LEADER 04136nam 2200577Ia 450 001 9910974250403321 005 20200520144314.0 010 0 $a0191524018 010 0 $a9780191524011 035 $a(MiAaPQ)EBC7036497 035 $a(CKB)24235107600041 035 $a(MiAaPQ)EBC3052867 035 $a(Au-PeEL)EBL3052867 035 $a(CaPaEBR)ebr10274556 035 $a(CaONFJC)MIL194396 035 $a(OCoLC)316582812 035 $a(OCoLC)1056826348 035 $a(FINmELB)ELB164780 035 $a(Au-PeEL)EBL7036497 035 $a(OCoLC)1336404428 035 $a(EXLCZ)9924235107600041 100 $a19990802d2000 uy 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aFoundations without foundationalism $ea case for second-order logic 205 $a1st ed. 210 $aOxford $cClarendon Press ;$aNew York $cOxford University Press$d2000 215 $axxii, 277 p 225 1 $aOxford logic guides ;$v17 300 $aOriginally published: Oxford: Clarendon, 1991. 320 $aIncludes bibliographical references (p. [263]-272) and index. 327 $aIntro -- PREFACE -- Contents -- PART I: ORIENTATION -- 1. Terms and questions -- 1.1 Orientation -- 1.2 What is the issue? -- 1.3 Sets and properties -- 2. Foundationalism and foundations of mathematics -- 2.1 Variations and metaphors -- 2.2 Foundations and psychologism -- 2.3 Two conceptions of logic -- 2.4 Marriage: Can there be harmony? -- 2.5 Divorce: Life without completeness -- 2.6 Logic and computation -- PART II: LOGIC AND MATHEMATICS -- 3. Theory -- 3.1 Language -- 3.2 Deductive systems -- 3.3 Semantics -- 4. Metatheory -- 4.1 First-order theories -- 4.2 Second-order-standard semantics -- 4.3 Non-standard semantics-Henkin and first-order -- 5. Second-order logic and mathematics -- 5.1 Mathematical notions -- 5.2 First-order theories-what goes wrong -- 5.3 Second-order languages and the practice of mathematics -- 5.4 Set theory -- 6. Advanced metatheory -- 6.1 A word on semantic theory -- 6.2 Reductions -- 6.3 Reflection: small large cardinals -- 6.4 Löwenheim-Skolem analogues: large large cardinals -- 6.5 Characterizations of first-order logic -- 6.6 Definability and other odds and ends -- PART III: HISTORY AND PHILOSOPHY -- 7. The historical 'triumph' of first-order languages -- 7.1 Introduction -- 7.2 Narrative -- 7.3 To the present -- 8. Second-order logic and rule-following -- 8.1 The regress -- 8.2 Options -- 8.3 Rules and logic -- 9. The competition -- 9.1 Other logics -- 9.2 Free relation variables -- 9.3 First-order set theory -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z. 330 $aThe central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics.Professor Shapiro demonstrates the prevalence of second-order notions in mathematics, and also the extent to which mathematical concepts can be formulated in second-order languages. He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics.Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. 410 0$aOxford logic guides ;$v17. 606 $aLogic, Symbolic and mathematical 606 $aMathematics 615 0$aLogic, Symbolic and mathematical. 615 0$aMathematics. 676 $a511.3 676 $a511.3 700 $aShapiro$b Stewart$f1951-$0447519 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910974250403321 996 $aFoundations without foundationalism$94464365 997 $aUNINA