Definition
Incompleteness is the property a formal system has when there exists a statement that is true but that the system cannot prove. A complete system proves every truth expressible in its language; an incomplete one leaves some truths permanently out of reach.
The notion only makes sense once truth and provability are kept apart. Provability is an internal property — being derivable from the axioms by the rules. Truth is about the subject matter the system describes. Incompleteness is the gap between the two.
Why it matters
How it works
A system is incomplete if some sentence in its language is true but is neither provable nor refutable from its axioms. You might hope to repair this by adding the missing truth as a new axiom — but the strengthened system, if still consistent and strong enough, simply generates a fresh unprovable truth of its own.
Priest presents incompleteness as the headline consequence of Godel's work. It is not a temporary shortfall awaiting better axioms; it is structural. Any consistent formal system rich enough to express elementary arithmetic will leave some arithmetical truth unproved, and no amount of patching can close the gap for good.