What Is Mathematical Logic?.

Format:

Book

Author/Creator:

Badia, Guillermo.

Language:

English

Physical Description:

1 online resource (161 pages)

Edition:

1st ed.

Place of Publication:

Oxford : Oxford University Press, Incorporated, 2026.

Summary:

Mathematical logic has grown into an indispensable tool in computer science as well as other parts of mathematics. This concise book presents the subject of mathematical logic in a lively and approachable fashion although logic can be a formidably abstruse topic, even for mathematicians.

Contents:

Cover

Title page

Copyright page

Dedication page

Preface

Contents

1 Historical Survey

1.1 Origins of formal logic

1.2 Origins of analysis

1.3 Set theory

1.4 The axiom of choice

1.5 Theorems about logic

1.6 The unprovability of consistency

1.7 Computability

1.8 Turing machines

1.9 Incompleteness in set theory

1.10 Computational complexity

1.11 Suggestions for further reading

2 The Completeness of First-order Logic

2.1 Formal languages

2.2 The formal language LOrder

2.3 Interpretations

2.4 Completeness problem

2.5 The Go-exdel-Henkin Completeness Theorem

2.6 Suggestions for further reading

3 Model Theory

3.1 Equality

3.2 Limitations of first-order logic

3.3 The Compactness and Lo-exwenheim-Skolem Theorems

3.4 Categoricity in power

3.5 Applications of Model Theory

3.6 Suggestions for further reading

4 Turing Machines

4.1 From Post to Turing

4.2 Turing machines

4.3 Computable (partial) functions

4.4 Standard description of Turing machines

4.5 An unsolvable problem

4.6 Universal Turing machines

4.7 Turing machines via word transformations

4.8 An aside on word problems in algebra

4.9 Turing machines and first-order logic

4.10 Turing completeness

4.11 Feasible computation

4.12 NP problems and NP-completeness

4.13 Suggestions for further reading

5 Go-exdel's Incompleteness Theorems

5.1 Peano Arithmetic

5.2 Go-exdel numbering (or arithmetization of syntax)

5.3 Representability of computable relations

5.4 Go- exdel's First Incompleteness Theorem

5.5 Robinson arithmetic and essential undecidability

5.6 Concrete incompleteness: the Paris-Harrington theorem

5.7 The Second Incompleteness Theorem

5.8 Suggestions for further reading

6 Computability in Practice

6.1 Introduction

6.2 Development.

6.3 Interactive theorem-proving

6.4 The Curry-Howard isomorphism

6.5 Quis custodet custodies?

6.6 A few of the very many results

6.7 Producing correct programs

6.8 Concluding remarks

6.9 Suggestions for further reading

7 Set Theory

7.1 Axiomatizing set theory

7.2 Other axioms of Zermelo-Fraenkel set theory

7.3 Ordinal numbers

7.4 Cardinal numbers

7.5 The Continuum Hypothesis

7.6 Consistency of the Axiom of Choice

7.7 Independence of Axiom of Choice and Continuum Hypothesis

7.8 Inaccessibles

7.9 Recursion

7.10 Suggestions for further reading

8 Beyond First-order Logic

8.1 Second-order logic

8.2 Infinitary logic

8.3 Cardinality quantifiers

8.4 Suggestions for further reading

9 Further Topics

9.1 Computability and logic

9.2 Intuitionist logic

9.3 Category theory and logic

9.4 Free logic

9.5 Modal logic

9.6 Final remarks

9.7 Suggestions for further reading

Index.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

0-19-893088-7

0-19-893087-9

9780198930877

OCLC:

1555344477