Hilbert's Program Research Articles

On an alleged refutation of Hilbert's Program using G�del's First Incompleteness Theorem

Le premier theoreme d'incompletude de Godel ne permet pas de demontrer l'irrealisabilite du programme de Hilbert. Certaines theories qui incorporent des contraintes de consistance dans les conditions memes de demontrabilite constituent des manieres plausibles de construire des theories ideales au sens de Hilbert

Jean-Yves Girard. Proof theory and logical complexity. Volume I. Studies in proof theory, no. 1. Bibliopolis, Naples 1987, also distributed by Humanities Press, Atlantic Highlands, N.J., 503 pp.

Introduction: Elementary Proof Theory. The Fall of Hilbert's Program. Hilbert's Program. Recursive Functions. The First Incompleteness Theorem. The Second Incompleteness Theorem. Exercises. Annex: Intuitionism. Part I: Sigma 0 1 Proof Theory. The Calculus of Sequents. Definitions. Completeness of the Sequent Calculus. The Cut-Elimination Theorem. The Subformula Property. Intuitionistic Sequent Calculus. Herbrand's Theorem. Generalization. Annex: Natural Deduction. The Church-Rosser Property. Strong Normalization. The Semantics of Sequent Calculus. Completeness of the Cut-Free Rules. Three-Valued Models. Three-Valued Logic. Annex: Takeuti's Conjecture. Limitations of Takeuti's Conjecture. Three-Valued Equivalence. Cut-Free Analysis. Three-Valued Semantics and Generalized Logics. Applications of the ``Hauptsatz''. The Interpolation Lemma. The Reflection Schema of PA. Elementary Consistency Proofs. 1-Consistency. Annex: The Hauptsatz in a Concrete Case. Normalization in HA. Normalization for NL 2 J. Part II: Pi 1 1 Proof Theory. Pi 1 1 Formulas and Well-Foundedness. The Projective Hierarchy. Well-Founded Trees. Well-Orders. Equivalents of (Sigma 0 1 -CA * ). Recursive Well-Orders. Hyperarithmetical Sets. Annex: Kleene's 0. Hierarchies Indexed by 0. Paths Through 0. The Classification Problem. The omega-Rule. omega-Logic. The Cut-Elimination Theorem. Bounds for Cut-Elimination. Equivalents for (Sigma 0 1 -CA * ). Annex: The Calculus Lomega 1 omega. Cut-Elimination in Lomega 1 omega. The Ordinal epsilon o and Arithmetic. Ordinal Analysis of PA. Extensions to other Systems. Ordinals and Theories. Annex: Godel's System T. Functional Interpretation. Spector's Interpretation. No Conterexample Interpretation. An Application. Bibliography. Analytical Index.

Partial realizations of Hilbert's program

§0. Introduction. What follows is a write-up of my contribution to the symposium “Hilbert's Program Sixty Years Later” which was sponsored jointly by the American Philosophical Association and the Association for Symbolic Logic. The symposium was held on December 29,1985 in Washington, D. C. The panelists were Solomon Feferman, Dag Prawitz and myself. The moderator was Wilfried Sieg. The research which I discuss here was partially supported by NSF Grant DMS-8317874.I am grateful to the organizers of this timely symposium on an important topic. As a mathematician I particularly value the opportunity to address an audience consisting largely of philosophers. It is true that I was asked to concentrate on the mathematical aspects of Hilbert's program. But since Hilbert's program is concerned solely with the foundations of mathematics, the restriction to mathematical aspects is really no restriction at all.Hilbert assigned a special role to a certain restricted kind of mathematical reasoning known as finitistic. The essence of Hilbert's program was to justify all of set-theoretical mathematics by means of a reduction to finitism. It is now well known that this task cannot be carried out. Any such possibility is refuted by Gödel's theorem. Nevertheless, recent research has revealed the feasibility of a significant partial realization of Hilbert's program. Despite Gödel's theorem, one can give a finitistic reduction for a substantial portion of infinitistic mathematics including many of the best-known nonconstructive theorems. My purpose here is to call attention to these modern developments.

