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The Logical Foundations of Mathematics
Foundations and Philosophy of Science and Technology Series
- 1st Edition - May 9, 2014
- Author: William S. Hatcher
- Editor: Mario Bunge
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 7 3 8 2 - 5
- 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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Request a sales quoteThe 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
- No. of pages: 330
- Language: English
- Edition: 1
- Published: May 9, 2014
- Imprint: Pergamon
- Paperback ISBN: 9781483173825
- eBook ISBN: 9781483189635
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