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Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics. In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second-order arithmetic.

Predicate calculus. --- Calculus, Predicate --- Quantification theory --- Logic, Symbolic and mathematical

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Mathematical logic --- Predicate calculus --- Unsolvability (Mathematical logic) --- 510.6 --- #WWIS:ALTO --- Degrees of unsolvability --- Turing degrees of unsolvability --- Recursive functions --- Calculus, Predicate --- Quantification theory --- Logic, Symbolic and mathematical --- Predicate calculus. --- Unsolvability (Mathematical logic). --- 510.6 Mathematical logic

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At the heart of the justification for the reasoning used in modern mathematics lies the completeness theorem for predicate calculus. This unique textbook covers two entirely different ways of looking at such reasoning. Topics include: - the representation of mathematical statements by formulas in a formal language; - the interpretation of formulas as true or false in a mathematical structure; - logical consequence of one formula from others; - formal proof; - the soundness and completeness theorems connecting logical consequence and formal proof; - the axiomatization of some mathematical theories using a formal language; - the compactness theorem and an introduction to model theory. This book is designed for self-study by students, as well as for taught courses, using principles successfully developed by the Open University and used across the world. It includes exercises embedded within the text with full solutions to many of these. In addition there are a number of exercises without answers so that students studying under the guidance of a tutor may be assessed on the basis of what has been taught. Some experience of axiom-based mathematics is required but no previous experience of logic. Propositional and Predicate Calculus gives students the basis for further study of mathematical logic and the use of formal languages in other subjects. Derek Goldrei is Senior Lecturer and Staff Tutor at the Open University and part-time Lecturer in Mathematics at Mansfield College, Oxford, UK.

Propositional calculus --- Predicate calculus --- Calculus, Predicate --- Quantification theory --- Logic, Symbolic and mathematical --- Calculus, Propositional --- Logic, Symbolic and mathematical. --- Mathematical Logic and Foundations. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Mathematical logic. --- Propositional calculus - Problems, exercises, etc. --- Predicate calculus - Problems, exercises, etc.

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Mathematical logic --- 510.6 --- Logic, Symbolic and mathematical --- Predicate calculus --- #WWIS:ALTO --- Calculus, Predicate --- Quantification theory --- Algebra of logic --- Logic, Universal --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Logic, Symbolic and mathematical. --- Predicate calculus. --- 510.6 Mathematical logic

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Predicate calculus --- Tense (Logic) --- Calcul des prédicats --- Temps (Logique) --- Logic, Tense --- Calculus, Predicate --- Quantification theory --- Predicate calculus. --- Tense (Logic). --- Calcul des prédicats --- Grammar, Comparative and general --- Languages --- -Pragmatics --- Semantics --- 801.56 --- Formal semantics --- Semasiology --- Semiology (Semantics) --- Pragmalinguistics --- Comparative grammar --- Grammar --- Grammar, Philosophical --- Grammar, Universal --- Language and languages --- Philosophical grammar --- Foreign languages --- 801.56 Syntaxis. Semantiek --- Syntaxis. Semantiek --- Philosophy --- Grammar, Comparative --- Logic --- Time --- Comparative linguistics --- Information theory --- Lexicology --- Meaning (Psychology) --- Logic, Symbolic and mathematical --- General semantics --- Semantics (Philosophy) --- Linguistics --- Philology --- Anthropology --- Communication --- Ethnology --- Tense --- Lexicology. Semantics --- Pragmatics --- Grammar, Comparative and general. --- Pragmatics. --- Langage et langues --- Sémantique --- Pragmatique --- Grammaire comparée et générale --- Philosophie