Skip to main content
SpringerLink
Log in
Book cover

A Course in Mathematical Logic

  • Textbook
  • © 1977

Overview

Authors:
  1. Yu. I. Manin

    1. V. A. Steklov Mathematical Institute of the Academy of Sciences, Moscow, USSR

    View author publications

    You can also search for this author in PubMed   Google Scholar

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

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (8 chapters)

  1. Front Matter

    Pages i-xiii
    Download chapter PDF

  2. Provability

    1. Front Matter

      Pages 1-1
      Download chapter PDF

    2. Introduction to formal languages

      • Yu. I. Manin
      Pages 3-19

    3. Truth and deducibility

      • Yu. I. Manin
      Pages 20-102

    4. The continuum problem and forcing

      • Yu. I. Manin
      Pages 103-148

    5. The continuum problem and constructible sets

      • Yu. I. Manin
      Pages 149-174

  3. Computability

    1. Front Matter

      Pages 175-175
      Download chapter PDF

    2. Recursive functions and Church’s thesis

      • Yu. I. Manin
      Pages 177-205

    3. Diophantine sets and algorithmic undecidability

      • Yu. I. Manin
      Pages 206-230

  4. Provability and Computability

    1. Front Matter

      Pages 231-231
      Download chapter PDF

    2. Gödel’s incompleteness theorem

      • Yu. I. Manin
      Pages 233-260

    3. Recursive groups

      • Yu. I. Manin
      Pages 261-283

  5. Back Matter

    Pages 285-288
    Download chapter PDF
Back to top

Keywords

About this book

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.

Reviews

"As one might expect from a graduate text on logic by a very distinguished algebraic geometer, this book assumes no previous acquaintance with logic, but proceeds at a high level of mathematical sophistication. Chapters I and II form a short course. Chapter I is a very informal introduction to formal languages, e.g., those of first order Peano arithmetic and of ZFC set theory. Chapter II contains Tarski's definition of truth, Gödel's completeness theorem, and the Löwenheim-Skolem theorem. The emphasis is on semantics rather than syntax. Some rarely-covered side topics are included (unique readability for languages with parentheses, Mostowski's transitive collapse lemma, formalities of introducing definable constants and function symbols). Some standard topics are neglected. (The compactness theorem is not mentioned!) The latter part of Chapter II contains Smullyan's quick proof of Tarski's theorem on the undefinability of truth in formal arithmetic, and an account of the Kochen-Specker "no hidden variables" theorem in quantum logic. There are digressions on philosophical issues (formal logic vs. ordinary language, computer proofs). A wealth of material is introduced in these first 100 pages of the book..."--MATHEMATICAL REVIEWS

Authors and Affiliations

  • V. A. Steklov Mathematical Institute of the Academy of Sciences, Moscow, USSR

    Yu. I. Manin

Bibliographic Information

  • Book Title: A Course in Mathematical Logic

  • Authors: Yu. I. Manin

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4757-4385-2

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1977

  • eBook ISBN: 978-1-4757-4385-2Published: 29 June 2013

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 1

  • Number of Pages: XIII, 288

  • Topics: Mathematical Logic and Foundations

Publish with us

Policies and ethics

Back to top

Search

Navigation