Logic of Mathematics: A Modern Course of Classical Logic

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John Wiley & Sons, 1997 - 260 pages
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A thorough, accessible, and rigorous presentation of the central theorems of mathematical logic . . . ideal for advanced students of mathematics, computer science, and logic

Logic of Mathematics combines a full-scale introductory course in mathematical logic and model theory with a range of specially selected, more advanced theorems. Using a strict mathematical approach, this is the only book available that contains complete and precise proofs of all of these important theorems:
* Gödel's theorems of completeness and incompleteness
* The independence of Goodstein's theorem from Peano arithmetic
* Tarski's theorem on real closed fields
* Matiyasevich's theorem on diophantine formulas

Logic of Mathematics also features:
* Full coverage of model theoretical topics such as definability, compactness, ultraproducts, realization, and omission of types
* Clear, concise explanations of all key concepts, from Boolean algebras to Skolem-Löwenheim constructions and other topics
* Carefully chosen exercises for each chapter, plus helpful solution hints

At last, here is a refreshingly clear, concise, and mathematically rigorous presentation of the basic concepts of mathematical logic-requiring only a standard familiarity with abstract algebra. Employing a strict mathematical approach that emphasizes relational structures over logical language, this carefully organized text is divided into two parts, which explain the essentials of the subject in specific and straightforward terms.

Part I contains a thorough introduction to mathematical logic and model theory-including a full discussion of terms, formulas, and other fundamentals, plus detailed coverage of relational structures and Boolean algebras, Gödel's completeness theorem, models of Peano arithmetic, and much more.

Part II focuses on a number of advanced theorems that are central to the field, such as Gödel's first and second theorems of incompleteness, the independence proof of Goodstein's theorem from Peano arithmetic, Tarski's theorem on real closed fields, and others. No other text contains complete and precise proofs of all of these theorems.

With a solid and comprehensive program of exercises and selected solution hints, Logic of Mathematics is ideal for classroom use-the perfect textbook for advanced students of mathematics, computer science, and logic.
 

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Table des matières

Introduction
1
Mathematical Structures and Their Theories
7
Relational Systems
9
Boolean Algebras
13
Subsystems and Homomorphisms
19
Operations on Relational Systems
25
Terms and Formulas
30
Theories and Models
47
Definability
86
Peano Arithmetic
94
SkolemLowenheim Theorems
104
Ultraproducts HI 16 Types of Elements
121
Supplementary Questions
136
Defining Functions in N
147
Total Functions
160
Arithmetical Consistency
182

Substitution of Terms
55
Theorems and Proofs
62
Theorems of the Logical Calculus
67
Generalization Rule and Elimination of Constants
75
The Completeness of the Logical Calculus
79
Independence of Goodsteins Theorem
201
Tarskis Theorem
223
Guide to Further Reading
252
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À propos de l'auteur (1997)

ZOFIA ADAMOWICZ, PhD, is a professor at the Institute of Mathematics of the Polish Academy of Sciences in Warsaw.

PAWEL ZBIERSKI, PhD, is a professor at the Department of Mathematics at Warsaw University and the coauthor of Hausdorff Gaps and Limits.

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