The Logical Foundations of Mathematics

The Logical Foundations of Mathematics

Foundations and Philosophy of Science and Technology Series

1st Edition - January 1, 1982

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  • Author: William S. Hatcher
  • eBook ISBN: 9781483189635

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The Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive foundational systems. The position of constructivism within the spectrum of foundational philosophies is discussed, along with the exact relationship between topos theory and set theory. Comprised of eight chapters, this book begins with an introduction to first-order logic. In particular, two complete systems of axioms and rules for the first-order predicate calculus are given, one for efficiency in proving metatheorems, and the other, in a "natural deduction" style, for presenting detailed formal proofs. A somewhat novel feature of this framework is a full semantic and syntactic treatment of variable-binding term operators as primitive symbols of logic. Subsequent chapters focus on the origin of modern foundational studies; Gottlob Frege's formal system intended to serve as a foundation for mathematics and its paradoxes; the theory of types; and the Zermelo-Fraenkel set theory. David Hilbert's program and Kurt Gödel's incompleteness theorems are also examined, along with the foundational systems of W. V. Quine and the relevance of categorical algebra for foundations. This monograph will be of interest to students, teachers, practitioners, and researchers in mathematics.

Table of Contents

  • Chapter 1. First-Order Logic

    Section 1. The Sentential Calculus

    Section 2. Formalization

    Section 3. The Statement Calculus as a Formal System

    Section 4. First-Order Theories

    Section 5. Models of First-Order Theories

    Section 6. Rules of Logic; Natural Deduction

    Section 7. First-Order Theories with Equality; Variable-Binding Term Operators

    Section 8. Completeness with Vbtos

    Section 9. An Example of a First-Order Theory

    Chapter 2. The Origin of Modern Foundational Studies

    Section 1. Mathematics as an Independent Science

    Section 2. The Arithmetization of Analysis

    Section 3. Constructivism

    Section 4. Frege and the Notion of a Formal System

    Section 5. Criteria for Foundations

    Chapter 3. Frege's System and the Paradoxes

    Section 1. The Intuitive Basis of Frege's System

    Section 2. Frege's System

    Section 3. The Theorem of Infinity

    Section 4. Criticisms of Frege's System

    Section 5. The Paradoxes

    Section 6. Brouwer and Intuitionism

    Section 7. Poincaré's Notion of Impredicative Definition

    Section 8. Russell's Principle of Vicious Circle

    Section 9. The Logical Paradoxes and the Semantic Paradoxes

    Chapter 4. The Theory of Types

    Section 1. Quantifying Predicate Letters

    Section 2. Predicative Type Theory

    Section 3. The Development of Mathematics in PT

    Section 4. The System TT

    Section 5. Criticisms of Type Theory as a Foundation for Mathematics

    Section 6. The System ST

    Section 7. Type Theory and First-Order Logic

    Chapter 5. Zermelo-Fraenkel Set Theory

    Section 1. Formalization of ZF

    Section 2. The Completing Axioms

    Section 3. Relations, Functions, and Simple Recursion

    Section 4. The Axiom of Choice

    Section 5. The Continuum Hypothesis; Descriptive Set Theory

    Section 6. The Systems of von Neumann-Bernays-Gödel and Mostowski-Kelley-Morse

    Section 7. Number Systems; Ordinal Recursion

    Section 8. Conway's Numbers

    Chapter 6. Hilbert's Program and Gödel's Incompleteness Theorems

    Section 1. Hilbert's Program

    Section 2. Gödel's Theorems and their Import

    Section 3. The Method of Proof of Gödel's Theorems; Recursive Functions

    Section 4. Nonstandard Models of S

    Chapter 7. The Foundational Systems of W. V. Quine

    Section 1. The System NF

    Section 2. Cantor's Theorem in NF

    Section 3. The Axiom of Choice in NF and the Theorem of Infinity

    Section 4. NF and ST; Typical Ambiguity

    Section 5. Quine's System ML

    Section 6. Further Results on NF; Variant Systems

    Section 7. Conclusions

    Chapter 8. Categorical Algebra

    Section 1. The Notion of a Category

    Section 2. The First-Order Language of Categories

    Section 3. Category Theory and Set Theory

    Section 4. Functors and Large Categories

    Section 5. Formal Development of the Language and Theory CS

    Section 6. Topos Theory

    Section 7. Global Elements in Toposes

    Section 8. Image Factorizations and the Axiom of Choice

    Section 9. A Last Look at CS

    Section 10. ZF and WT

    Section 11. The Internal Logic of Toposes

    Section 12. The Internal Language of a Topos

    Section 13. Conclusions

    Selected Bibliography


Product details

  • No. of pages: 330
  • Language: English
  • Copyright: © Pergamon 1982
  • Published: January 1, 1982
  • Imprint: Pergamon
  • eBook ISBN: 9781483189635

About the Author

William S. Hatcher

About the Editor

Mario Bunge

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