Challenges to Predicative Foundations of Arithmetic

Introduction. This is a sequel to our article “Predicative foundations of arithmetic” (1995), referred to in the following as [PFA]; here we review and clarify what was accomplished in [PFA], present some improvements and extensions, and respond to several challenges. The classic challenge to a program of the sort exemplified by [PFA] was issued by Charles Parsons in a 1983 paper, subsequently revised and expanded as Parsons (1992). Another critique is due to Daniel Isaacson (1987). Most recently, Alexander George and Daniel Velleman (1996) have examined [PFA] closely in the context of a general discussion of different philosophical approaches to the foundations of arithmetic. The plan of the present paper is as follows. Section 1 reviews the notions and results of [PFA], in a bit less formal terms than there and without the supporting proofs, and presents an improvement communicated to us by Peter Aczel. Then Section 2 elaborates on the structuralist perspective which guided [PFA]. It is in Section 3 that we take up the challenge of Parsons. Finally, Section 4 deals with the challenges of George and Velleman, and thereby, that of Isaacson as well. The paper concludes with an appendix by Geoffrey Hellman, which verifies the predicativity, in the sense of [PFA], of a suggestion credited to Michael Dummett for another definition of the natural number concept.