Definition
Provability is the property a statement has when it can be derived within a formal system — that is, when there is a finite chain of steps leading to it from the system's axioms, each step licensed by a rule of inference.
Provability is therefore relative to a system. A statement is not provable simply or absolutely; it is provable in this system or that one. Change the axioms or the rules and the set of provable statements changes with them.
Why it matters
How it works
A proof in a formal system is a finite sequence of statements ending in the target, where every statement is either an axiom or follows from earlier ones by an inference rule. Whether such a sequence counts as a proof is decidable by inspection: you can check each step purely formally.
Priest stresses the consequence Godel made vivid. Because provability is mechanical and bounded by a fixed system, while truth answers to the whole subject matter, the two need not coincide. In any consistent system strong enough for arithmetic there are true statements with no proof. Provability captures what a system can establish; it does not exhaust what is the case.