Overview
- Presents an introduction to formal mathematical logic and set theory
- Presents simple yet nontrivial results in modern model theory
- Provides introductory remarks to all results, including a historical background
Part of the book series: Springer Graduate Texts in Philosophy (SGTP, volume 3)
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Table of contents (16 chapters)
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Logic, Sets, and Numbers
-
Relations, Structures, Geometry
Keywords
- first-order logic introduction
- Abstract symmetries
- Number system development
- Set theory mathematics
- Model theory
- Formal arithmetic
- Tameness mathematical structures
- Axiomatic set theory
- Compactness Theorem
- Logical visibility
- first-order logic applications
- complexity mathematical structures
- Applied compactness theorem
- geometry definable sets
- mathematical structure relations
- logical visibility
- language of modern mathematics
- recent developents model theory
- Ramsey theory
About this book
This book, presented in two parts, offers a slow introduction to mathematical logic, and several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions.
Its first part, Logic Sets, and Numbers, shows how mathematical logic is used to develop the number structures of classical mathematics. The exposition does not assume any prerequisites; it is rigorous, but as informal as possible. All necessary concepts are introduced exactly as they would be in a course in mathematical logic; but are accompanied by more extensive introductory remarks and examples to motivate formal developments.
The second part, Relations, Structures, Geometry, introduces several basic concepts of model theory, such as first-order definability, types, symmetries, and elementary extensions, and shows how they are used to study and classify mathematical structures. Although more advanced, this second part is accessible to the reader who is either already familiar with basic mathematical logic, or has carefully read the first part of the book. Classical developments in model theory, including the Compactness Theorem and its uses, are discussed. Other topics include tameness, minimality, and order minimality of structures.
The book can be used as an introduction to model theory, but unlike standard texts, it does not require familiarity with abstract algebra. This book will also be of interest to mathematicians who know the technical aspects of the subject, but are not familiar with its history and philosophical background.
Reviews
“This fun book can be viewed as a very gentle introduction to the notion of mathematical structure, and hence to model theory. … Each chapter concludes with a selection of exercises of varying degrees of difficulty, often asking the reader to establish facts. Apart from the uses suggested on the book's cover, I can well imagine teaching an introduction to proof class with this textbook.” (Jana Maříková, Mathematical Reviews, March 2021)
“The author has made a significant effort to present the (not so easy) material in an understandable way … . I am sure that readers of this well-written book will experience many such satisfying moments.” (Temur Kutsia, Computing Reviews, September 11, 2019)
“Such modesty and humility. Wow. Here is an outstanding book. In the beginning, we learn of the difficulties the author encountered as a student while learning some of the very topics he writes about in this book. So successfully has the author conquered his youthful difficulties that model theory is now his research specialty and is also an important component of this book.” (Dennis W. Gordon, MAA Reviews, May 19, 2019)
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Mathematical Logic
Book Subtitle: On Numbers, Sets, Structures, and Symmetry
Authors: Roman Kossak
Series Title: Springer Graduate Texts in Philosophy
DOI: https://doi.org/10.1007/978-3-319-97298-5
Publisher: Springer Cham
eBook Packages: Religion and Philosophy, Philosophy and Religion (R0)
Copyright Information: Springer International Publishing AG part of Springer Nature 2018
Softcover ISBN: 978-3-030-07331-2Published: 19 January 2019
eBook ISBN: 978-3-319-97298-5Published: 03 October 2018
Series ISSN: 2627-6046
Series E-ISSN: 2627-6054
Edition Number: 1
Number of Pages: XIII, 186
Number of Illustrations: 28 b/w illustrations
Topics: Philosophy of Mathematics, Mathematical Logic and Foundations, Arithmetic and Logic Structures, Logic, Applications of Mathematics