The status of a formal sentence relative to a given axiom system when neither the sentence nor its negation can be derived from that system; commonly described as independence from the axioms or unprovability within the chosen theory.

Principle

Provability is relative to the deductive apparatus and axioms: a sentence is unprovable in a theory precisely when no finite derivation from those axioms yields it, and independence is often shown by constructing models in which the sentence holds and others in which it fails.

Demonstration

The Continuum Hypothesis is unprovable (independent) from Zermelo–Fraenkel set theory with Choice (ZFC): one can build models of ZFC where the hypothesis holds and models where it fails, establishing neither side is derivable from ZFC alone.

Misapplication

Interpreting unprovability as evidence that a sentence is false or meaningless; unprovability only reflects the relation to a fixed axiomatic basis, not absolute truth or semantic content outside that basis.

Consequence

Unprovability motivates extending the axiom system, adopting new axioms, or accepting pluralism about which extensions to accept; it also directs metamathematical study of relative consistency and model construction.

Reversal

Provability: the sentence (or its negation) can be derived from the axioms via the accepted inference rules, leaving no independence from that axiomatic system.

Boundary

Unprovability is defined with respect to a chosen theory, language, and deductive system; it does not by itself determine truth in particular models unless complemented by semantic or model-theoretic analysis.

Semantic Tension

Tension exists between proof-theoretic unprovability and semantic undecidability: a sentence can be unprovable yet true in some models and false in others, raising questions about criteria for adopting new axioms.

Synthesis

Unprovability describes a sentence's independence from a specified axiomatic basis; it is a relative, model-sensitive phenomenon that prompts choices about extending theories, changing languages, or accepting multiple coexisting mathematical frameworks.