Product Description
Logic forms the basis of mathematics and is a fundamental part of any mathematics course. This book provides students with a clear and accessible introduction to this important subject, using the concept of model as the main focus and covering a wide area of logic. The chapters of the book cover propositional calculus, boolean algebras, predicate calculus and completelness theorems with answeres to all of the exercises and the end of the volume. This is an ideal introduction to mathematics and logic for the advanced undergraduate student. ... Read more

Customer Reviews (1)

OK but Hard
You'll find this very hard unless you are a competent math major at one of the better universities. Similar to Elliot Mendelson's text, but not quite as good. Good chapter on Boolean algebra as a
piece of pure math; Halmos and Givant is gentler, though.

Interesting topic covered: the resolution so dear to the AI crowd. Unlike most mathematicians, Cori and Lascar have time for
the way computer scientists think. At the same time, this book does not cover tableau methods (see Smullyan), natural deduction, Genzen's ideas, and so on. For pure logic at the advanced undergrad level, you're better off with Bostock.

Haven't seen Part II, so cannot comment on the treatment of set theory. This is something Mendelson and Machover already do well. ... Read more

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Boolean algebras have historically played a special role in the development of the theory of general or "universal" algebraic systems, providing important links between algebra and analysis, set theory, mathematical logic, and computer science. It is not surprising then that focusing on specific properties of Boolean algebras has lead to new directions in universal algebra.In the first unified study of polynomial completeness, Polynomial Completeness in Algebraic Systems focuses on and systematically extends another specific property of Boolean algebras: the property of affine completeness. The authors present full proof that all affine complete varieties are congruence distributive and that they are finitely generated if and only if they can be presented using only a finite number of basic operations. In addition to these important findings, the authors describe the different relationships between the properties of lattices of equivalence relations and the systems of functions compatible with them. An introductory chapter surveys the appropriate background material, exercises in each chapter allow readers to test their understanding, and open problems offer new research possibilities. Thus Polynomial Completeness in Algebraic Systems constitutes an accessible, coherent presentation of this rich topic valuable to both researchers and graduate students in general algebraic systems. ... Read more

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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. ... Read more

Customer Reviews (1)

"Show that any finite poset is a lattice."
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. ... Read more

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Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Entscheidungsproblem, Gödel's Completeness Theorem, Compactness Theorem, Gödel's Incompleteness Theorems, Löwenheim-skolem Theorem, Deduction Theorem, Tarski's Undefinability Theorem, Herbrand's Theorem, Gentzen's Consistency Proof, Löb's Theorem, Metatheorem, Lindström's Theorem, Barwise Compactness Theorem. Excerpt:In mathematical logic , the Barwise compactness theorem , named after Jon Barwise , is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967. Statement of the theorem References (URLs online) Websites (URLs online) A hyperlinked version of this chapter is at In mathematical logic , the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory , as it provides a useful method for constructing models of any set of sentences that is finitely consistent . The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces ; hence, the theorem 's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along with the downward Löwenheim Skolem theorem , that is used in Lindström's theorem to characterize first-order logic. Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them. Applicat... ... Read more

Product Description
Logic forms the basis of mathematics and is a fundamental part of any mathematics course. This book provides students with a clear and accessible introduction to this important subject, using the concept of model as the main focus and covering a wide area of logic. The chapters of the book cover propositional calculus, boolean algebras, predicate calculus and completelness theorems with answeres to all of the exercises and the end of the volume. This is an ideal introduction to mathematics and logic for the advanced undergraduate student. ... Read more

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Recursive Functions and Metamathematics deals withproblems of the completeness and decidability of theories, using asits main tool the theory of recursive functions. This theory is firstintroduced and discussed. Then Gödel's incompleteness theoremsare presented, together with generalizations, strengthenings, and thedecidability theory. The book also considers the historical andphilosophical context of these issues and their philosophical andmethodological consequences. Recent results and trends have beenincluded, such as undecidable sentences of mathematical content,reverse mathematics. All the main results are presented in detail. Thebook is self-contained and presupposes only some knowledge ofelementary mathematical logic. There is an extensive bibliography.
Readership: Scholars and advanced students of logic,mathematics, philosophy of science. ... Read more

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In algorithmic information theory, the Kolmogorov complexity of an object such as a piece of text is a measure of the computational resources needed to specify the object. The first string admits a short English language description, namely "ab 32 times", which consists of 11 characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, which has 64 characters. More formally, the complexity of a string is the length of the string's shortest description in some fixed universal description language. The sensitivity of complexity relative to the choice of description language is discussed below. It can be shown that the Kolmogorov complexity of any string cannot be too much larger than the length of the string itself. Strings whose Kolmogorov complexity is small relative to the string's size are not considered to be complex. The notion of Kolmogorov complexity is surprisingly deep and can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem. ... Read more

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High Quality Content by WIKIPEDIA articles! The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal.For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. ... Read more