This two-volume work bridges the gap between introductory expositions of logic (or set theory) and the research literature. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly lecture style that makes them equally effective for self-study or class use. Volume I includes formal proof techniques, applications of compactness (including nonstandard analysis), computability and its relation to the completeness phenonmenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen.
Author(s): George Tourlakis
Series: Cambridge Studies in Advanced Mathematics 82
Publisher: Cambridge University Press
Year: 2003
Language: English
Pages: 328
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 11
I Basic Logic......Page 15
I.1. First Order Languages......Page 19
Logical Symbols......Page 21
Nonlogical Symbols......Page 22
I.2. A Digression into the Metatheory: Informal Induction and Recursion......Page 33
I.3. Axioms and Rules of Inference......Page 42
I.4. Basic Metatheorems......Page 56
I.5. Semantics; Soundness, Completeness, Compactness......Page 66
I.6. Substructures, Diagrams, and Applications......Page 89
I.7. Defined Symbols......Page 126
I.8. Computability and Uncomputability......Page 137
I.9. Arithmetic, Definability, Undefinability, and Incompletableness......Page 169
I.10. Exercises......Page 205
II The Second Incompleteness Theorem......Page 219
II.1. Peano Arithmetic......Page 220
II.2. A Formal Beta-Function......Page 246
II.3. Formal Primitive Recursion......Page 262
II.4. The Boldface Delta and Sigma......Page 270
II.5. Arithmetization......Page 279
II.6. Derivability Conditions; Fixed Points......Page 286
II.7. Exercises......Page 330
Bibliography......Page 333
List of Symbols......Page 335
Index......Page 337
