Ancestral arithmetic and Isaacson's Thesis

Abstract

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 Godel sentence for PA. Second, there are sentences like the arithmetization of Good- stein's Theorem. In the first sort of case, we can come to appreciate the truth of the Godelian undecidable sentences by reflecting on PA's consistency or by coming to accept the instances of the 1 reflection schema for PA. And those routes involve deploying ideas beyond those involved in accepting PA as true. To reason to the truth of the Godel sentence, we need not just to be able to do basic arithmetic, but to be able to reflect on our practice. In the second sort of case, we come to appreciate the truth of the sentences which are undecidable in PA by deploying transfinite induction or other infinitary ideas. So the reasoning again involves ideas which go beyond what's involved in grasping basic arithmetic. Thinking about these sorts of cases suggests a plausible general conjecture. Given the arguments of Daniel Isaacson (1987, 1992), let's call it