Set Theory and the Continuum Problem

Set Theory and the Continuum Problem is a novel introduction to set theory, including axiomatic development, consistency, and independence results. It is self-contained and covers all the set theory that a mathematician should know. Part I introduces set theory, including basic axioms, development of the natural number system, Zorn's Lemma and other maximal principles. Part II proves the consistency of the continuum hypothesis and the axiom of choice, with material on collapsing mappings, model-theoretic results, and constructible sets. Part III presents a version of Cohen's proofs of the independence of the continuum hypothesis and the axiom of choice. It also presents, for the first time in a textbook, the double induction and superinduction principles, and Cowen's theorem. The book will interest students and researchers in logic and set theory.

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.

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