| Description |
1 online resource |
| Series |
Encyclopedia of mathematics and its applications |
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Encyclopedia of mathematics and its applications.
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| Contents |
Cover -- Half-title Page -- Series Page -- Title Page -- Copyright Page -- Contents -- List of Contributors -- Preface -- Part I Introductory -- 1 An Introduction to Intuitionistic Logic -- 1.1 Introduction -- 1.2 Constructive Existence -- 1.2.1 Counterexamples from Analysis -- 1.3 The Brouwer-Heyting-Kolmogorov Interpretation -- 1.4 Natural Deductions -- 1.5 A Hilbert-Style System for Intuitionistic Logic -- 1.6 Realizability -- 1.7 The Curry-Howard Correspondence -- References -- 2 An Introduction to Constructive Set Theory: An Appetizer -- 2.1 Introduction -- 2.1.1 Overview |
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2.1.2 Background: the 'Intellectual' Landscape -- 2.2 The Axiomatic Framework -- 2.2.1 The Axioms of CZF -- 2.3 Elementary Mathematics in CZF -- 2.3.1 Operations on Sets and Classes -- 2.3.2 Reasoning Intuitionistically, for Example, Russell's Paradox -- 2.3.3 Class Relations and Functions -- 2.3.4 Some Consequences of Replacement -- 2.3.5 Cartesian Products and Sums of Classes -- 2.3.6 Quotients -- 2.3.7 The Naturals in ECST -- 2.3.8 Exponentiation and Subset Collection -- 2.3.9 The Cauchy Real Numbers, R[sub(c)] -- 2.3.10 The Constructive Dedekind Reals |
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2.4 The Development of Set Theory in CZF -- 2.4.1 Transfinite Recursion -- 2.4.2 Ordinals -- 2.4.3 Inductive Definitions of Classes -- 2.4.4 Class Inductive Definition Theorem -- 2.4.5 Bounded Inductive Definitions -- 2.5 Large Sets in CZF -- 2.5.1 Other Notions of Large Set -- 2.5.2 An 'Algebraic' Characterization of 'Inaccessibility' -- 2.6 Axioms of Choice in Constructive Set Theory -- 2.7 CZF and the Limited Principle of Omniscience -- 2.8 Models of CZF and Axiomatic Freedom -- References -- 3 Bishop's Mathematics: A Philosophical Perspective -- 3.1 Introduction -- 3.2 Bishop on Brouwer |
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3.3 Brouwer's Mathematics -- 3.4 Persuasion and Dialogue -- 3.5 Formalisation -- 3.6 Philosophy -- 3.7 Traditional Philosophical Arguments for Intuitionistic Logic -- 3.8 Philosophical Objections -- 3.9 Too Strong -- 3.10 Concluding Remarks -- Acknowledgements -- References -- Part II Algebra and Geometry -- 4 Algebra in Bishop's style: A Course in Constructive Algebra -- 4.1 Introduction -- 4.2 Revisiting Bishop's Set Theory -- 4.3 The Corpus of Classical Abstract Algebra Treated in the Book -- 4.4 Principal Ideal Domains and Finitely Generated Modules on these Rings |
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4.5 Factorization Problems -- 4.6 Noetherian Rings, Primary Decompositions and the Principal Ideal Theorem -- 4.6.1 Hilbert-Basis Theorem -- 4.6.2 Primary Decomposition Theorem -- 4.6.3 Principal Ideal Theorem -- 4.6.4 Fully Lasker-Noether Rings -- 4.7 Wedderburn Structure Theorem for Finite-Dimensional k-Algebras -- 4.8 Dedekind Domains -- Acknowledgements -- References -- 5 Constructive Algebra: The Quillen-Suslin Theorem -- 5.1 Introduction -- 5.2 Quillen's Proof of Serre's Problem -- 5.2.1 Finitely Generated Projective Modules -- 5.2.2 Finitely Generated Stably Free Modules |
| Summary |
Constructive mathematics - mathematics in which 'there exists' always means 'we can construct' - is enjoying a renaissance. fifty years on from Bishop's groundbreaking account of constructive analysis, constructive mathematics has spread out to touch almost all areas of mathematics and to have profound influence in theoretical computer science. This handbook gives the most complete overview of modern constructive mathematics, with contributions from leading specialists surveying the subject's myriad aspects. Major themes include: constructive algebra and geometry, constructive analysis, constructive topology, constructive logic and foundations of mathematics, and computational aspects of constructive mathematics. A series of introductory chapters provides graduate students and other newcomers to the subject with foundations for the surveys that follow. Edited by four of the most eminent experts in the field, this is an indispensable reference for constructive mathematicians and a fascinating vista of modern constructivism for the increasing number of researchers interested in constructive approaches |
| Bibliography |
Includes index |
| Notes |
Description based on online resource; title from digital title page (viewed on May 12, 2023) |
| Subject |
Constructive mathematics.
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Constructive mathematics
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| Genre/Form |
handbooks.
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Handbooks and manuals
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Handbooks and manuals.
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Guides et manuels.
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| Form |
Electronic book
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| Author |
Bridges, D. S. (Douglas S.), 1945- editor.
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Ishihara, Hajime, editor
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Rathjen, Michael, editor.
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Schwichtenberg, Helmut, 1942- editor.
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| ISBN |
9781009039888 |
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1009039881 |
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9781009041416 |
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100904141X |
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