence and syntactic provability.

Semantic consequence means that a formula holds true in all possible interpretations or models of the theory. Syntactic provability means that the formula can be derived through a finite sequence of logical inference rules starting from axioms. The "if" part of the theorem, that is, if a formula is provable, then it is logically valid, is generally straightforward to demonstrate and is often referred to as soundness. The more significant and challenging part is the "only if" direction: if a formula is logically valid, then it is provable.

Gödel's Completeness Theorem has profound implications for mathematics and logic. It demonstrates that formal systems of first-order logic are "complete" in the sense that they can capture all valid logical truths. This means that there are no valid statements in first-order logic that cannot be proven within the system. It provides a strong foundation for the consistency and expressive power of formal deductive systems.