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The logical foundations of mathematics /
The Logical Foundations of Mathematics.
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Burlington :
Pergamon Press/Elsevier Science,
[2014], ©1982.
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| Colección: | Foundations & philosophy of science & technology.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Front Cover; The Logical Foundations of Mathematics; Copyright Page; Dedication; Preface; Table of Contents; Chapter 1. First-order Logic; 1.1. The sentential calculus; 1.2. Formalization; 1.3. The statement calculus as a formal system; 1.4. First-order theories; 1.5. Models of first-order theories; 1.6. Rules of logic; natural deduction; 1.7. First-order theories with equality; variable-binding term operators; 1.8. Completeness with vbtos; 1.9. An example of a first-order theory; Chapter 2. The Origin of Modern Foundational Studies; 2.1. Mathematics as an independent science.
- 2.2. The arithmetization of analysis2.3. Constructivism; 2.4. Frege and the notion of a formal system; 2.5. Criteria for foundations; Chapter 3. Frege's System and the Paradoxes; 3,1. The intuitive basis of Frege's system; 3.2. Frege's system; 3.3. The theorem of infinity; 3.4. Criticisms of Frege's system; 3.5. The paradoxes; 3.6. Brouwer and intuitionism; 3.7. Poincare'snotion of im predicative definition; 3.8. Russell's principle of vicious circle; 3.9. The logical paradoxes and the semantic paradoxes; Chapter 4. The Theory of Types; 4.1. Quantifying predicate letters.
- 4.2. Predicative type theory4.3. The development of mathematics in PT; 4.4. The system TT; 4.5. Criticisms of type theory as a foundation for mathematics; 4.6. The system ST; 4.7. Type theory and first-order logic; Chapter 5. Zermelo-Fraenkel Set Theory; 5.1. Formalization of ZF; 5.2. The completing axioms; 5.3. Relations, functions, and simple recursion; 5.4. The axiom of choice; 5.5. The continuum hypothesis; descriptive set theory; 5.6. The systems of vonNeumann-Bernays-Godel and Mostowski-Kelley-Morse; 5.7. Number systems; ordinal recursion; 5.8. Conway's numbers.
- Chapter 6. Hilbert's Program and Godel's IncompletenessTheorems6.1. Hilbert's program; 6.2. Godel's theorems and their import; 6.3. The method of proof of Godel's theorems; recursive functions; 6.4. Nonstandard models of S; Chapter 7. The Foundational Systems of W.V. Quine; 7.1. The system NF; 7.2. Cantor's theorem in NF; 7.3. The axiom of choice in NF and the theorem of infinity; 7.4. NF and ST; typical ambiguity; 7.5. Quine's system ML; 7.6. Further results on NF; variant systems; 7.7. Conclusions; Chapter 8. Categorical Algebra; 8.1. The notion of a category.
- 8.2. The first-order language of categories8.3. Category theory and set theory; 8.4. Functors and large categories; 8.5. Formal development of the language and theory CS; 8.6. Topos theory; 8.7. Global elements in toposes; 8.8. Image factorizations and the axiom of choice; 8.9. A last look at CS; 8.10. ZF andWT; 8.11. The internal logic of toposes; 8.12. The internal language of a topos; 8.13. Conclusions; Selected Bibliography; Index.