Set Theory and the Continuum Problem (Oxford Logic Guides)

This book 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. 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. 1. General Background -- 2. Some Basics Of Class-set Theory -- 3. The Natural Numbers -- 4. Superinduction, Well Ordering And Choice -- 5. Ordinal Numbers -- 6. Order Isomorphism And Transfinite Recursion -- 7. Rank -- 8. Foundation, [epsilon]-induction, And Rank -- 9. Cardinals -- 10. Mostowski-shepherdson Mappings -- 11. Reflection Principles -- 12. Constructible Sets -- 13. L Is A Well Founded First-order Universe -- 14. Constructibility Is Absolute Over L -- 15. Constructibility And The Continuum Hypothesis -- 16. Forcing, The Very Idea -- 17. The Construction Of S4 Models For Zf -- 18. The Axiom Of Constructibility Is Independent -- 19. Independence Of The Continuum Hypothesis -- 20. Independence Of The Axiom Of Choice -- 21. Constructing Classical Models -- 22. Forcing Background. Raymond M. Smullyan And Melvin Fitting. Includes Bibliographical References And Indexes. 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. This text forms an 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.