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About this book
This book first examines the following foundational question: are all theorems in classic mathematics expressible in second-order arithmetic provable in second-order arithmetic? The author gives a counterexample for this question and isolates this counterexample from the Martin-Harrington Theorem in set theory. It shows that the statement “Harrington's principle implies zero sharp" is not provable in second-order arithmetic. This book further examines what is the minimal system in higher-order arithmetic to prove the theorem “Harrington's principle implies zero sharp" and shows that it is neither provable in second-order arithmetic or third-order arithmetic, but provable in fourth-order arithmetic. The book also examines the large cardinal strength of Harrington's principle and its strengthening over second-order arithmetic and third-order arithmetic.
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Table of contents (6 chapters)
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Bibliographic Information
Book Title: Incompleteness for Higher-Order Arithmetic
Book Subtitle: An Example Based on Harrington’s Principle
Authors: Yong Cheng
Series Title: SpringerBriefs in Mathematics
DOI: https://doi.org/10.1007/978-981-13-9949-7
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019
Softcover ISBN: 978-981-13-9948-0Published: 11 September 2019
eBook ISBN: 978-981-13-9949-7Published: 30 August 2019
Series ISSN: 2191-8198
Series E-ISSN: 2191-8201
Edition Number: 1
Number of Pages: XIV, 122
Number of Illustrations: 1 b/w illustrations