# A Course in Mathematical Logic

• Yu. I. Manin
Book

Part of the Graduate Texts in Mathematics book series (GTM, volume 53)

1. Front Matter
Pages i-xiii
2. ### Provability

1. Front Matter
Pages 1-1
2. Yu. I. Manin
Pages 3-19
3. Yu. I. Manin
Pages 20-102
4. Yu. I. Manin
Pages 103-148
5. Yu. I. Manin
Pages 149-174
3. ### Computability

1. Front Matter
Pages 175-175
2. Yu. I. Manin
Pages 177-205
3. Yu. I. Manin
Pages 206-230
4. ### Provability and Computability

1. Front Matter
Pages 231-231
2. Yu. I. Manin
Pages 233-260
3. Yu. I. Manin
Pages 261-283
5. Back Matter
Pages 285-288

### Introduction

1. This book is above all addressed to mathematicians. It is intended to be a textbook of mathematical logic on a sophisticated level, presenting the reader with several of the most significant discoveries of the last ten or fifteen years. These include: the independence of the continuum hypothe­ sis, the Diophantine nature of enumerable sets, the impossibility of finding an algorithmic solution for one or two old problems. All the necessary preliminary material, including predicate logic and the fundamentals of recursive function theory, is presented systematically and with complete proofs. We only assume that the reader is familiar with "naive" set theoretic arguments. In this book mathematical logic is presented both as a part of mathe­ matics and as the result of its self-perception. Thus, the substance of the book consists of difficult proofs of subtle theorems, and the spirit of the book consists of attempts to explain what these theorems say about the mathematical way of thought. Foundational problems are for the most part passed over in silence. Most likely, logic is capable of justifying mathematics to no greater extent than biology is capable of justifying life. 2. The first two chapters are devoted to predicate logic. The presenta­ tion here is fairly standard, except that semantics occupies a very domi­ nant position, truth is introduced before deducibility, and models of speech in formal languages precede the systematic study of syntax.

### Keywords

Logic Mathematica Mathematische Logik boundary element method computability forcing form formal language formal languages function language mathematical logic presentation theorem theory of complexity

#### Authors and affiliations

• Yu. I. Manin
• 1
1. 1.V. A. Steklov Mathematical Institute of the Academy of SciencesMoscowUSSR

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4757-4385-2
• Copyright Information Springer-Verlag New York 1977
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-1-4757-4387-6
• Online ISBN 978-1-4757-4385-2
• Series Print ISSN 0072-5285