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Books > Science & Mathematics > Mathematics > Mathematical foundations
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The Real Numbers - An Introduction to Set Theory and Analysis (Hardcover, 2013 ed.)
Loot Price: R1,433
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The Real Numbers - An Introduction to Set Theory and Analysis (Hardcover, 2013 ed.)
Series: Undergraduate Texts in Mathematics
Expected to ship within 9 - 17 working days
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While 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-Schroeder-Bernstein
theorem, continuous functions, uniform convergence, Zorn's lemma,
Borel sets, Baire functions, Lebesgue measure, and Riemann
integrable functions.
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