The real numbers an introduction to set theory and analysis John Stillwell

Format:

Book

Author/Creator:

Stillwell, John, author.

Series:

Undergraduate texts in mathematics 0172-6056

Undergraduate Texts in Mathematics 0172-6056

Language:

English

Subjects (All):

Mathematics.

Logic, Symbolic and mathematical.

applied mathematics.

mathematics.

Physical Description:

1 online resource

Place of Publication:

Cham Springer 2013

System Details:

text file

PDF

Summary:

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-Schroder-Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions

Contents:

The 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

Notes:

Includes bibliographical references and index

Online resource; title from PDF title page (SpringerLink, viewed October 21, 2013)

Other Format:

Printed edition:

ISBN:

9783319015774

331901577X

3319015761

9783319015767

OCLC:

862974177

Access Restriction:

Restricted for use by site license