provably

Provably is an adverb used to describe something that can be demonstrated by a proof within a formal system. It indicates that the truth of a statement can be established in a rigorous way according to the accepted rules of inference. In mathematics and theoretical computer science, a result or property may be described as provably true or provably correct. Provable statements are those that have a deduction from the axioms through a finite sequence of logical steps. A program may be said to be provably correct if there is a formal proof that its outputs meet its specification for all valid inputs; provable bounds or running times provide guarantees anchored in proofs.

Provability is relative to a chosen formal theory. Gödel's incompleteness theorem shows that in any reasonably expressive, consistent system, there are true statements that are unprovable within the system. The concept also leads to related ideas such as provability predicates and provability logic, which study how the assertion “this is provable” behaves within a formal framework.

In ordinary language, “provably” can be used to emphasize that there exists a justification or proof. In formal discourse it carries a precise meaning: derivability within a specified axiomatic basis. The word derives from the verb prove, via Latin probare, and is widely used in mathematics, logic, and computability theory.