A course on mathematical logic / S. M. Srivastava.

Format:

Book

Author/Creator:

Srivastava, S. M. (Shashi Mohan)

Contributor:

Gödel, Kurt.

Series:

Universitext

Language:

English

Subjects (All):

Logic, Symbolic and mathematical.

Physical Description:

x, 140 pages : illustrations, portrait ; 23 cm.

Place of Publication:

New York : Springer, 2008.

Summary:

This is a short, distinctive, modern, and motivated introduction to mathematical logic for senior undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in knowing what logic is concerned with and who would like to learn Godel's incompleteness theorems should find this book particularly convenient. The treatment is thoroughly mathematical, and the entire subject has been approached like a branch of mathematics. Serious efforts have been made to make the book suitable for the classroom as well as for self-reading. The book does not strive to be a comprehensive encyclopedia of logic. Still, it gives essentially all the basic concepts and results in mathematical logic. The book 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. The main prerequisite for this book is the willingness to work at a reasonable level of mathematical rigor and generality.

Contents:

1 Syntax of First-Order Logic 1

1.1 First-Order Languages 1

1.2 Terms of a Language 4

1.3 Formulas of a Language 6

1.4 First-Order Theories 10

2 Semantics of First-Order Languages 15

2.1 Structures of First-Order Languages 16

2.2 Truth in a Structure 17

2.3 Model of a Theory 19

2.4 Embeddings and Isomorphisms 20

3 Propositional Logic 29

3.1 Syntax of Propositional Logic 30

3.2 Semantics of Propositional Logic 31

3.3 Compactness Theorem for Propositional Logic 33

3.4 Proof in Propositional Logic 37

3.5 Metatheorems in Propositional Logic 38

3.6 Post Tautology Theorem 42

4 Proof and Metatheorems in First-Order Logic 45

4.1 Proof in First-Order Logic 45

4.2 Metatheorems in First-Order Logic 46

4.3 Some Metatheorems in Arithmetic 59

4.4 Consistency and Completeness 62

5 Completeness Theorem and Model Theory 65

5.1 Completeness Theorem 65

5.2 Interpretations in a Theory 70

5.3 Extension by Definitions 72

5.4 Compactness Theorem and Applications 74

5.5 Complete Theories 77

5.6 Applications in Algebra 79

6 Recursive Functions and Arithmetization of Theories 83

6.1 Recursive Functions and Recursive Predicates 84

6.2 Semirecursive Predicates 93

6.3 Arithmetization of Theories 96

6.4 Decidable Theories 103

7 Incompleteness Theorems and Recursion Theory 107

7.1 Representability 107

7.2 First Incompleteness Theorem 115

7.3 Arithmetical Sets 116

7.4 Recursive Extensions of Peano Arithemetic 125

7.5 Second Incompleteness Theorem 131.

Notes:

"This book is written on the occasion of the birth centenary year of Kurt Gödel (1906-1978)."--Preface.

Includes bibliographical references (page [135]) and index.

ISBN:

9780387762753

0387762752

OCLC:

180470520