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Incompleteness for Higher-Order Arithmetic

An Example Based on Harrington’s Principle

  • Yong Cheng
Book

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Yong Cheng
    Pages 1-31
  3. Yong Cheng
    Pages 33-52
  4. Back Matter
    Pages 99-122

About this book

Introduction

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. 

Keywords

Incompleteness higher order arithmetic Harrington's Principle Set Theory Martin-Harrington Theorem L-cardinals

Authors and affiliations

  • Yong Cheng
    • 1
  1. 1.School of PhilosophyWuhan UniversityWuhanChina

Bibliographic information

  • DOI https://doi.org/10.1007/978-981-13-9949-7
  • Copyright Information The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2019
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-981-13-9948-0
  • Online ISBN 978-981-13-9949-7
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • Buy this book on publisher's site