Set Theory and the Continuum Problem
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Set Theory and the Continuum Problem

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A lucid, elegant, and complete survey of set theory, this volume is drawn from the authors' substantial teaching experience. The first of three parts focuses on axiomatic set theory. The second part explores the consistency of the continuum hypothesis, and the final section examines forcing and independence results.
Part One's focus on axiomatic set theory features nine chapters that examine problems related to size comparisons between infinite sets, basics of class theory, and natural numbers. Additional topics include author Raymond Smullyan's double induction principle, super induction, ordinal numbers, order isomorphism and transfinite recursion, and the axiom of foundation and cardinals. The six chapters of Part Two address Mostowski-Shepherdson mappings, reflection principles, constructible sets and constructibility, and the continuum hypothesis. The text concludes with a seven-chapter exploration of forcing and independence results. This treatment is noteworthy for its clear explanations of highly technical proofs and its discussions of countability, uncountability, and mathematical induction, which are simultaneously charming for experts and understandable to graduate students of mathematics.
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Product details

  • Paperback | 336 pages
  • 150 x 226 x 20mm | 386g
  • New York, United States
  • English
  • Revised
  • Revised
  • Illustrations, unspecified
  • 0486474844
  • 9780486474847
  • 93,395

Table of contents

Preface to the Revised 2010 EditionPrefaceI Axiomatic Set Theory1. General Background2. Some Basics of Class-Set Theory3. The Natural Numbers4. Superinduction, Well Ordering and Choice5. Ordinal Numbers6. Order Isomorphism and Transfinite Recursion7. Rank8. Foundation, Induction and Rank9. CardinalsII Consistency of the Continuum Hypothesis10. Mostowski-Shepherdson Mappings11. Reflection Principles12. Constructible Sets13. L is a Well-Founded First-Order Universe14. Constructibility is Absolute Over L15. Constructibility and the Continuum HypothesisIII Forcing and Independence Results16. Forcing, the Very Idea17. The Construction of S 4 Models for ZF18. The Axiom of Constructibility is Independent19. Independence in the Continuum Hypothesis20. Independence of the Axiom of Choice21. Constructing Classical Models22. Forcing BackwardBibliographyIndexList of Notation
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Review quote

"Smullyan and Fitting. . .achieve miraculous clarity in a subject crowded with intimidating espositions; in particular their book meets the very high standard of exposition set by Smullyan's previous works." --Choice
"This text is a general introduction to NBG (von Neumann-Bernays-Godel class-set theory), and to Godel and Cohen proofs of the relative consistency and the independence of the generalized continuum hypothesis (GCH) and the axiom of choice (AC). . . .The authors write with admirable lucidity. There
are some truly charming set pieces on countability and uncountability and on mathematical induction--I intend to appropriate them for my classes. . . .this is an excellent book for anyone interested in set theory and foundations."--Mathematical Reviews
"A well-written discussion of set theory, and readers will need a solid background in mathematics to fully appreciate its contents. The book is self-contained and intended for advanced undergraduates and graduate students in mathematics and computer science, especially those interested in set theory
and its relationship to logic." --Computing Reviews
"Intended as a text for advanced undergraduates and graduate students. Essentially self-contained."--The Bulletin of Mathematics Books
"The book under review is a textbook for a beginning graduate course on set theory. The structure is fairly standard, with the book divided into three main sections; after an introductory section developing the basic facts about the universe of set theory, there is a section on constructibility and
a section on forcing. The main goals of the book are to give proofs that the axiom of choice (AC) and the generalised continuumhypothesis (GCH) are consistent with and independent of the axioms of Zermelo-Fraenkel set theory (ZF). . . . The distinctive features of this book are the use of class set
theory, the treatment of induction, and the use of modal logic in the treatment of forcing. The writing is lucid and accurate, and the main theorems are proved in an efficient way."--Journal of Symbolic Logic
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About Raymond M. Smullyan

Raymond Smullyan received his PhD from Princeton University and taught at Dartmouth, Princeton, Indiana University, and New York's Lehman College. Best known for his mathematical and creative logic puzzles and games, he was also a concert pianist and a magician. He wrote over a dozen books of logic puzzles and texts on mathematical logic.Melvin Fitting, a former student of Dr. Smullyan, is Professor of Mathematics and Computer Science at Lehman College, City University of New York.
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Rating details

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