The Logical Foundations of 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 TheoryChapter 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 FoundationsChapter 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 ParadoxesChapter 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 LogicChapter 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 NumbersChapter 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 SChapter 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. ConclusionsChapter 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. ConclusionsSelected BibliographyIndex