What’s so Special About the Gödel Sentence$$\mathcal {G}$$ ?

Abstract

The very fact that the Gödel sentence$$\mathcal {G}$$ is independent of Peano Arithmetic fuels controversy over our access to the truth of$$\mathcal {G}$$. In particular, does the truth of$$\mathcal {G}$$ (of the form$$\forall x\varphi (x)$$ ) precede the truth of its numerical instances$$\varphi (0)$$,$$\varphi (1)$$,$$\varphi (2), \ldots $$, as the so-called standard argument induces one to believe? This paper offers a shift in perspective on this old problem. We start by reassessing Michael Dummett’s 1963 argument which seems to speak in favour of the priority of the truth of the numerical instances of$$\mathcal {G}$$ over the truth of$$\mathcal {G}$$ itself. In opposition to some recent criticisms of Dummett’s argument, we argue that the latter is not reducible to the standard one. We then point out its prototypical nature in the sense individuated by Jacques Herbrand. This shift in perspective brings us to the claim that the controversy over the priority between$$\mathcal {G}$$ and its numerical instances endures only because the problem is ultimately ill-posed. An encompassing moral about the epistemological mechanism of prototype proofs is also drawn.