Hilbert's Program Research Articles

THE COLLAPSE OF THE HILBERT PROGRAM: A VARIATION ON THE GÖDELIAN THEME

AbstractThe Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to $\exists xA(x)$ , or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent ( ${\Sigma}_1^0$ -correct). Here we show that if the result is supposed to be provable within S, a statement about all ${\Pi}_2^0$ statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel’s but arises naturally out of the Hilbert program itself.

Arithmetic Formulated Relevantly


 
 
 The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that (1) it is trivial that relevant arithmetic is absolutely consistent, but (2) classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under (1), I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, of course, is exactly the formula whose unprovability was sought in the Hilbert program for proving arithmetic consistent. Under (2), I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. 
 
 
 
 The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment. 
 
 
 
 
 

The Consistency of Arithmetic


 
 
 This paper offers an elementary proof that formal arithmetic is consistent. The system that will be proved consistent is a first-order theory R♯, based as usual on the Peano postulates and the recursion equations for + and ×. However, the reasoning will apply to any axiomatizable extension of R♯ got by adding classical arithmetical truths. Moreover, it will continue to apply through a large range of variation of the un- derlying logic of R♯, while on a simple and straightforward translation, the classical first-order theory P♯ of Peano arithmetic turns out to be an exact subsystem of R♯. Since the reasoning is elementary, it is formalizable within R♯ itself; i.e., we can actually demonstrate within R♯ (or within P♯, if we care) a statement that, in a natural fashion, asserts the consistency of R♯ itself. 
 
 
 
 The reader is unlikely to have missed the significance of the remarks just made. In plain English, this paper repeals Goedel’s famous second theorem. (That’s the one that asserts that sufficiently strong systems are inadequate to demonstrate their own consistency.) That theorem (or at least the significance usually claimed for it) was a mis- take—a subtle and understandable mistake, perhaps, but a mistake nonetheless. Accordingly, this paper reinstates the formal program which is often taken to have been blasted away by Goedel’s theorems— namely, the Hilbert program of demonstrating, by methods that everybody can recognize as effective and finitary, that intuitive mathematics is reliable. Indeed, the present consistency proof for arithmetic will be recognized as correct by anyone who can count to 3. (So much, indeed, for the claim that the reliability of arithmetic rests on transfinite induction up to ε0, and for the incredible mythology that underlies it.) 
 
 
 
 
 

Review of The Foundations of Computability Theory (Second Edition) by Borut Robič

Computability theory forms the foundation for much of theoretical computer science. Many of our great unsolved questions stem from the need to understand what problems can even be solved. The greatest question of computer science, P vs. NP, even sidesteps this entirely, asking instead how efficiently we can find solutions for the problems that we know are solvable. For many students both at the undergraduate and graduate level, a first exposure to computability theory follows a standard sequence on data structures and algorithms and students often marvel at the first results they see on undecidability - how could we possibly prove that we can never solve a problem? This book, in contrast with other books that are often used as first exposures to computability, finite automata, Turing machines, and the like, focuses very specifically on the notion of what is computable and how computability theory, as a science unto itself, fits into the grander scheme. The book is appropriate for advanced undergraduates and beginning graduate students in computer science or mathematics who are interested in theoretical computer science. Robič sidesteps the standard theoretical computer science progression - understanding finite automata and pushdown automata before moving into Turing machines - by setting the stage with Hilbert's program and mathematical prerequisites before introducing the Turing machine absent the usual prerequisites, and then introducing advanced topics often absent in introductory texts. Most chapters are relatively short and contain problem sets, making it appropriate for both a classroom text or for self-study.

The proof-theoretic strength of Ramsey's theorem for pairs and two colors

Ramsey's theorem for n-tuples and k-colors (RTkn) asserts that every k-coloring of [N]n admits an infinite monochromatic subset. We study the proof-theoretic strength of Ramsey's theorem for pairs and two colors, namely, the set of its Π10 consequences, and show that RT22 is Π30 conservative over IΣ10. This strengthens the proof of Chong, Slaman and Yang that RT22 does not imply IΣ20, and shows that RT22 is finitistically reducible, in the sense of Simpson's partial realization of Hilbert's Program. Moreover, we develop general tools to simplify the proofs of Π30-conservation theorems.

