# Mathematical Logic

## On Numbers, Sets, Structures, and Symmetry

• Roman Kossak
Textbook

Part of the Springer Graduate Texts in Philosophy book series (SGTP, volume 3)

1. Front Matter
Pages i-xiii
2. ### Logic, Sets, and Numbers

1. Front Matter
Pages 1-1
2. Roman Kossak
Pages 3-17
3. Roman Kossak
Pages 19-31
4. Roman Kossak
Pages 33-39
5. Roman Kossak
Pages 41-56
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Pages 57-70
7. Roman Kossak
Pages 71-79
3. ### Relations, Structures, Geometry

1. Front Matter
Pages 81-81
2. Roman Kossak
Pages 83-95
3. Roman Kossak
Pages 97-104
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Pages 105-113
5. Roman Kossak
Pages 115-129
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11. Roman Kossak
Pages 167-168
4. Back Matter
Pages 169-186

### Introduction

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.

### 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

#### Authors and affiliations

• Roman Kossak
• 1
1. 1.City University of New YorkNew YorkUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-97298-5
• Copyright Information Springer International Publishing AG part of Springer Nature 2018
• Publisher Name Springer, Cham
• eBook Packages Religion and Philosophy
• Print ISBN 978-3-319-97297-8
• Online ISBN 978-3-319-97298-5
• Buy this book on publisher's site