The Logical Foundations of Mathematics - 1st Edition - ISBN: 9780080258003, 9781483189635

The Logical Foundations of Mathematics

1st Edition

Foundations and Philosophy of Science and Technology Series

Authors: William S. Hatcher
Editors: Mario Bunge
eBook ISBN: 9781483189635
Imprint: Pergamon
Published Date: 1st January 1982
Page Count: 330
Sales tax will be calculated at check-out Price includes VAT/GST
54.95
43.99
72.95
Unavailable
Price includes VAT/GST
× DRM-Free

Easy - Download and start reading immediately. There’s no activation process to access eBooks; all eBooks are fully searchable, and enabled for copying, pasting, and printing.

Flexible - Read on multiple operating systems and devices. Easily read eBooks on smart phones, computers, or any eBook readers, including Kindle.

Open - Buy once, receive and download all available eBook formats, including PDF, EPUB, and Mobi (for Kindle).

Institutional Access

Secure Checkout

Personal information is secured with SSL technology.

Free Shipping

Free global shipping
No minimum order.

Description

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 Complet

Details

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

About the Author

William S. Hatcher

About the Editor

Mario Bunge

Ratings and Reviews