nonderivable

Nonderivable is a term used in logic and philosophy to describe propositions that cannot be derived from the axioms and inference rules of a given formal system. A proposition is nonderivable in a theory if no finite sequence of applications of the theory’s rules yields that proposition from its axioms. This notion is relative to the specified framework: a statement may be nonderivable in one theory but derivable in a stronger or differently axiomatized theory, highlighting that nonderivability is not an absolute truth but a property of a particular formal context.

A central motivator for discussing nonderivability is Gödel’s incompleteness theorem. In any consistent, effectively axiomatized theory that is sufficiently expressive to capture basic arithmetic, there exist true statements that are nonderivable within the theory. Such statements are often described as independent from the theory, and, in some contexts, as undecidable or unprovable, though terminology varies by emphasis. Undecidable can refer to problems whose truth value cannot be decided by a given algorithm, while nonderivable focuses on the absence of a proof within the theory’s rules.

In practice, identifying nonderivable propositions motivates the search for new axioms, extensions, or alternative frameworks. It is a foundational concept in proof theory and model theory, illuminating the limits of formal systems and the relationship between syntactic provability and semantic truth. The concept presumes a well-defined theory and does not apply when the theory is inconsistent, since in that case every proposition becomes derivable.