 Actually, a simple special case of \(\omega\)-consistency suffices here; namely, the assumption is only needed with respect to what logicians call \(\Sigma^{0}_1\)-formulas; these are, roughly, the purely existential formulas; more exactly, formulas of the form \(\exists x_1\exists x_2 \ldots \exists x_n A\), where \(A\) does not contain any unbounded quantifiers \((A\) may contain bounded universal quantifiers \(\forall x \lt t\) and bounded existential quantifiers \(\exists x \lt t)\). This restricted \(\omega\)-consistency is called 1-consistency.