Unprovability

Unprovability refers to the property that a statement cannot be derived from the axioms of a formal system using its rules of inference. A statement is provable if a finite proof exists within the system; it is unprovable if no such proof can be found. A statement may be true in the intended interpretation yet remain unprovable in a given theory.

Gödel’s incompleteness theorems show fundamental limits of formal systems. If a theory T is consistent and sufficiently expressive to formalize basic arithmetic, then there are true mathematical statements that cannot be proved within T. The first theorem guarantees the existence of such undecidable statements. A common presentation uses a sentence G that says “G is not provable in T”; if T is consistent, G is true but unprovable in T. The second theorem states that T cannot prove its own consistency, assuming consistency.

Independence is a related concept. A statement is independent of a theory if neither the statement nor its negation is provable from the theory. Classical examples include the continuum hypothesis in ZFC, assuming ZFC is consistent; many other natural mathematical statements are independent of weaker theories like PA. Independence results are established through methods such as forcing and model theory.

The study of unprovability highlights the limits of formalizing all of mathematics within a single system. It also motivates views of relative consistency, where one shows that if a theory is consistent, then its stronger extensions obtained by adding independent statements are also consistent. Unprovability thus delineates the boundary between provability and truth within formal frameworks.