Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction

by Pavel Pudlák

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.

Genres: Mathematics, Computer Science, Logic

709 pages, Hardcover

First published May 20, 2013

Reviews

[Jan Jaap]

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I found this title in the skip list, comparing books with https://www.goodreads.com/user/show/5... Tom Schulte. That was today (2024 0318 mo 12 h CET).

https://link.springer.com/book/10.100... publisher
https://www.springer.com/series/3733 part of SMM

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[Rose Hossain · Rating 5 out of 5]

This book is framed in such a spectacular way. It is one of those learning curves where you feel as if you understand nothing until you have finished the section or the book, and you feel as if now that it is done you understand it too well. The author uses sentence and word examples, relating to how logical foundations are applicable to sentence structure, input and output models and helped my understanding once I learned coding.

[Tom Schulte · Rating 5 out of 5]

This book lives up to its title. It spans the historical, logical, and at times philosophical underpinnings of mathematical complexity. Students of mathematics seeking a transition to higher mathematics will find this helpful, as will mathematicians with expertise in other areas. Philosophers and the philosophically inclined with a background in mathematics may be drawn to many chapters here. This is an excellent choice for a first text in studying complexity, or as a clarifying adjunct to any assigned text in this area...


[Read my entire review at MAA Reviews.]