This book is intended to provide an account of those parts of contemporary set theory which are of direct relevance to other areas of pure mathematics. The intended reader is either an advanced level undergraduate, or a beginning graduate student in mathematics, or else an accomplished mathematician who desires or needs a familiarity with modern set theory. The book is written in a fairly easy going style, with a minimum of formalism (a format characteristic of contemporary set theory) In Chapter I the basic principles of set theory are developed in a "naive" tl manner. Here the notions of "set I II union " , "intersection", "power set" I "relation" I "function" etc. are defined and discussed. One assumption in writing this chapter has been that whereas the reader may have met all of these concepts before, and be familiar with their usage, he may not have considered the various notions as forming part of the continuous development of a pure subject (namely set theory) Consequently, our development is at the same time rigorous and fast. Chapter II develops the theory of sets proper. Starting with the naive set theory of Chapter I, we begin by asking the question "What is a set?" Attempts to give a rLgorous answer lead naturally to the axioms of set theory introduced by Zermelo and Fraenkel, which is the system taken as basic in this book.
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Verlagsort
Verlagsgruppe
Dateigröße
ISBN-13
978-1-4684-0084-7 (9781468400847)
DOI
10.1007/978-1-4684-0084-7
Schweitzer Klassifikation
I. NAIVE SET THEORY.- 1. What is a set?.- 2. Operations on sets..- 3. Notation for sets..- 4. Sets of sets..- 5. Relations..- 6. Functions..- 7. Well-orderings and ordinals..- II. THE ZERMELO-FRAENKEL AXIOMS.- 1. The language of set theory..- 2. The cumulative hierarchy of sets..- 3. Zermelo-Fraenkel set theory..- 4. Axioms for set theory..- 5. Summary of the Zermelo-Fraenkel axioms..- 6. Classes..- 7. Set theory as an axiomatic theory..- 8. The recursion principle..- 9. The axiom of choice..- III. , ORDINAL AND CARDINAL NUMBERS.- 1. Ordinal numbers..- 2. Addition of ordinals..- 3. Multiplication of ordinals..- 4. Sequences of ordinals..- 5. Ordinal exponentiation..- 6. Cardinality. Cardinal numbers..- 7. Arithmetic of cardinal numbers..- 8. Cofinality. Singular and regular cardinals..- 9. Cardinal exponentiation..- 10. Inaccessible cardinals..- IV. SOME TOPICS IN PURE SET THEORY..- 1. The Borel hierarchy..- 2. Closed unbounded sets..- 3. Stationary sets and regressive functions..- 4. Trees..- 5. Extensions of Lebesgue measure..- 6. A result about the GCH..- V. THE AXIOM OF CONSTRUCTIBILITY..- 1. Constructible sets..- 2. The constructible hierarchy..- 3. The axiom of constructibility..- 4. The consistency of constructible set theory..- 5. Use of the axiom of constructibility..- VI. INDEPENDENCE PROOFS IN SET THEORY..- 1. Some examples of undecidable statements..- 2. The idea of a boolean-valued universe..- 3. The boolean-valued universe..- 4. VB and V..- 5. Boolean-valued sets and independence proofs..- 6. The non-provability of CH..- GLOSSARY OF NOTATION.