Hilbert's sixth problem and the failure of the Boltzmann to Euler limit.

This paper addresses the main issue of Hilbert's sixth problem, namely the rigorous passage of solutions to the mesoscopic Boltzmann equation to macroscopic solutions of the Euler equations of compressible gas dynamics. The results of the paper are that (i) in general Hilbert's program will fail because of the appearance of van der Waals-Korteweg capillarity terms in a macroscopic description of motion of a gas, and (ii) the van der Waals-Korteweg theory itself might satisfy Hilbert's quest for a map from the 'atomistic view' to the laws of motion of continua.This article is part of the theme issue 'Hilbert's sixth problem'.

Bernays and the Completeness Theorem

A well-known result in Reverse Mathematics is the equivalence of the formalized version of the Godel completeness theorem [8] – i.e. every countable, consistent set of first-order sentences has a model – and Weak Konig's Lemma [WKL] – i.e. every infinite tree of 0-1 sequences contains an infinite path– over the base theory RCA0. It is less well known how the Completeness Theorem came to be studied in the setting of second-order arithmetic and computability theory. The first goal of this note will be to recount these developments against the backdrop of the latter phases of the Hilbert program, culminating in the publication of the second volume of Hilbert and Bernays’s [13] Grundlagen der Mathematiks in 1939. This work contains a detailed formalization of the Completeness Theorem in a system similar to first-order Peano arithmetic [PA] – a result which has come to be known as the Arithmetized Completeness Theorem. Its second goal will be to illustrate how reflection on this result informed Bernays’s views about the philosophy of mathematics, in particular in regard to his engagement with the maxim “consistency implies existence”.

Algunas consecuencias filosóficas del trabajo de Kurt Gödel

<p class='p1'>El artículo presenta los argumentos que le permiten a Gödel defender el realismo platónico a partir de sus resultados de incompletitud. Se sostiene que hay un estrecho paralelismo con lo que podríamos denominar el fracaso del programa cartesiano. El programa cartesiano fracasó al querer mostrar la autonomía plena de la razón y tuvo que reconocer en la intuición sensorial otra fuente del conocimiento objetivo de los objetos externos. De la misma manera, el programa de Hilbert, como se deduce de los teoremas de incompletitud de Gödel, fracasó al querer mostrar la plena autonomía de la razón formalizada. En ese sentido, el único recurso que le queda al matemático para salvar el carácter descriptivo de sus proposiciones, a juicio de Gödel, es apoyarse en la intuición de los conceptos abstractos. El argumento de Gödel exige que la mente humana no se pueda reducir a un mecanismo. Esta segunda exigencia es también paralela a la defensa del dualismo cartesiano.</p>

On A.Ya. Khinchin's paper ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ (1926): A translation with introduction and commentary

The translation into English of Aleksandr Yakovlevich Khinchin's (1894–1959) 1926 paper entitled ‘Ideas of intuitionism and the struggle for a subject matter in contemporary mathematics’ is made available for the first time. Here, Khinchin presented the famous foundational debate between L.E.J. Brouwer and David Hilbert of the 1920s in terms of a search for a mathematics with content. His main aim seems to have been to make intuitionism ideologically acceptable to his audience at the Communist Academy by means of the claim that insofar as Brouwer's intuitionism had a clear ‘subject matter’ and Hilbert's new program was a concession to intuitionism, the alleged victory of intuitionism not only implied the defeat of ‘empty’ formalism, but also showed the compatibility and affinity of Marxism with the newest developments in modern mathematics. This introduction provides a tentative exploration of the issue of what was tactical (or due to ideological pressure) and what was real scientific interest (or due to ignorance) (or what was both) in Khinchin's 1926 paper in the form of a detailed commentary, especially, on the tactical side of his presentation of the positions of Brouwer and Hilbert.

