The Mathematics of Logic
A Guide to Completeness Theorems and their Applications

This textbook rescues students from traditional, dry and uninspiring introductory courses in logic, quickly providing context using genuine mathematical applications.

Richard W. Kaye (Author)

9780521708777, Cambridge University Press

Paperback, published 12 July 2007

216 pages, 4 b/w illus. 141 exercises
22.5 x 15.3 x 1 cm, 0.304 kg

"Kaye (pure mathematics, U. of Birmingham) gives undergraduate and first-year graduates key materials for a first course in logic, including a full mathematical account of the Completeness Theorem for first-order logic. As he builds a series of systems increasing in complexity, and proving and discussing the Completeness Theorem for each, Kaye keeps unfamiliar terminology to a minimum and provides proofs of all the required set theoretical results. He covers K<:o>nig's Lemma (including two ways of looking at mathematics), posets and maximal elements (including order), formal systems (including post systems and compatibility as bonuses), deduction in posets (including proving statements about a poset), Boolean algebras, propositional logic (including a system for proof about propositions), valuations (including semantics for propositional logic), filters and ideals (including the algebraic theory of Boolean algebras), first-order logic, completeness and compactness, model theory (including countable models) and nonstandard analysis (including infinitesimal numbers)." --Book News

This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology is kept to a minimum, no background in formal set-theory is required, and the book contains proofs of all the required set theoretical results. The reader is taken on a journey starting with König's Lemma, and progressing via order relations, Zorn's Lemma, Boolean algebras, and propositional logic, to completeness and compactness of first-order logic. As applications of the work on first-order logic, two final chapters provide introductions to model theory and nonstandard analysis.

Preface
How to read this book
1. König's lemma
2. Posets and maximal elements
3. Formal systems
4. Deductions in posets
5. Boolean algebras
6. Propositional logic
7. Valuations
8. Filters and ideals
9. First-order logic
10. Completeness and compactness
11. Model theory
12. Nonstandard analysis
Bibliography
Index.

Subject Areas: Mathematical foundations [PBC]