 In his original proof, Gödel used his specific notion of \(\omega\)-consistency, and for some purposes, it is still convenient to follow Gödel’s original approach. A formalized theory \(F\) is \(\omega\)-consistent if it is not the case that for some formula \(A(x)\), both \(F \vdash \neg A(\underline{n})\) for all  \(\mathbf{n}\), and \(F \vdash \exists xA(x)\). Naturally this implies normal consistency, and follows from the assumption that the natural numbers satisfy the axioms of \(F\).