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Incompleteness for Higher-Order Arithmetic

An Example Based on Harrington’s Principle

  • Book
  • © 2019

Overview

Authors:
  1. Yong Cheng

    1. School of Philosophy, Wuhan University, Wuhan, China

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Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Introduction and Preliminaries

    • Yong Cheng
    Pages 1-31

  3. A Minimal System

    • Yong Cheng
    Pages 33-52

  4. The Boldface Martin-Harrington Theorem in \(\mathsf{Z_2}\)

    • Yong Cheng
    Pages 53-56

  5. Strengthenings of Harrington’s Principle

    • Yong Cheng
    Pages 57-68

  6. Forcing a Model of Harrington’s Principle Without Reshaping

    • Yong Cheng
    Pages 69-88

  7. The Strong Reflecting Property for L-Cardinals

    • Yong Cheng
    Pages 89-97

  8. Back Matter

    Pages 99-122
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Keywords

About this book

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 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.

Authors and Affiliations

  • School of Philosophy, Wuhan University, Wuhan, China

    Yong Cheng

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

  • Topics: Mathematical Logic and Foundations

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