UID:
almafu_9960788382502883
Umfang:
xxii, 277 p.
Ausgabe:
1st ed.
ISBN:
0191524018
,
9780191524011
Serie:
Oxford logic guides ; 17
Inhalt:
The 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.
Anmerkung:
Originally published: Oxford: Clarendon, 1991.
,
Intro -- 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.
Sprache:
Englisch
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