id: 06210195
dt: b
an: 2014e.00633
au: Stillwell, John
ti: The real numbers. An introduction to set theory and analysis.
so: Undergraduate Texts in Mathematics. Cham: Springer (ISBN
978-3-319-01576-7/hbk; 978-3-319-01577-4/ebook). xvi, 244~p. (2013).
py: 2013
pu: Cham: Springer
la: EN
cc: I15 E65
ut: real number; set theory; axiom of choice; ordinals; continuity;
measurability
ci:
li: doi:10.1007/978-3-319-01577-4
ab: Despite its subtitle this book will be fully appreciated by either
professional mathematicians or those students, who already have passed
a course in analysis or set theory. It is not the case that the text
cannot be used as the first reading. It can, but the added value here
provides a wider overview above the real numbers theory and links to
other subjects ‒ mostly mathematical, but also historical. The
introductory chapter is an excellent example of the art how to make the
text attractive ‒ its subsections are e.g. Why does $ab=ba$? What are
numbers? What is a line? What is geometry? What are functions? What is
continuity? What is a measure? These questions are in different aspects
discussed later in the book and summarized again in the last chapter.
The study of the reals and their properties is done in close connection
with the study of related concepts from analysis, as it is obvious from
the chapter titles: From discrete to continuous, Infinite sets,
Functions and limits, Open sets and continuity, Ordinals, The axiom of
choice, Borel sets, Measure theory. Therefore calculus is an absolutely
necessary prerequisite for the reader. The book contains a quantity of
motivation examples, worked-out examples and exercises, what makes it
suitable also for self-study.
rv: Vladimír Janiš (Banská Bystrica)