Mathematical Logic and Formalized Theories (eBook)

A Survey of Basic Concepts and Results

Robert L. Rogers (Autor)

eBook Download: PDF

2014 | 2. Auflage
248 Seiten
Elsevier Science (Verlag)
978-1-4832-5797-6 (ISBN)

Infos

Mathematical Logic and Formalized Theories: A Survey of Basic Concepts and Results focuses on basic concepts and results of mathematical logic and the study of formalized theories. The manuscript first elaborates on sentential logic and first-order predicate logic. Discussions focus on first-order predicate logic with identity and operation symbols, first-order predicate logic with identity, completeness theorems, elementary theories, deduction theorem, interpretations, truth, and validity, sentential connectives, and tautologies. The text then tackles second-order predicate logic, as well as second-order theories, theory of definition, and second-order predicate logic F2. The publication takes a look at natural and real numbers, incompleteness, and the axiomatic set theory. Topics include paradoxes, recursive functions and relations, Gödel's first incompleteness theorem, axiom of choice, metamathematics of R and elementary algebra, and metamathematics of N. The book is a valuable reference for mathematicians and researchers interested in mathematical logic and formalized theories.

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Mathematical Logic and Formalized Theories: A Survey of Basic Concepts and Results focuses on basic concepts and results of mathematical logic and the study of formalized theories. The manuscript first elaborates on sentential logic and first-order predicate logic. Discussions focus on first-order predicate logic with identity and operation symbols, first-order predicate logic with identity, completeness theorems, elementary theories, deduction theorem, interpretations, truth, and validity, sentential connectives, and tautologies. The text then tackles second-order predicate logic, as well as second-order theories, theory of definition, and second-order predicate logic F2. The publication takes a look at natural and real numbers, incompleteness, and the axiomatic set theory. Topics include paradoxes, recursive functions and relations, Godel's first incompleteness theorem, axiom of choice, metamathematics of R and elementary algebra, and metamathematics of N. The book is a valuable reference for mathematicians and researchers interested in mathematical logic and formalized theories.

Inhalt

Front Cover 1
Mathematical logic and Formalized Theories: A Survey of Basic Concepts and Results 4
Copyright Page 5
Table of Contents 10
Dedication 6
PREFACE 8
CHAPTER I. THE SENTENTIAL LOGIC 14
1.1. Introduction 14
1.2. Sentential Connectives 16
1.3. The Sentential Logic P. Symbols and Formulas 24
1.4. Tautologies 27
1.5. Axiom Schemata of P. Rules of Inference and Theorems 34
1.6. Metamathematical Properties of P 38
CHAPTER II. THE FIRST-ORDER PREDICATE LOGIC: I 41
2.1. The First-Order Predicate Logic F1. Symbols, Quantifiers and Formulas 42
2.2. Interpretations. Truth and Validity 47
2.3. Axiom Schemata of F . Rules of Inference and Theorems. Consistency of F 56
2.4. The Deduction Theorem 61
CHAPTER III. THE FIRST-ORDER PREDICATE LOGIC: II 65
3.1. Elementary Theories 65
3.2. Completeness Theorems 73
3.3. Further Corollaries. Decision Problem 81
3.4. The First-Order Predicate Logic With Identity 84
3.5. The First-Order Predicate Logic With Identity and Operation Symbols 91
CHAPTER IV. THE SECOND-ORDER PREDICATE LOGIC.THEORY OF DEFINITION 96
4.1. Introduction 96
4.2. The Second-Order Predicate Logic F2 98
4.3. Second-Order Theories 109
4.4. Theory of Definition 111
CHAPTER V. THE NATURAL NUMBERS 120
5.1. Introduction 120
5.2. Elementary Arithmetic: The Theory N 122
5.3. The Metamathematics of N 129
5.4. Second-Order Arithmetic: The Theory N2 133
5.5. The Metamathematics of N2 137
CHAPTER VI. THE REAL NUMBERS 140
6.1. The Theory R 142
6.2. The Metamathematics of R and of Elementary Algebra 146
6.3. Second-Order Real Number Theory: The Theory R2 150
6.4. The Metamathematics of R2 153
CHAPTER VII. AXIOMATIC SET THEORY 155
7.1. Paradoxes 156
7.2. The Zermelo-Fraenkel Axioms 162
7.3. The Axiom of Choice 178
7.4. The Metamathematics of ZF 185
7.5. Strengthened Forms of ZF 195
CHAPTER VIIII. NCOMPLETENESS. UNDECIDABILITY 199
8.1. Introduction 199
8.2. Recursive Functions and Relations. Representability 201
8.3. Arithmetization 209
8.4. Gödel's First Incompleteness Theorem 212
8.5. Gödel's Second Incompleteness Theorem 219
8.6. Tarski's Theorem 223
8.7. Decision Problem. Church's Thesis. Recursively Enumerable Sets 228
8.8. Undecidability 233
AUTHOR INDEX 243
SUBJECT INDEX 245

Produkt-Details

• Erscheint lt. Verlag: 12.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Technik
• ISBN-10: 1-4832-5797-5 / 1483257975
• ISBN-13: 978-1-4832-5797-6 / 9781483257976