Hilbert program of formalism as a working philosophical direction for consideration of the bases of mathematics

Hilbert program of formalism as a working philosophical direction for consideration of the bases of mathematics

Hilbert's objectivity

Detlefsen (1986) reads Hilbert's program as a sophisticated defense of instrumentalism, but Feferman (1998) has it that Hilbert's program leaves significant ontological questions unanswered. One such question is of the reference of individual number terms. Hilbert's use of admittedly “meaningless” signs for numbers and formulae appears to impair his ability to establish the reference of mathematical terms and the content of mathematical propositions (Weyl, 2009/1949; Kitcher, 1976). The paper traces the history and context of Hilbert's reasoning about signs, which illuminates Hilbert's account of mathematical objectivity, axiomatics, idealization, and consistency.

Kurt Godel and the foundations of mathematics: horizons of truth

Part I. Historical Context - Godel's Contributions and Accomplishments: 1. The impact of Godel's incompleteness theorems on mathematics Angus Macintyre 2. Logical hygiene, foundations, and abstractions: diversity among aspects and options Georg Kreisel 3. The reception of Godel's 1931 incompletability theorems by mathematicians, and some logicians, to the early 1960s Ivor Grattan-Guinness 4. 'Dozent Godel will not lecture' Karl Sigmund 5. Godel's thesis: an appreciation Juliette C. Kennedy 6. Lieber Herr Bernays!, Lieber Herr Godel! Godel on finitism, constructivity, and Hilbert's program Solomon Feferman 7. Computation and intractability: echoes of Kurt Godel Christos H. Papadimitriou 8. From the entscheidungsproblem to the personal computer - and beyond B. Jack Copeland 9. Godel, Einstein, Mach, Gamow, and Lanczos: Godel's remarkable excursion into cosmology Wolfgang Rindler 10. Physical unknowables Karl Svozil Part II. A Wider Vision - the Interdisciplinary, Philosophical, and Theological Implications of Godel's Work: 11. Godel and physics John D. Barrow 12. Godel, Thomas Aquinas, and the unknowability of God Denys A. Turner 13. Godel's mathematics of philosophy Piergiorgio Odifreddi 14. Godel's ontological proof and its variants Petr Hajek 15. The Godel theorem and human nature Hilary Putnam 16. Godel, the mind, and the laws of physics Roger Penrose Part III. New Frontiers - Beyond Godel's Work in Mathematics and Symbolic Logic: 17. Godel's functional interpretation and its use in current mathematics Ulrich Kohlenbach 18. My forty years on his shoulders Harvey M. Friedman 19. My interaction with Kurt Godel: the man and his work Paul J. Cohen 20. The transfinite universe W. Hugh Woodin 21. The Godel phenomena in mathematics: a modern view Avi Wigderson.

非可述性の分析としての証明論(<特集>あたらしい数理論理学の揺籃:証明論的な順序数と集合論的な順序数)

Proof theory was initiated by Hilbert for the foundation of mathematics, especially for carrying out "Hilbert's program". After Godel's incompleteness theorems, proof theory (ordinal analysis) has been developed as pure mathematics. It has been widely held that it is quite difficult to explain what has been gained in proof theory (ordinal analysis). In this paper we review the development of proof theory and its recent results for impredicative subsystems of analysis from the viewpoint of "analysis of impredicativity". We point out the main difficulties to explain conceptual significance of proof theory. Our conclusion is that such difficulties should be addressed by developing (not only proof theory as pure mathematics) philosophy of mathematics in future.

Primitive Recursion and the Chain Antichain Principle

Let the chain antichain principle (CAC) be the statement that each partial order on $\mathbb{N}$ possesses an infinite chain or an infinite antichain. Chong, Slaman, and Yang recently proved using forcing over nonstandard models of arithmetic that CAC is $\Pi^1_1$-conservative over $\text{RCA}_0+\Pi^0_1\text{-CP}$ and so in particular that CAC does not imply $\Sigma^0_2$-induction. We provide here a different purely syntactical and constructive proof of the statement that CAC (even together with WKL) does not imply $\Sigma^0_2$-induction. In detail we show using a refinement of Howard's ordinal analysis of bar recursion that $\text{WKL}_0^\omega+\text{CAC}$ is $\Pi^0_2$-conservative over PRA and that one can extract primitive recursive realizers for such statements. Moreover, our proof is finitary in the sense of Hilbert's program. CAC implies that every sequence of $\mathbb{R}$ has a monotone subsequence. This Bolzano-Weierstras}-like principle is commonly used in proofs. Our result makes it possible to extract primitive recursive terms from such proofs. We also discuss the Erdős-Moser principle, which—taken together with CAC—is equivalent to $\text{RT}^2_2$.

