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The logical foundations of mathematics / / by William S. Hatcher
| The logical foundations of mathematics / / by William S. Hatcher |
| Autore | Hatcher William S. |
| Edizione | [First edition.] |
| Pubbl/distr/stampa | Oxford, England : , : Pergamon Press, , 1982 |
| Descrizione fisica | 1 online resource (331 p.) |
| Disciplina | 510/.1 |
| Collana | Foundations and Philosophy of Science and Technology Series |
| Soggetto topico | Mathematics - Philosophy |
| ISBN | 1-4831-8963-5 |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
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 |
| Record Nr. | UNINA-9910786641503321 |
Hatcher William S.
|
||
| Oxford, England : , : Pergamon Press, , 1982 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
The logical foundations of mathematics / / by William S. Hatcher
| The logical foundations of mathematics / / by William S. Hatcher |
| Autore | Hatcher William S. |
| Edizione | [First edition.] |
| Pubbl/distr/stampa | Oxford, England : , : Pergamon Press, , 1982 |
| Descrizione fisica | 1 online resource (331 p.) |
| Disciplina | 510/.1 |
| Collana | Foundations and Philosophy of Science and Technology Series |
| Soggetto topico | Mathematics - Philosophy |
| ISBN | 1-4831-8963-5 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
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 |
| Record Nr. | UNINA-9910811257203321 |
|
Hatcher William S.
|
||
| Oxford, England : , : Pergamon Press, , 1982 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||