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Ordinal Analysis with an Introduction to Proof Theory

  • Textbook
  • © 2020

Overview

Authors:
  1. Toshiyasu Arai

    1. University of Tokyo, Tokyo, Japan

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  • Covers ordinal analysis at an advanced level, and proof theory at an elementary level
  • Approaches ordinal analysis in a contemporary fashion
  • Includes a wealth of exercises and answers, making it suitable for student use

Part of the book series: Logic in Asia: Studia Logica Library (LIAA)

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xvi
    Download chapter PDF

  2. Introduction

    • Toshiyasu Arai
    Pages 1-27

  3. Calculi for Classical Logic

    • Toshiyasu Arai
    Pages 29-64

  4. Cut-Elimination with Depths

    • Toshiyasu Arai
    Pages 65-96

  5. Epsilon Numbers

    • Toshiyasu Arai
    Pages 97-173

  6. Iterations

    • Toshiyasu Arai
    Pages 175-218

  7. Collapsings

    • Toshiyasu Arai
    Pages 219-290

  8. Answers to Exercises

    • Toshiyasu Arai
    Pages 291-308

  9. Back Matter

    Pages 309-313
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Keywords

About this book

This book provides readers with a guide to both ordinal analysis, and to proof theory. It mainly focuses on ordinal analysis, a research topic in proof theory that is concerned with the ordinal theoretic content of formal theories. However, the book also addresses ordinal analysis and basic materials in proof theory of first-order or omega logic, presenting some new results and new proofs of known ones.
Primarily intended for graduate students and researchers in mathematics, especially in mathematical logic, the book also includes numerous exercises and answers for selected exercises, designed to help readers grasp and apply the main results and techniques discussed.

Reviews

“Arai's book represents an important contribution to the field of ordinal analysis, and is recommended to any mathematician interested in the classification of the proof-theoretic strength of mathematical theories. … the book is well written, well organized, and concise … . Arai is to be commended for this significant resource for anyone who wishes to study ordinal analysis, and gain a better understanding of the proof-theoretic strength of theories.” (Matthew Jura, Mathematical Reviews, December, 2023)

Authors and Affiliations

  • University of Tokyo, Tokyo, Japan

    Toshiyasu Arai

About the author

Toshiyasu Arai received a PhD from Tsukuba University. His thesis was on an extension of G. Takeuti’s consistency proofs.
He got positions in Nagoya University, Hiroshima University, Kobe University, Chiba University, and University of Tokyo. His research interest is in proof theory. He received the MSJ Autumn Prize from the Mathematical Society of Japan in 2004 for the research on the Hilbert’s second problem.

Bibliographic Information

  • Book Title: Ordinal Analysis with an Introduction to Proof Theory

  • Authors: Toshiyasu Arai

  • Series Title: Logic in Asia: Studia Logica Library

  • DOI: https://doi.org/10.1007/978-981-15-6459-8

  • Publisher: Springer Singapore

  • eBook Packages: Religion and Philosophy, Philosophy and Religion (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2020

  • Hardcover ISBN: 978-981-15-6458-1Published: 12 August 2020

  • Softcover ISBN: 978-981-15-6461-1Published: 13 August 2021

  • eBook ISBN: 978-981-15-6459-8Published: 11 August 2020

  • Series ISSN: 2364-4613

  • Series E-ISSN: 2364-4621

  • Edition Number: 1

  • Number of Pages: XVI, 313

  • Number of Illustrations: 20 b/w illustrations

  • Topics: Logic, Mathematical Logic and Formal Languages

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