 A theory is called \(\omega\)-consistent if the following holds for every formula \(\phi(x)\) containing \(x\) as its only free variable: If \(\vdash \neg \phi(n)\) for every natural number \(n\) then \(\not\vdash \exists x\phi(x). \omega\)-consistency is a stronger condition than ordinary consistency, so any \(\omega\)-consistent theory will also be consistent. A theory is incomplete if it contains a formula which can neither be proved nor disproved.