This is a sequel to our article “Predicative foundations of arithmetic” (Feferman and Hellman [1995], reproduced as Chapter 7 in this volume), 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 [1998] have examined PFA closely in the context of a general discussion of different philosophical approaches to the foundations of arithmetic.

Predicative mathematics in the sense originating with Poincaré and Weyl begins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or set-theoretic standpoint, this appears problematic, for, as the story is usually told, impredicative principles seem to play an essential role in the foundations of arithmetic itself. It is the main purpose of this paper to show that this appearance is illusory: as will emerge, a predicatively acceptable axiomatization of the natural number system can be formulated, and both the existence of structures of the relevant type and the categoricity of the relevant axioms can be proved in a predicatively acceptable way.

Summary. From 1931 until late in his life (at least 1970) Gödel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Gödel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of infinity. In particular, he speculated (in his 1946 Princeton remarks) that there might be a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the “higher infinite” in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences.

In this paper, I present a new very general notion of the “unfolding” closure of schematically axiomatized formal systems S which provides a uniform systematic means of expanding in an essential way both the language and axioms (and hence theorems) of such systems S. Reporting joint work with T. Strahm, a characterization is given in more familiar terms in the case that S is a basic system of non-finitist arithmetic. When reflective closure is applied to suitable systems of set theory, one is able to derive cardinal axioms as theorems. It is an open question how these may be characterized in terms of current notions in that subject.

Why new axioms?

Gödel's published statements over the years (from 1931 to 1972) pointing to the need for new axioms to settle both undecided number-theoretic and set-theoretic propositions are rather well known. They are most easily cited by reference to the first two volumes of the edition of his Collected Works. A number of less familiar statements of a similar character from his unpublished essays and lectures are now available in the third volume of that edition.

Given the ready accessibility of these sources, there is no need for extensive quotation, though several representative passages are singled out below for special attention.

Introduction

We offer here some historical notes on the conceptual routes taken in the development of recursion theory over the last 60 years, and their possible significance for computational practice. These illustrate, incidentally, the vagaries to which mathematical ideas may be susceptible on the one hand, and – once keyed into a research program – their endless exploitation on the other.

At the hands primarily of mathematical logicians, the subject of effective computability, or recursion theory as it has come to be called (for historical reasons to be explained in the next section), has developed along several interrelated but conceptually distinctive lines. While this began with what were offered as analyses of the absolute limits of effective computability, the immediate primary aim was to establish negative results of the effective unsolvability of various problems in logic and mathematics. From this the subject turned to refined classifications of unsolvability for which a myriad of techniques were developed. The germinal step, conceptually, was provided by Turing's notion of computability relative to an ‘oracle’. At the hands of Post, this provided the beginning of the subject of degrees of unsolvability, which became a massive research program of great technical difficulty and combinatorial complexity. Less directly provided by Turing's notion, but implicit in it, were notions of uniform relative computability, which led to various important theories of recursive functionals. Finally the idea of computability has been relativized by extension, in various ways, to more or less arbitrary structures, leading to what has come to be called generalized recursion theory. Marching in under the banner of degree theory, these strands were to some extent woven together by the recursion theorists, but the trend has been to pull the subject of effective computability even farther away from questions of actual computation. The rise in recent years of computation theory as a subject with that as its primary concern forces a reconsideration of notions of computability theory both in theory and practice. Following the historical sections, I shall make the case for the primary significance for practice of the various notions of relative (rather than absolute) computability, but not of most methods or results obtained thereto in recursion theory.

Gödel, Bernays, and Hilbert

The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel's Collected Works (1986–2003) devoted to his letters of (primarily) scientific, philosophical, and historical interest. Ranging from 1930 to 1975, except for one long break, this correspondence engages a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and post-Kantian philosophy. In addition, Gödel's side of the exchange includes his thoughts on many topics that are not expressed elsewhere and testify to the lifelong warm, personal relationship that he shared with Bernays. I have given a detailed synopsis of the Bernays-Gödel correspondence, with explanatory background, in my introductory note to it in CW IV (pp. 41–79). My purpose here is to focus on only one group of interrelated topics from these exchanges, namely, the light that this correspondence – together with assorted published and unpublished articles and lectures by Gödel – throws on his perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert's program. In that connection, this piece has an important subtext, namely, the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career.

Let me explain. Hilbert and Ackermann (1928) posed the fundamental problem of the completeness of the first-order predicate calculus in their logic text; Gödel (1929) settled that question in the affirmative in his dissertation a year later.

The final two volumes, numbers IV and V, of the Oxford University Press edition of the Collected Works of Kurt Gödel appeared in 2003, thus completing a project that started over twenty years earlier. What I mainly want to do here is trace, from the vantage point of my personal involvement, the at some times halting and at other times intense development of the Gödel editorial project from the first initiatives following Gödel's death in 1978 to its completion last year. It may be useful to scholars mounting similar editorial projects for other significant figures in our field to learn how and why various decisions were made and how the work was carried out, though of course much is particular to who and what we were dealing with.

My hope here is also to give the reader who is not already familiar with the Gödel Works a sense of what has been gained in the process, and to encourage dipping in according to interest. Given the absolute importance of Gödel for mathematical logic, students should also be pointed to these important source materials to experience first hand the exercise of his genius and the varied ways of his thought and to see how scholarly and critical studies help to expand their significance.

Though indeed much has been gained in our work there is still much that can and should be done; besides some indications below, for that the reader is referred to.

The year 2006 marked the centennial of Kurt Gödel, who was born on 28 April 1906. The importance of Gödel's work for nearly all areas of logic and foundations of mathematics hardly needs to be explained to our readers.

The year 2006 saw several centennial observances. In particular, the program committee for the 2006 Association for Symbolic Logic annual meeting, which took place on 17–21 May at the Université du Québec à Montréal, commissioned a subcommittee to arrange a portion of the program that would commemorate the Gödel centennial. The subcommittee arranged three one-hour lectures, by Jeremy Avigad, Sy-David Friedman, and Akihiro Kanamori. It also arranged a two-hour special session on Gödel's philosophy of mathematics, with lectures by Steve Awodey, John Burgess, and William Tait. All of the lectures have led to papers in this volume. The volume contains one other new paper, “The Gödel hierarchy and reverse mathematics,” by Stephen G. Simpson. Other papers included in the volume are reprinted, in all but one case from The Bulletin of Symbolic Logic. We have included the papers presented at the 2004 ASL annual meeting at Carnegie-Mellon University, in a special session organized by the editors of Gödel's Collected Works, by Martin Davis, John W. Dawson, Jr., Cheryl A. Dawson, Solomon Feferman, Warren Goldfarb, Donald A. Martin, Wilfried Sieg, and William Tait. These appeared in the June 2005 Bulletin. Also reprinted are papers by Mark van Atten and Juliette Kennedy and by Charles Parsons that appeared in earlier issues of the Bulletin, as well as a paper by Peter Koellner that appeared in Philosophia Mathematica.