LEADER 04413nam 2200577 450 001 9910786641503321 005 20230120014530.0 010 $a1-4831-8963-5 035 $a(CKB)3710000000199910 035 $a(EBL)1901360 035 $a(SSID)ssj0001267023 035 $a(PQKBManifestationID)12564639 035 $a(PQKBTitleCode)TC0001267023 035 $a(PQKBWorkID)11255015 035 $a(PQKB)11516658 035 $a(MiAaPQ)EBC1901360 035 $a(EXLCZ)993710000000199910 100 $a20150119h19821982 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 14$aThe logical foundations of mathematics /$fby William S. Hatcher 205 $aFirst edition. 210 1$aOxford, England :$cPergamon Press,$d1982. 210 4$dİ1982 215 $a1 online resource (331 p.) 225 1 $aFoundations and Philosophy of Science and Technology Series 300 $aDescription based upon print version of record. 311 $a1-322-55676-8 311 $a0-08-025800-X 320 $aIncludes bibliographical references and index. 327 $aFront 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 327 $a2.2. 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 327 $a4.2. 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 327 $aChapter 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 327 $a8.2. 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 330 $aThe Logical Foundations of Mathematics 410 0$aFoundations & philosophy of science & technology. 606 $aMathematics$xPhilosophy 615 0$aMathematics$xPhilosophy. 676 $a510/.1 700 $aHatcher$b William S.$047598 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910786641503321 996 $aLogical foundations of mathematics$979103 997 $aUNINA