Logic of Mathematics: A Modern Course of Classical LogicJohn Wiley & Sons, 26 sept. 2011 - 272 pages A thorough, accessible, and rigorous presentation of the central theorems of mathematical logic . . . ideal for advanced students of mathematics, computer science, and logic Logic of Mathematics combines a full-scale introductory course in mathematical logic and model theory with a range of specially selected, more advanced theorems. Using a strict mathematical approach, this is the only book available that contains complete and precise proofs of all of these important theorems: * Gödel's theorems of completeness and incompleteness * The independence of Goodstein's theorem from Peano arithmetic * Tarski's theorem on real closed fields * Matiyasevich's theorem on diophantine formulas Logic of Mathematics also features: * Full coverage of model theoretical topics such as definability, compactness, ultraproducts, realization, and omission of types * Clear, concise explanations of all key concepts, from Boolean algebras to Skolem-Löwenheim constructions and other topics * Carefully chosen exercises for each chapter, plus helpful solution hints At last, here is a refreshingly clear, concise, and mathematically rigorous presentation of the basic concepts of mathematical logic-requiring only a standard familiarity with abstract algebra. Employing a strict mathematical approach that emphasizes relational structures over logical language, this carefully organized text is divided into two parts, which explain the essentials of the subject in specific and straightforward terms. Part I contains a thorough introduction to mathematical logic and model theory-including a full discussion of terms, formulas, and other fundamentals, plus detailed coverage of relational structures and Boolean algebras, Gödel's completeness theorem, models of Peano arithmetic, and much more. Part II focuses on a number of advanced theorems that are central to the field, such as Gödel's first and second theorems of incompleteness, the independence proof of Goodstein's theorem from Peano arithmetic, Tarski's theorem on real closed fields, and others. No other text contains complete and precise proofs of all of these theorems. With a solid and comprehensive program of exercises and selected solution hints, Logic of Mathematics is ideal for classroom use-the perfect textbook for advanced students of mathematics, computer science, and logic. |
Table des matières
1 | |
A Modern Course of Classical Logic PART I Mathematical Structures and Their Theories | 7 |
A Modern Course of Classical Logic PART II Selected Topics | 145 |
A Modern Course of Classical Logic Guide to Further Reading | 252 |
Autres éditions - Tout afficher
Expressions et termes fréquents
arithmetical formula Assume assumption atomic formulas Boolean algebra bounded formula called class PR completes the proof condition Cons(T consistent set constants countable model deduction theorem definable defined defined by induction definition denote diagonal lemma diophantine distinguished elements elimination of quantifiers equivalent Example Exercise F E Fm filter find finite finite set finite subset first Fm(L formula F free variables function f function symbols G PR Goodstein Goodstein sequence Hence holds homomorphism ifand implies integer isomorphic language Let f Lindenbaum algebra mathematical Mod(T modus ponens o-minimal obtain open formula operations f parametrically definable Peano arithmetic polynomials prove real closed fields relational systems Remark satisfies sentence F set of formulas set of sentences set theory Show Similarly Skolem Skolem function substitution Tarski ultrafilter ultrapower ultraproduct Vf(F whence