Partial realizations of Hilbert's program

§0. Introduction. What follows is a write-up of my contribution to the symposium “Hilbert's Program Sixty Years Later” which was sponsored jointly by the American Philosophical Association and the Association for Symbolic Logic. The symposium was held on December 29,1985 in Washington, D. C. The panelists were Solomon Feferman, Dag Prawitz and myself. The moderator was Wilfried Sieg. The research which I discuss here was partially supported by NSF Grant DMS-8317874.I am grateful to the organizers of this timely symposium on an important topic. As a mathematician I particularly value the opportunity to address an audience consisting largely of philosophers. It is true that I was asked to concentrate on the mathematical aspects of Hilbert's program. But since Hilbert's program is concerned solely with the foundations of mathematics, the restriction to mathematical aspects is really no restriction at all.Hilbert assigned a special role to a certain restricted kind of mathematical reasoning known as finitistic. The essence of Hilbert's program was to justify all of set-theoretical mathematics by means of a reduction to finitism. It is now well known that this task cannot be carried out. Any such possibility is refuted by Gödel's theorem. Nevertheless, recent research has revealed the feasibility of a significant partial realization of Hilbert's program. Despite Gödel's theorem, one can give a finitistic reduction for a substantial portion of infinitistic mathematics including many of the best-known nonconstructive theorems. My purpose here is to call attention to these modern developments.

Hilbert's program relativized; Proof-theoretical and foundational reductions

In the Symposium on Hilbert's Program to which the following was contributed, I was asked to talk about the metamathematical aspects of Hilbert's Program (H.P.), while the other two speakers (Simpson and Prawitz) were to deal with the mathematical and philosophical aspects respectively. However, more so than for other foundational schemes, these three aspects of H.P., both as originally conceived and in its subsequent developments, are intimately linked.Here I shall survey a body of proof-theoretical results stemming from H.P., but organized in a way that is closely tied to various reductive foundational aims, albeit going beyond those advanced by Hilbert. I believe this view of reductive proof-theory (not original with me) helps one to better understand what has been achieved thereby than other, more familiar accounts.

Hilbert's program relativized; Proof-theoretical and foundational reductions

In the Symposium on Hilbert's Program to which the following was contributed, I was asked to talk about the metamathematical aspects of Hilbert's Program (H.P.), while the other two speakers (Simpson and Prawitz) were to deal with the mathematical and philosophical aspects respectively. However, more so than for other foundational schemes, these three aspects of H.P., both as originally conceived and in its subsequent developments, are intimately linked.Here I shall survey a body of proof-theoretical results stemming from H.P., but organized in a way that is closely tied to various reductive foundational aims, albeit going beyond those advanced by Hilbert. I believe this view of reductive proof-theory (not original with me) helps one to better understand what has been achieved thereby than other, more familiar accounts.

Hilbert's program sixty years later

On June 4, 1925, Hilbert delivered an address to the Westphalian Mathematical Society in Miinster; that was, as a quick calculation will convince you, almost exactly sixty years ago. The address was published in 1926 under the title Über dasUnendlicheand is perhaps Hilbert's most comprehensive presentation of his ideas concerning the finitist justification of classical mathematics and the role his proof theory was to play in it. But what has become of the ambitious program for securing all of mathematics, once and for all? What of proof theory, the very subject Hilbert invented to carry out his program? The Hilbertian ambition reached out too far: in its original form, the program was refuted by Gödel's Incompleteness Theorems. And even allowing more than finitist means in metamathematics, the Hilbertian expectations for proof theory have not been realized: a constructive consistency proof for second-order arithmetic is still out of reach. (And since that theory provides a formal framework for analysis, it was considered by Hilbert and Bernays as decisive for proof theory.) Nevertheless, remarkable progress has been made. Two separate, but complementary directions of research have led to surprising insights: classical analysis can be formally developed in conservative extensions of elementary number theory; relative consistency proofs can be given by constructive means for impredicative parts of second order arithmetic. The mathematical and metamathematical developments have been accompanied by sustained philosophical reflections on the foundations of mathematics. This indicates briefly the main themes of the contributions to the symposium; in my introductory remarks I want to give a very schematic perspective, that is partly historical and partly systematic.

