Skip to main content
  • Book
  • © 2013

Logical Foundations of Mathematics and Computational Complexity

A Gentle Introduction

Authors:

  • Presents a wide range of results in logic and computational complexity

  • Explains the topic informally and then in more detail for the advanced reader

  • Presents the ideas behind the theoretical concepts

  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Monographs in Mathematics (SMM)

Buying options

eBook USD 169.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-00119-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 219.99
Price excludes VAT (USA)
Hardcover Book USD 219.99
Price excludes VAT (USA)

This is a preview of subscription content, access via your institution.

Table of contents (7 chapters)

  1. Front Matter

    Pages I-XIV
  2. Mathematician’s World

    • Pavel Pudlák
    Pages 1-64
  3. Language, Logic and Computations

    • Pavel Pudlák
    Pages 65-155
  4. Set Theory

    • Pavel Pudlák
    Pages 157-253
  5. Proofs of Impossibility

    • Pavel Pudlák
    Pages 255-364
  6. The Complexity of Computations

    • Pavel Pudlák
    Pages 365-493
  7. Proof Complexity

    • Pavel Pudlák
    Pages 495-581
  8. Consistency, Truth and Existence

    • Pavel Pudlák
    Pages 583-665
  9. Back Matter

    Pages 667-695

About this book

The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability.

Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs.

Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.

Keywords

  • Complexity of Proofs
  • Computational Complexity
  • Foundations of Mathematics
  • Incompleteness Theorems
  • Mathematical Logic
  • Set-Theoretical Foundations
  • algorithmic complexity
  • algorithm analysis and problem complexity

Reviews

“This monograph by the outstanding Czech logician Pavel Pudlák provides a broad but also deep survey of work in logic and computer science relevant to foundational issues, interpreted in a wide sense. … This is a fine overview of logic and complexity theory that can be confidently recommended to anybody who would like to orient themselves in an increasingly intricate and difficult field.” (Alasdair Urquhart, Philosophia Mathematica, Vol. 23 (3), October, 2015)

“For the non-expert it offers indeed a ‘gentle introduction’ to logic that is well selected and excellently explained. And for the logician it certainly offers some of the best introductions to those topics outside their area of direct expertise. … it contains plenty of informal explanations, intuition and motivation. … It is truly a gift to the logic and wider communities … . This book is very enjoyable to read and I wish it all success.” (Olaf Beyersdorff, Mathematical Reviews, August, 2014)

“It spans the historical, logical, and at times philosophical underpinnings of the theory of computational complexity. Students of mathematics seeking a transition to higher mathematics will find it helpful, as will mathematicians with expertise in other areas. … an excellent choice for a first text in studying complexity, or as a clarifying adjunct to any assigned text in this area. … a compact guide for graduate students with a need for or interest in computational complexity and its foundations.” (Tom Schulte, MAA Reviews, July, 2014)

“This book, exactly as indicated by its title, deals with the main philosophical, historical, logical and mathematical aspects … in a quite approachable and attractive way. … the prospective readers of this book are mathematicians with an interest in the foundations, philosophers with a good background in mathematics, and also philosophically minded scientists. Due to the author’s nice style, the book will be a very good choice for the first text in studying this subject.” (Branislav Boričić, zbMATH, Vol. 1270, 2013)

Authors and Affiliations

  • ASCR, Praha 1, Czech Republic

    Pavel Pudlák

Bibliographic Information

Buying options

eBook USD 169.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-00119-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 219.99
Price excludes VAT (USA)
Hardcover Book USD 219.99
Price excludes VAT (USA)