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BL
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| Record Number
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890519
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| Main Entry
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Cheng, Yong
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| Title & Author
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Incompleteness for higher-order arithmetic : : an example based on Harrington's principle /\ Yong Cheng.
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| Publication Statement
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Singapore :: Springer,, [2019]
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, ©2019
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| Series Statement
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SpringerBriefs in mathematics,
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| Page. NO
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1 online resource
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| ISBN
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9789811399497
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: 9789811399503
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: 9789811399510
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: 9811399492
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: 9811399506
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: 9811399514
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9789811399480
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9811399484
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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Intro; Preface; Acknowledgements; Contents; 1 Introduction and Preliminaries; 1.1 Preliminaries; 1.1.1 Basics of Incompleteness; 1.1.2 Basics of Reverse Mathematics; 1.1.3 Overview of Incompleteness for Higher-Order Arithmetic; 1.1.4 Basics of Set Theory; 1.2 Introduction; References; 2 A Minimal System; 2.1 Preliminaries; 2.1.1 Almost Disjoint Forcing; 2.1.2 The Large Cardinal Notion 0; 2.1.3 Remarkable Cardinal; 2.2 The Strength of Z2+ Harrington's Principle; 2.3 The Strength of Z3+ Harrington's Principle; 2.4 Z4 + Harrington's Principle Implies that 0 Exists; References
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3 The Boldface Martin-Harrington Theorem in Z23.1 The Boldface Martin Theorem in Z2; 3.2 The Boldface Harrington Theorem in Z2; References; 4 Strengthenings of Harrington's Principle; 4.1 Overview; 4.2 Harrington's Club Shooting Forcing; 4.3 The Strength of Z2 + HP(); 4.4 Subcomplete Forcing; 4.5 The Strength of Z3 + HP(); References; 5 Forcing a Model of Harrington's Principle Without Reshaping; 5.1 Introduction; 5.2 The Notion of Strong Reflecting Property for L-Cardinals; 5.3 Baumgartner's Forcing; 5.4 The First Step; 5.5 The Second Step; 5.6 The Third Step; 5.7 The Fourth Step; References
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| Abstract
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The book examines the following foundation question: are all theorems in classic mathematics which are expressible in second order arithmetic provable in second order arithmetic? In this book, the author gives a counterexample for this question and isolates this counterexample from Martin-Harrington theorem in set theory. It shows that the statement "Harrington's principle implies zero sharp" is not provable in second order arithmetic. The book also examines what is the minimal system in higher order arithmetic to show that Harrington's principle implies zero sharp and the large cardinal strength of Harrington's principle and its strengthening over second and third order arithmetic.
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| Subject
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Incompleteness theorems.
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| Subject
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Incompleteness theorems.
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| Dewey Classification
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511.3
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| LC Classification
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QA9.54.C54 2019eb
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