Ancestral arithmetic and Isaacson's Thesis
P. Smith · Analysis · 2008
First-order Peano Arithmetic (PA) is incomplete. So the question naturally arises: what kinds of sentences belonging to PA's language LA can we actually establish to be true even though they are unprovable in PA? There are two familiar classes of cases. First, there are sentences like the canonical Gödel sentence for PA. Second, there are sentences like the arithmetization of Goodstein's Theorem. In the first sort of case, we can come to appreciate the truth of the Gödelian undecidable sentences by reflecting on PA's consistency or by coming to accept the instances of the Π1 reflection schema for PA. And these routes involve deploying ideas beyond those involved in accepting PA as true. To reason to the truth of the Gödel sentence, we not only need to be able to do basic arithmetic, but need to be able to reflect on our practice. In the second...