Hilbert versus Hindman

We show that a statement HIL, which is motivated by a lemma of Hilbert and close in formulation to Hindman's theorem, is actually much weaker than Hindman's theorem. In particular, HIL is finitistically reducible in the sense of Hilbert's program, while Hindman's theorem is not.

Fundamental and Real-World Challenges in Economics

In the same way as the Hilbert Program was a response to the foundational crisis of mathematics, this article tries to formulate a research program for the socio-economic sciences. The aim of this contribution is to stimulate research in order to close serious knowledge gaps in mainstream economics that the recent financial and economic crisis has revealed. By identifying weak points of conventional approaches in economics, we identify the scientific problems which need to be addressed. We expect that solving these questions will bring scientists in a position to give better decision support and policy advice. We also indicate, what kinds of insights can be contributed by scientists from other research fields such as physics, biology, computer and social science. In order to make a quick progress and gain a systemic understanding of the whole interconnected socio-economic-environmental system, using the data, information and computer systems available today and in the near future, we suggest a multi-disciplinary collaboration as most promising research approach.

The autonomy of mathematical knowledge: Hilbert's program revisited

Preface 1. A new science 2. David Hilbert's naturalism 3. Arithmetization 4. Intensionality 5. Interpreting G2 for Q 6. Autonomy in context Bibliography Index.

Hilbert's idea of a physical axiomatics: the analytical apparatus of quantum mechanics

We discuss the Hilbert program for the axiomatization of physics in the context of what Hilbert and von Neumann came to call the analytical apparatus and its conditions of reality. We suggest that the idea of a physical logic is the basis for a physical mathematics and we use quantum mechanics as a paradigm case for axiomatics in the sense of Hilbert. Finite probability theory requires nite derivations in the measurement theory of QM and we give a polynomial formulation of local complementation for the metric induced on the topology of the Hilbert space. The conclusion hints at a constructivist physics.

On a Relationship between Gödel's Second Incompleteness Theorem and Hilbert's Program

On a Relationship between Gödel's Second Incompleteness Theorem and Hilbert's Program

Tracking the origins of transformational generative grammar

Tracking the main influences of 19thand 20th-century mathematics, logic and philosophy on pre-1958 American linguistics and especially on early Transformational Generative Grammar (TGG) is an ambitious crossdisciplinary endeavour. Ideally it would call for expertise in the methods of intellectual historiography, the history and content of 20th-century American linguistics, the history and philosophy of science (including logic and mathematics), the tools and results of mathematical logic, and the theory of computable functions. Scholars fully versed in all of these fields are rare indeed. If Marcus Tomalin makes some mistakes in his book (henceforth LFS), that should not be surprising. What is surprising is how much progress he makes in furthering intellectually serious work on the history of modern linguistics, and how wide his reading in the relevant technical literature has been. LFS locates the intellectual roots of TGG in the methods developed by 19thand 20th-century mathematics and logic for exhibiting the conceptual structure of theories and constructing rigorous proofs of theorems. Tomalin discusses the methods developed by Augustin-Louis Cauchy for the rigorisation of the calculus in the 1820s; Whitehead & Russell's use of the axiomatic method in Principia Mathematica (1910-1913); the Hilbert program (in the 1920s) to prove all of mathematics consistent; Bloomfield's early axiomatisation of a general linguistic theory (1926); Carnap's logical empiricist proposals for the logical reconstruction of science on an experiential basis in the 1920s and 1930s; and Goodman's (1951) adaptation and revision of Carnap (1928). Earlier histories of TGG have not investigated