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Titlebook: Incompleteness for Higher-Order Arithmetic; An Example Based on Yong Cheng Book 2019 The Author(s), under exclusive license to Springer Na

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发表于 2025-3-21 19:18:29 | 显示全部楼层 |阅读模式
书目名称Incompleteness for Higher-Order Arithmetic
副标题An Example Based on
编辑Yong Cheng
视频video
丛书名称SpringerBriefs in Mathematics
图书封面Titlebook: Incompleteness for Higher-Order Arithmetic; An Example Based on Yong Cheng Book 2019 The Author(s), under exclusive license to Springer Na
描述Gödel‘s true-but-unprovable sentence from the first incompleteness theorem is purely logical in nature, i.e. not mathematically natural or interesting. An interesting problem is to find mathematically natural and interesting statements that are similarly unprovable. A lot of research has since been done in this direction, most notably by Harvey Friedman. A lot of examples of concrete incompleteness with real mathematical content have been found to date. This brief contributes to Harvey Friedman‘s research program on concrete incompleteness for higher-order arithmetic and gives a specific example of concrete mathematical theorems which is expressible in second-order arithmetic but the minimal system in higher-order arithmetic to prove it is fourth-order arithmetic..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 arithmet
出版日期Book 2019
关键词Incompleteness; higher order arithmetic; Harrington‘s Principle; Set Theory; Martin-Harrington Theorem; L
版次1
doihttps://doi.org/10.1007/978-981-13-9949-7
isbn_softcover978-981-13-9948-0
isbn_ebook978-981-13-9949-7Series ISSN 2191-8198 Series E-ISSN 2191-8201
issn_series 2191-8198
copyrightThe Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019
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发表于 2025-3-21 20:36:38 | 显示全部楼层
The Boldface Martin-Harrington Theorem in , ., . exists. In this chapter, I prove the Boldface Martin-Harrington Theorem in . . In Sect. ., I prove in . that if for any real . exists, then . holds. In Sect. ., I prove in . that . implies that for any real ., . exists.
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Introduction and Preliminaries,., . and .. This should provide the reader with a good picture of the background and put the main results in this book into perspective. In Sect. ., I review some of the notions and facts from Set Theory used in this book. In Sect. ., I introduce the main research problems and outline the structure of this book.
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The Boldface Martin-Harrington Theorem in , ., . exists. In this chapter, I prove the Boldface Martin-Harrington Theorem in . . In Sect. ., I prove in . that if for any real . exists, then . holds. In Sect. ., I prove in . that . implies that for any real ., . exists.
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A Minimal System,In this chapter, we prove the following results..As a corollary, “.implies that .exists” is neither provable in .nor in ., i.e. .is the minimal system of higher-order arithmetic for proving that “.implies that . exists”.
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