Book:Alan G. Hamilton/Logic for Mathematicians/Second Edition

Subject Matter

 * Logic
 * Mathematical Logic

Contents

 * Preface


 * 1 Informal statement calculus
 * 1.1 Statements and connectives
 * 1.2 Truth functions and truth tables
 * 1.3 Rules for manipulation and substitution
 * 1.4 Normal forms
 * 1.5 Adequate sets of connectives
 * 1.6 Arguments and validity


 * 2 Formal statement calculus
 * 2.1 The formal system $L$
 * 2.2 The Adequacy Theorem for $L$


 * 3 Informal predicate calculus
 * 3.1 Predicates and quantifiers
 * 3.2 First order languages
 * 3.3 Interpretations
 * 3.4 Satisfaction, truth
 * 3.5 Skolemisation


 * 4 Formal predicate calculus
 * 4.1 The formal system $K_\mathcal L$
 * 4.2 Equivalence, substitution
 * 4.3 Prenex form
 * 4.4 The Adequacy Theorem for $K$
 * 4.5 Models


 * 5 Mathematical systems
 * 5.1 Introduction
 * 5.2 First order systems with equality
 * 5.3 The theory of groups
 * 5.4 First order arithmetic
 * 5.5 Formal set theory
 * 5.6 Consistency and models


 * 6 The Gödel Incompleteness Theorem
 * 6.1 Introduction
 * 6.2 Expressibility
 * 6.3 Recursive functions and relations
 * 6.4 Gödel numbers
 * 6.5 The incompleteness proof


 * 7 Computability, unsolvability, undecidability
 * 7.1 Algorithms and computability
 * 7.2 Turing machines
 * 7.3 Word problems
 * 7.4 Undecidability of formal systems


 * Appendix: Countable and uncountable sets


 * Hints and solutions to selected exercises


 * References and further reading


 * Glossary of symbols


 * Index