Hilbert's program sixty years later

On June 4, 1925, Hilbert delivered an address to the Westphalian Mathematical Society in Miinster; that was, as a quick calculation will convince you, almost exactly sixty years ago. The address was published in 1926 under the title Über das Unendliche and is perhaps Hilbert's most comprehensive presentation of his ideas concerning the finitist justification of classical mathematics and the role his proof theory was to play in it. But what has become of the ambitious program for securing all of mathematics, once and for all? What of proof theory, the very subject Hilbert invented to carry out his program? The Hilbertian ambition reached out too far: in its original form, the program was refuted by Gödel's Incompleteness Theorems. And even allowing more than finitist means in metamathematics, the Hilbertian expectations for proof theory have not been realized: a constructive consistency proof for second-order arithmetic is still out of reach. (And since that theory provides a formal framework for analysis, it was considered by Hilbert and Bernays as decisive for proof theory.) Nevertheless, remarkable progress has been made. Two separate, but complementary directions of research have led to surprising insights: classical analysis can be formally developed in conservative extensions of elementary number theory; relative consistency proofs can be given by constructive means for impredicative parts of second order arithmetic. The mathematical and metamathematical developments have been accompanied by sustained philosophical reflections on the foundations of mathematics. This indicates briefly the main themes of the contributions to the symposium; in my introductory remarks I want to give a very schematic perspective, that is partly historical and partly systematic.

Mathematical Intuition and Husserl's Phenomenology

In this paper I shall discuss the role of the concept of intuition in mathematics from a somewhat special point of view. Much of the interest in the notion has derived from its place in Hilbert's program for the foundation of mathematics and, more recently, from some of Godel's views on the subject.1 However, I shall orient my discussion around some rather strong views that were advanced by another philosopher, Husserl. Husserl did not develop anything that could specifically be called a theory of mathematical intuition, but he did attempt to give a general theory of intuition which should have application in different domains of knowledge. In particular, in the Sixth Investigation of the Logical Investigations, and in Part II of Formal and Transcendental Logic, Husserl attempted to give an account of intuition (Anschauung) suitable to the kinds of abstract objects which are typical to mathematics and logic. For the most part, Husserl's investigations are concerned with what might be thought of as necessary, but not sufficient, conditions for mathematical knowledge, such as are provided by elementary observations on the grammar of formal languages, and on the consistency and definiteness of axiom systems.2 On the specific question of what constitutes evidence (Evidenz) for mathematical objects, or what it means to intuit or have representing content for mathematical objects, Husserl's own observations leave much to be desired. The problem is most acute in the case of the kinds of abstract objects for which it is not thought to be possible to find concrete, space-time instances. The situation is exacerbated by the fact that Husserl did not attempt to keep abreast of developments in the foundations of mathematics, so that his works on logic are not as well-informed as they might be. Husserl nonetheless did claim that there was intuition of abstract objects, that the evidence for objects of intuition could come in different degrees, with apodicticity and adequacy giving a maximal measure on evidence,

The Reception of Gödel's Incompleteness Theorems

“Die Arbeit über formal unentscheidbare Sätze wurde wie ein Erdbeben empfunden; insbesondere auch von Carnap.” - - (Popper 1980).“Kurt Gödel's achievement in modern logic … is a landmark which will remain visible far in space and time.” - - John von NeumannIt is natural to invoke geological metaphors to describe the impact and the lasting significance of Gödel's incompleteness theorems. Indeed, how better to convey the impact of those results - - whose effect on Hilbert's program was so devastating and whose philosophical reverberations have yet to subside - - than to speak of tremors and shock waves? The image of shaken foundations is irresistible.

Remarks on the development of computability

The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized and the theory of computability was developed. It is suggested in Section 3 that a central item was the problem of generalizing Godel's incompleteness theorem. It is shown that this involved both the characterization of recursiveness and the attempt to clarify and formulate the notion of an effective process as it relates to the syntax of deductive systems. Section 4 concerns the decision problems which grew from the Hilbert program. Section 5 is devoted to the development of an informal' technique in the theory of...

On interpreting G�del's Second Theorem

In this paper I have considered various attempts to attribute significance to G2.25 Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilbert's Program), I have argued, are literally false. Two others (BCR and Resnik's Interpretation), I have argued, are groundless.

