Notes on Set Theory (Undergraduate Texts in Mathematics)
Publisher: Springer
Published: December 8, 2005
Language: eng
Number of pages: 284
ISBN-13: 9780387287232
Description: Whatthisbookisabout. Thetheoryofsetsisavibrant,excitingmathematical theory, with its own basic notions, fundamental results and deep open pr- lems,andwithsigni?cantapplicationstoothermathematicaltheories. Atthe sametime,axiomaticsettheoryisoftenviewedasafoundationofmathematics: it is allegedthat all mathematical objectsare sets, and theirpropertiescan be derived from the relatively few and elegant axioms about sets. Nothing so simple-minded can be quite true, but there is little doubt that in standard, current mathematical practice, “making a notion precise” is essentially s- onymouswith“de?ningitinsettheory”. Settheoryistheo?ciallanguageof mathematics,just asmathematicsisthe o?ciallanguageof science. Like most authors of elementary, introductory books about sets, I have triedtodojusticetobothaspectsofthesubject. From straight set theory, these Notes cover the basic facts about “abstract sets”, includingthe Axiom of Choice, trans?nite recursion, and cardinal and ordinal numbers. Somewhat less common is the inclusion of a chapter on “pointsets” which focuses on results of interest to analysts and introduces the reader to the Continuum Problem, central to set theory from the very beginning. There is also some novelty in the approach to cardinal numbers, whichare brought in very early (following Cantor, but somewhatdeviously), so that the basic formulas of cardinal arithmetic can be taught as quickly as possible. AppendixAgivesamoredetailed“construction”oftherealnumbers thaniscommonnowadays,whichinadditionclaimssomenoveltyofapproach and detail. Appendix B is a somewhat eccentric, mathematical introduction to the study of natural models of various set theoretic principles, including Aczel’s Antifoundation. It assumes no knowledge of logic, but should drive theseriousreaderto studyit. About set theory as a foundation of mathematics, there are two aspects of these Notes which are somewhat uncommon.