Mathematical Logic / by Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas.

Format:

Book

Author/Creator:

Ebbinghaus, Heinz-Dieter, 1939- author.

Flum, Jörg, author.

Thomas, Wolfgang, 1947- author.

Series:

Graduate Texts in Mathematics, 2197-5612 ; 291

Language:

English

Subjects (All):

Logic, Symbolic and mathematical.

Computer science--Mathematics.

Computer science.

Mathematical Logic and Foundations.

Mathematics of Computing.

Local Subjects:

Mathematical Logic and Foundations.

Mathematics of Computing.

Physical Description:

1 online resource (305 pages)

Edition:

3rd ed. 2021.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2021.

Summary:

This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science. The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function. Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.

Contents:

A

I Introduction

II Syntax of First-Order Languages

III Semantics of First-Order Languages

IV A Sequent Calculus

V The Completeness Theorem

VI The Löwenheim–Skolem and the Compactness Theorem

VII The Scope of First-Order Logic

VIII Syntactic Interpretations and Normal Forms

B

IX Extensions of First-Order Logic

X Computability and Its Limitations

XI Free Models and Logic Programming

XII An Algebraic Characterization of Elementary Equivalence

XIII Lindström’s Theorems

References

List of Symbols

Subject Index.

Notes:

Includes bibliographical references and index.

ISBN:

3-030-73839-6

OCLC:

1253630465