03701nam a22004935i 4500001001800000003000900018005001700027007001500044008004100059020003700100024003500137050001200172072001700184072002300201072002300224082001400247100003000261245011200291264007500403300004400478336002600522337002600548338003600574347002400610490005100634505023800685520178300923650001702706650003302723650001302756650002402769650001702793650002002810650004002830650003802870710003402908773002002942776003602962830005102998856004803049912001403097999001903111952007703130978-3-319-01577-4DE-He21320180115171602.0cr nn 008mamaa131016s2013 gw | s |||| 0|eng d a97833190157749978-3-319-01577-47 a10.1007/978-3-319-01577-42doi 4aQA331.5 7aPBKB2bicssc 7aMAT0340002bisacsh 7aMAT0370002bisacsh04a515.82231 aStillwell, John.eauthor.14aThe Real Numbersh[electronic resource] :bAn Introduction to Set Theory and Analysis /cby John Stillwell. 1aCham :bSpringer International Publishing :bImprint: Springer,c2013. aXVI, 244 p. 62 illus.bonline resource. atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda1 aUndergraduate Texts in Mathematics,x0172-60560 aThe Fundamental Questions -- From Discrete to Continuous -- Infinite Sets -- Functions and Limits -- Open Sets and Continuity -- Ordinals -- The Axiom of Choice -- Borel Sets -- Measure Theory -- Reflections -- Bibliography -- Index. aWhile most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions. 0aMathematics. 0aFunctions of real variables. 0aHistory. 0aMathematical logic.14aMathematics.24aReal Functions.24aMathematical Logic and Foundations.24aHistory of Mathematical Sciences.2 aSpringerLink (Online service)0 tSpringer eBooks08iPrinted edition:z9783319015767 0aUndergraduate Texts in Mathematics,x0172-605640uhttp://dx.doi.org/10.1007/978-3-319-01577-4 aZDB-2-SMA c371277d371277 001040708EBookaelibbelibd2018-01-15r2018-01-15w2018-01-15yEBOOK