unprovable

Unprovable is an adjective describing a proposition or claim that cannot be derived as a theorem from a given set of axioms by the rules of a formal proof system. Whether a statement is provable depends on the chosen framework, so a claim may be unprovable in one theory and provable in another.

In mathematics and logic, Gödel's incompleteness theorems show that any consistent, effectively axiomatized system that is strong enough to encompass arithmetic will contain true statements that are unprovable within the system. The second theorem also implies that such a system cannot prove its own consistency from within.

Independence results illustrate unprovability: if both a statement and its negation are unprovable from a theory, the statement is independent of that theory. The continuum hypothesis is a famous example: it is independent of ZFC, meaning neither CH nor its negation can be proved from ZFC alone.

Unprovable does not mean false or unknowable. It indicates a limit of the given formal framework; the statement may be true, but true only relative to a stronger or different axiomatization, or it may require a different method of justification.

In broader contexts, "unprovable" can describe claims in epistemology, philosophy, or science that lack formal proofs within current methodologies, relying instead on evidence, reasoning, or consensus.

See also: undecidable proposition; Gödel's incompleteness theorem; independence (mathematics); axiomatic system.