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A Concise Introduction to Mathematical Logic

Wolfgang Rautenberg
Publisher: 
Springer Verlag
Publication Date: 
2006
Number of Pages: 
256
Format: 
Paperback
Edition: 
2
Series: 
Universitext
Price: 
49.95
ISBN: 
0387302948
Category: 
Textbook
We do not plan to review this book.

Introduction XIII

Notation XVI

1 Propositional Logic 1

1.1 Boolean Functions and Formulas . . . . . . . . . . . . . . . . . . . . 2

1.2 Semantic Equivalence and Normal Forms . . . . . . . . . . . . . . . . 9

1.3 Tautologies and Logical Consequence . . . . . . . . . . . . . . . . . . 14

1.4 A Complete Calculus for . . . . . . . . . . . . . . . . . . . . . . . . 18

1.5 Applications of the Compactness Theorem . . . . . . . . . . . . . . . 25

1.6 Hilbert Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Predicate Logic 33

2.1 Mathematical Structures . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2 Syntax of Elementary Languages . . . . . . . . . . . . . . . . . . . . 43

2.3 Semantics of Elementary Languages . . . . . . . . . . . . . . . . . . . 49

2.4 General Validity and Logical Equivalence . . . . . . . . . . . . . . . . 58

2.5 Logical Consequence and Theories . . . . . . . . . . . . . . . . . . . . 62

2.6 Explicit Definitions—Expanding Languages . . . . . . . . . . . . . . 67

3 G¨odel’s Completeness Theorem 71

3.1 A Calculus of Natural Deduction . . . . . . . . . . . . . . . . . . . . 72

3.2 The Completeness Proof . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.3 First Applications—Nonstandard Models . . . . . . . . . . . . . . . . 81

3.4 ZFC and Skolem’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5 Enumerability and Decidability . . . . . . . . . . . . . . . . . . . . . 92

3.6 Complete Hilbert Calculi . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.7 First-Order Fragments and Extensions . . . . . . . . . . . . . . . . . 99

IX

X Contents

4 The Foundations of Logic Programming 105

4.1 Term Models and Horn Formulas . . . . . . . . . . . . . . . . . . . . 106

4.2 Propositional Resolution . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.3 Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.4 Logic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

4.5 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . 129

5 Elements of Model Theory 131

5.1 Elementary Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.2 Complete and κ-Categorical Theories . . . . . . . . . . . . . . . . . . 137

5.3 Ehrenfeucht’s game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.4 Embedding and Characterization Theorems . . . . . . . . . . . . . . 145

5.5 Model Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

5.6 Quantifier Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.7 Reduced Products and Ultraproducts . . . . . . . . . . . . . . . . . . 163

6 Incompleteness and Undecidability 167

6.1 Recursive and Primitive Recursive Functions . . . . . . . . . . . . . . 169

6.2 Arithmetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.3 Representability of Arithmetical Predicates . . . . . . . . . . . . . . . 182

6.4 The Representability Theorem . . . . . . . . . . . . . . . . . . . . . . 189

6.5 The Theorems of G¨odel, Tarski, Church . . . . . . . . . . . . . . . . 194

6.6 Transfer by Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 200

6.7 The Arithmetical Hierarchy . . . . . . . . . . . . . . . . . . . . . . . 205

7 On the Theory of Self-Reference 209

7.1 The Derivability Conditions . . . . . . . . . . . . . . . . . . . . . . . 210

7.2 The Theorems of G¨odel and L¨ob. . . . . . . . . . . . . . . . . . . . . 217

7.3 The Provability Logic G . . . . . . . . . . . . . . . . . . . . . . . . . 221

7.4 The Modal Treatment of Self-Reference . . . . . . . . . . . . . . . . . 223

7.5 A Bimodal Provability Logic for PA . . . . . . . . . . . . . . . . . . . 226

7.6 Modal Operators in ZFC . . . . . . . . . . . . . . . . . . . . . . . . . 228

Hints to the Exercises 231

Literature 241

Contents XI

Index of Terms and Names 247

Index of Symbols 255

Dummy View - NOT TO BE DELETED