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Searching... | 30000000237499 | QA9 H29 1978 | Open Access Book | Book | Searching... |
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Summary
Summary
Intended for logicians and mathematicians, this text is based on Dr. Hamilton's lectures to third and fourth year undergraduates in mathematics at the University of Stirling. With a prerequisite of first year mathematics, the author introduces students and professional mathematicians to the techniques and principal results of mathematical logic. In presenting the subject matter without bias towards particular aspects, applications or developments, it is placed in the context of mathematics. To emphasize the level, the text progresses from informal discussion to the precise description and use of formal mathmematical and logical systems. The revision of this very successful textbook includes new sections on skolemization and the application of well-formed formulae to logic programming; numerous corrections have been made and extra exercises added.
Table of Contents
| Preface | p. vii |
| 1 Informal statement calculus | |
| 1.1 Statements and connectives | p. 1 |
| 1.2 Truth functions and truth tables | p. 4 |
| 1.3 Rules for manipulation and substitution | p. 10 |
| 1.4 Normal forms | p. 15 |
| 1.5 Adequate sets of connectives | p. 19 |
| 1.6 Arguments and validity | p. 22 |
| 2 Formal statement calculus | |
| 2.1 The formal system L | p. 27 |
| 2.2 The Adequacy Theorem for L | p. 37 |
| 3 Informal predicate calculus | |
| 3.1 Predicates and quantifiers | p. 45 |
| 3.2 First order languages | p. 49 |
| 3.3 Interpretations | p. 57 |
| 3.4 Satisfaction, truth | p. 59 |
| 3.5 Skolemisation | p. 70 |
| 4 Formal predicate calculus | |
| 4.1 The formal system K[subscript se] | p. 73 |
| 4.2 Equivalence, substitution | p. 80 |
| 4.3 Prenex form | p. 86 |
| 4.4 The Adequacy Theorem for K | p. 92 |
| 4.5 Models | p. 100 |
| 5 Mathematical systems | |
| 5.1 Introduction | p. 105 |
| 5.2 First order systems with equality | p. 106 |
| 5.3 The theory of groups | p. 112 |
| 5.4 First order arithmetic | p. 116 |
| 5.5 Formal set theory | p. 120 |
| 5.6 Consistency and models | p. 125 |
| 6 The Godel Incompleteness Theorem | |
| 6.1 Introduction | p. 128 |
| 6.2 Expressibility | p. 130 |
| 6.3 Recursive functions and relations | p. 137 |
| 6.4 Godel numbers | p. 146 |
| 6.5 The incompleteness proof | p. 150 |
| 7 Computability, unsolvability, undecidability | |
| 7.1 Algorithms and computability | p. 156 |
| 7.2 Turing machines | p. 164 |
| 7.3 Word problems | p. 183 |
| 7.4 Undecidability of formal systems | p. 189 |
| Appendix Countable and uncountable sets | p. 199 |
| Hints and solutions to selected exercises | p. 203 |
| References and further reading | p. 219 |
| Glossary of symbols | p. 220 |
| Index | p. 224 |
