The Logical Foundations of Mathematics

Foundations and Philosophy of Science and Technology Series

1st Edition - January 1, 1982
Author: William S. Hatcher
Editor: Mario Bunge
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 1 - 8 9 6 3 - 5

The Logical Foundations of Mathematics offers a study of the foundations of mathematics, stressing comparisons between and critical analyses of the major non-constructive… Read more

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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.

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

Index