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Godel's Proof

Godel's Proof

Ernest Nagel
James R. Newman
Edited and with a New Foreword by Douglas R. Hofstadter
Copyright Date: 2001
Published by: NYU Press
Pages: 160
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  • Book Info
    Godel's Proof
    Book Description:

    In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions ofPrincipia Mathematicaand Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences-perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times."

    However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.

    New York University Press is proud to publish this special edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

    eISBN: 978-0-8147-5903-5
    Subjects: Mathematics, Political Science

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Foreword to the New Edition
    (pp. ix-xxii)
    Douglas R. Hofstadter

    In August 1959, my family returned to Stanford, California, after a year in Geneva. I was fourteen, newly fluent in French, in love with languages, entranced by writing systems, symbols, and the mystery of meaning, and brimming with curiosity about mathematics and how the mind works.

    One evening, my father and I went to a bookstore where I chanced upon a little book with the enigmatic titleGödel’s Proof. Flipping through it, I saw many intriguing figures and formulas, and was particularly struck by a footnote about quotation marks, symbols, and symbols symbolizing other symbols. Intuitively sensing thatGödel’s Proof...

  4. Acknowledgments
    (pp. xxiii-xxvi)
  5. I Introduction
    (pp. 1-6)

    In 1931 there appeared in a German scientific periodical a relatively short paper with the forbidding title “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” (“On Formally Undecidable Propositions of Principia Mathematica and Related Systems”). Its author was Kurt Gödel, then a young mathematician of 25 at the University of Vienna and since 1938 a permanent member of the Institute for Advanced Study at Princeton. The paper is a milestone in the history of logic and mathematics. When Harvard University awarded Gödel an honorary degree in 1952, the citation described the work as one of the most important...

  6. II The Problem of Consistency
    (pp. 7-24)

    The nineteenth century witnessed a tremendous expansion and intensification of mathematical research. Many fundamental problems that had long withstood the best efforts of earlier thinkers were solved; new areas of mathematical study were created; and in various branches of the discipline new foundations were laid, or old ones entirely recast with the help of more precise techniques of analysis. To illustrate: the Greeks had proposed three problems in elementary geometry: with compass and straight-edge to trisect any angle, to construct a cube with a volume twice the volume of a given cube, and to construct a square equal in area...

  7. III Absolute Proofs of Consistency
    (pp. 25-36)

    The limitations inherent in the use of models for establishing consistency, and the growing apprehension that the standard formulations of many mathematical systems might all harbor internal contradictions, led to new attacks upon the problem. An alternative to relative proofs of consistency was proposed by Hilbert. He sought to construct “absolute” proofs, by which the consistency of systems could be established without assuming the consistency of some other system. We must briefly explain this approach as a further preparation for understanding Gödel’s achievement.

    The first step in the construction of an absolute proof, as Hilbert conceived the matter, is the...

  8. IV The Systematic Codification of Formal Logic
    (pp. 37-44)

    There are two more bridges to cross before entering upon Gödel’s proof itself. We must indicate how and why Whitehead and Russell’sPrincipia Mathematicacame into being; and we must give a short illustration of the formalization of a deductive system—we shall take a fragment ofPrincipia—and explain how its absolute consistency can be established.

    Ordinarily, even when mathematical proofs conform to accepted standards of professional rigor, they suffer from an important omission. They embody principles (or rules) of inference not explicitly formulated, of which mathematicians are frequently unaware. Take Euclid’s proof that there is no greatest prime...

  9. V An Example of a Successful Absolute Proof of Consistency
    (pp. 45-56)

    We must now attempt the second task mentioned at the outset of the preceding section, and familiarize ourselves with an important, though easily understandable, example of an absolute proof of consistency. By mastering the proof, the reader will be in a better position to appreciate the significance of Gödel’s paper of 1931.

    We shall outline how a small portion ofPrincipia, the elementary logic of propositions, can be formalized. This entails the conversion of the fragmentary system into a calculus of uninterpreted signs. We shall then develop an absolute proof of consistency.

    The formalization proceeds in four steps. First, a...

  10. VI The Idea of Mapping and Its Use in Mathematics
    (pp. 57-67)

    The sentential calculus is an example of a mathematical system for which the objectives of Hilbert’s theory of proof are fully realized. To be sure, this calculus codifies only a fragment of formal logic, and its vocabulary and formal apparatus do not suffice to develop even elementary arithmetic. Hilbert’s program, however, is not so limited. It can be carried out successfully for more inclusive systems, which can be shown by meta-mathematical reasoning to be both consistent and complete. By way of example, an absolute proof of consistency is available for a formal system in which axioms for addition but not...

  11. VII Gödel’s Proofs
    (pp. 68-108)

    Gödel’s paper is difficult. Forty-six preliminary definitions, together with several important preliminary propositions, must be mastered before the main results are reached. We shall take a much easier road; nevertheless, it should afford the reader glimpses of the ascent and of the crowning structure.

    Gödel described a formalized calculus, which we shall call “PM,” within which all the customary arithmetical notations can be expressed and familiar arithmetical relations established.¹⁵ The formulas of the calculus are constructed out of a class of elementary signs, which constitute the fundamental vocabulary. A set of primitive formulas (or axioms) are the underpinning, and the...

  12. VIII Concluding Reflections
    (pp. 109-113)

    The import of Gödel’s conclusions is far-reaching, though it has not yet been fully fathomed. These conclusions show that the prospect of finding for every deductive system (and, in particular, for a system in which the whole of number theory can be expressed) an absolute proof of consistency that satisfies the finitistic requirements of Hilbert’s proposal, though not logically impossible, is most unlikely.³⁹ They show also that there are an endless number of true arithmetical statements which cannot be formally deduced from any given set of axioms by a closed set of rules of inference. It follows that an axiomatic...

  13. Appendix: Notes
    (pp. 114-124)
  14. Brief Bibliography
    (pp. 125-126)
  15. Index
    (pp. 127-130)
  16. Back Matter
    (pp. 131-131)