G_F is provably equivalent to the universal formula ∀x¬Prf_F(x, ⌈G_F⌉) when the provability predicate Prov_F(x) is chosen as a Σ⁰₁-formula

In fact, in favourable circumstances, it can be shown that \(G_F\) is true, provided that \(F\) is indeed consistent. This is the case if, for example, the provability predicate \(\Prov_F (x)\) has been chosen as a \(\Sigma^{0}_1\)-formula: The Gödel sentence is then provably equivalent to the universal formula \(\forall x\neg\Prf_F (x, \ulcorner G_F\urcorner)\). Such formulas can be proved false whenever they in fact are false: if false, there would be a number \(\boldsymbol{n}\) such that \(F