The Mathematics of Logic: A Guide to Completeness Theorems and their Applications [Paperback]

Richard W. Kaye (Author)

Book Description

Publication Date: July 30, 2007 | ISBN-10: 052170877X | ISBN-13: 978-0521708777 | Edition: 1

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.

Editorial Reviews

Review

"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

Book Description

Undergraduate textbook covering the key material for a typical first course in logic, including a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. The author ensures that the number of new concepts at each stage is manageable, whilst providing lively mathematical applications throughout.

See all Editorial Reviews

Product Details

• Paperback: 216 pages
• Publisher: Cambridge University Press; 1 edition (July 30, 2007)
• Language: English
• ISBN-10: 052170877X
• ISBN-13: 978-0521708777
• Product Dimensions: 8.8 x 5.9 x 0.4 inches
• Shipping Weight: 11.4 ounces (View shipping rates and policies)
• Average Customer Review: 2.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #1,828,801 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

7 of 8 people found the following review helpful:

2.0 out of 5 stars "Show that any finite poset is a lattice.", May 18, 2009

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This review is from: The Mathematics of Logic: A Guide to Completeness Theorems and their Applications (Paperback)

First off, this book has a wider and more interesting variety of formal systems than the other logic books I have gone through. What's more, Kaye's formal systems are much easier and more intuitive. Sadly, the text is absolutely riddled with errors. Check the errata on his site and you will see two or three corrections, read the book yourself and you will find that that list is far from complete. Most of them are just minor annoyances, but some of them really cause a lot of confusion. One of the questions was so bad that even my professor could not find a way to salvage it. Also, Kaye seems at times somewhat confused about his target audience. He feels the need to define simple things such as countability, but expects the reader to have decent exposure to both algebra and topology (I know little about either). Chapter eight is pretty much incomprehensible if you're not good with algebra, fortunately that chapter is optional. I also found sections 10.4, 11.3 and 12.3 (all optional) too difficult for me to get through. Overall I would say the book has a lot of potential, but I can't recommend it to anyone in the condition it's in now, especially if you don't have a professor to help walk you through it. Does not cover incompleteness.