Hilbert's Epistemology

Hilbert's program attempts to show that our mathematical knowledge can be certain because we are able to know for certain the truths of elementary arithmetic. I argue that, in the absence of a theory of mathematical truth, Hilbert does not have a complete theory of our arithmetical knowledge. Further, while his deployment of a Kantian notion of intuition seems to promise an answer to scepticism, there is no way to complete Hilbert's epistemology which would answer to his avowed aims.

Topological Group Foundations of Rigid Space Geometry

1. Hilbert, after building up geometry from a point of view which relegated considerations to the background [4], built up plane geometry afresh [5] on the foundation of groups of homeomorphisms of the number plane, both continuity concepts. It is this latter point of view with which we are concerned in this paper. Hilbert carried out this program only for the plane but he hinted that it might be possible to carry it out in a somewhat similar way for three-space. Kerekjarto took up the problem [6] for threespace and made a great deal of progress with it, but, as he wrote before the recent developments in topological groups, he found it necessary to employ a stronger set of axioms than is necessary now. Relying on the theory of topological groups we recently characterized the rotation group of three-space [9], and in commenting on that work P. A. Smith [12] suggested that it might provide the basis for an extension of Hilbert's program to three-space. The purpose of this paper is twofold. In the first place we shall characterize the classical space geometries on the basis of a fairly weak set of axioms, and in the second place we shall show that Hilbert's axioms for the case of the plane can be weakened by replacing what might be called his three-point condition by a two-point condition. We achieve this latter purpose more or less incidentally to the first. In comparing our set of three-space axioms with Hilbert's axfoms for the plane we find that the first axiom is the same in both cases. The third axiom of this paper is weaker than Hilbert's, and our second purpose above is to show that this weaker axiom also suffices in Hilbert's case. Our second axiom is weaker than Hilbert's second axiom in that it relates to the subgroup leaving a single point fixed, but it is incomparably stronger in what it asks of that one subgroup. We do not, in this paper, settle the question of whether or not Hilbert's second axiom is adequate for three-space geometries. This question is bound up with an unsolved problem concerning transformation groups. Finally we wish to remark that instead of assuming that the space we are dealing with is ordinary three-space, it is only necessary to make certain topological assumptions on the space from which it follows by virtue of the

Topological group foundations of rigid space geometry

1. Hilbert, after building up geometry from a point of view which relegated continuity considerations to the background [4], built up plane geometry afresh [5] on the foundation of groups of homeomorphisms of the number plane, both "continuity" concepts. It is this latter point of view with which we are concerned in this paper. Hilbert carried out this program only for the plane but he hinted that it might be possible to carry it out in a somewhat similar way for three-space. Kerekjarto took up the problem [6] for threespace and made a great deal of progress with it, but, as he wrote before the recent developments in topological groups, he found it necessary to employ a stronger set of axioms than is necessary now. Relying on the theory of topological groups we recently characterized the rotation group of three-space [9], and in commenting on that work P. A. Smith [12] suggested that it might provide the basis for an extension of Hilbert's program to three-space. The purpose of this paper is twofold. In the first place we shall characterize the classical space geometries on the basis of a fairly weak set of axioms, and in the second place we shall show that Hilbert's axioms for the case of the plane can be weakened by replacing what might be called his "three-point condition" by a two-point condition. We achieve this latter purpose more or less incidentally to the first. In comparing our set of three-space axioms with Hilbert's axfoms for the plane we find that the first axiom is the same in both cases. The third axiom of this paper is weaker than Hilbert's, and our second purpose above is to show that this weaker axiom also suffices in Hilbert's case. Our second axiom is weaker than Hilbert's second axiom in that it relates to the subgroup leaving a single point fixed, but it is incomparably stronger in what it asks of that one subgroup. We do not, in this paper, settle the question of whether or not Hilbert's second axiom is adequate for three-space geometries. This question is bound up with an unsolved problem concerning transformation groups. Finally we wish to remark that instead of assuming that the space we are dealing with is ordinary three-space, it is only necessary to make certain topological assumptions on the space from which it follows by virtue of the