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A Course on Mathematical Logic

  • Textbook
  • © 2013
  • Latest edition

Overview

Authors:
  1. Shashi Mohan Srivastava

    1. Theoretical Statistics & Math. Unit, Indian Statistical Institute, Kolkata, India

  • New edition extensively revised and updated
  • Includes a new chapter on model theory, and several new sections on topics such as ultraproducts, quantifier eliminations, real closed and algebraically closed fields, definability, partial elementary maps, and homogenous structures
  • Contains numerous exercises, examples, and applications such as Chevalley's theorem, Hilbert's Nullstellensatz, and the solution to Hilbert's 17th problem
  • Employs Gödel’s completeness and incompleteness theorems to motivate the entire text
  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

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About this book

This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, recursion theory, and computability.

In this new edition, many small and large changes have been made throughout the text.  The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic.  Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry.  Some proofs, such as the proof of the very important completeness theorem,  have been completely rewritten in a more clear and concise manner.  The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more.

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Syntax of First-Order Logic

    • Shashi Mohan Srivastava
    Pages 1-13

  3. Semantics of First-Order Languages

    • Shashi Mohan Srivastava
    Pages 15-40

  4. Propositional Logic

    • Shashi Mohan Srivastava
    Pages 41-55

  5. Completeness Theorem for First-Order Logic

    • Shashi Mohan Srivastava
    Pages 57-84

  6. Model Theory

    • Shashi Mohan Srivastava
    Pages 85-140

  7. Recursive Functions and Arithmetization of Theories

    • Shashi Mohan Srivastava
    Pages 141-164

  8. Representability and Incompleteness Theorems

    • Shashi Mohan Srivastava
    Pages 165-191

  9. Back Matter

    Pages 193-198
    Download chapter PDF
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Reviews

From the reviews:

"In this work, which provides an introduction to mathematical logic, Srivastava … indicates that his main goal is to ‘state and prove Gödel’s completeness and incompleteness theorems in precise mathematical terms.’ … the author presents the material in a clear fashion, with consistent and understandable notation. The book includes a number of exercises for the student to attempt and examples from a variety of areas in mathematics for the student to review. … Summing Up: Recommended. Advanced upper-division undergraduates, graduate students, faculty." (S. L. Sullivan, Choice, Vol. 46 (4), December, 2008)

"This is an introductory textbook on modern mathematical logic, aimed at upper-level undergraduates. … The book is well-equipped with examples … ." (Allen Stenger, MathDL, July, 2008)

"The main goal of this book is to give a motivated introduction to mathematical logic for graduated and advanced undergraduate students of logic, set theory, recursion theory and computer science. Its intended audience includes also all mathematicians who are interested in knowing what mathematical logic is dealing with. … All results included in the book are very carefully selected and proved. The author’s manner of writing is excellent, which will surely make this book useful to many categories of readers." (Marius Tarnauceanu, Zentralblatt MATH, Vol. 1140, 2008)

From the reviews of the second edition:

“This very sophisticated work deals with what the author presents as proof of Gödel’s completeness and incompleteness theorems in precise mathematical terms. … This rigorous book is appropriate for graduate students or very talented undergraduates.” (James Van Speybroeck, Computing Reviews, April, 2013)

Authors and Affiliations

  • Theoretical Statistics & Math. Unit, Indian Statistical Institute, Kolkata, India

    Shashi Mohan Srivastava

About the author

Shashi Mohan Srivastava is a professor at the Indian Statistical Institute, Kolkata, India. He is also the author of A Course on Borel Sets (GTM 180), and recipient of the Indian National Science Academy Medal for Young Scientists in 1981.

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