"Attractive and well-written introduction." — Journal of Symbolic Logic
The logic that mathematicians use to prove their theorems is itself a part of mathematics, in the same way that algebra, analysis, and geometry are parts of mathematics. This attractive and well-written introduction to mathematical logic is aimed primarily at undergraduates with some background in college-level mathematics; however, little or no acquaintance with abstract mathematics is needed.
Divided into three chapters, the book begins with a brief encounter of naïve set theory and logic for the beginner, and proceeds to set forth in elementary and intuitive form the themes developed formally and in detail later. In Chapter Two, the predicate calculus is developed as a formal axiomatic theory. The statement calculus, presented as a part of the predicate calculus, is treated in detail from the axiom schemes through the deduction theorem to the completeness theorem. Then the full predicate calculus is taken up again, and a smooth-running technique for proving theorem schemes is developed and exploited.
Chapter Three is devoted to first-order theories, i.e., mathematical theories for which the predicate calculus serves as a base. Axioms and short developments are given for number theory and a few algebraic theories. Then the metamathematical notions of consistency, completeness, independence, categoricity, and decidability are discussed, and the predicate calculus is proved to be complete. The book concludes with an outline of Godel's incompleteness theorem.
Ideal for a one-semester course, this concise text offers more detail and mathematically relevant examples than those available in elementary books on logic. Carefully chosen exercises, with selected answers, help students test their grasp of the material. For any student of mathematics, logic, or the interrelationship of the two, this book represents a thought-provoking introduction to the logical underpinnings of mathematical theory.
"An excellent text." — Mathematical Reviews
Table of Contents1. Introduction
1. Rules of Inference
2. Set Theory
3. Axiomatic Theories
4. Predicates and Quantifiers
5. Statement Connectives
6. The Interpretation of Predicates and Quantifiers
7. The Predicate Calculus and First Order Theories
8. The Omission of Parentheses
9. Substitution of a Term for a Variable
10. Removing and Inserting Quantifiers
2. The Predicate Calculus
13. The Statement Calculus
14. The Deudction Theorem
15. The Completeness Theorem for the Statement Calculus
16. Applications of the Completeness Theorem for the Statement Calculus
18. Equivalence and Replacement
19. Theorem Schemes
20. Normal Forms
3. First Order Theories
22. Definition and Examples
24. Number Theory
25. Consistency and Completeness
27. The Completeness Theorem
29. Completeness and Categoricity
31. Gödel's Theorem
Notes; References; Addendum; Index of Symbols; Subject Index