Item description for The Number Systems of Analysis by C. H. C. Little, K. L. Teo & B. Van Brunt...
Although students of analysis are familiar with real and complex numbers, few treatments of analysis deal with the development of such numbers in any depth. An understanding of number systems at a fundamental level is necessary for a deeper grasp of analysis. Beginning with elementary concepts from logic and set theory, this book develops in turn the natural numbers, the integers and the rational, real and complex numbers. The development is motivated by the need to solve polynomial equations, and the book concludes by proving that such equations have solutions in the complex number system.
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Studio: World Scientific Publishing Company
Est. Packaging Dimensions: Length: 0.75" Width: 6.25" Height: 9.25" Weight: 1.2 lbs.
Publisher World Scientific Publishing Company
ISBN 9812386068 ISBN13 9789812386069
Availability 0 units.
More About C. H. C. Little, K. L. Teo & B. Van Brunt
Reviews - What do customers think about The Number Systems of Analysis?
Where do numbers come from? Jul 31, 2004
That question is absolutely fundamental to mathematics and philosophy, and this book deserves to become the classic answer for our time. Everyone specializing in analysis, fields, foundations, or the philosophy of mathematics should learn this material.
A little over 100 years ago, Dedekind, Frege, Peano, and Russell-Whitehead tackled this fundamental intellectual question. The culimination of these endeavours was Principia Mathematica, the mathematical equivalent of Battlestar Galactica. What came out of this vast enterprise was numbers as equivalence classes, given some flavour of set theory.
In 1930, the Gottingen mathematician Edmund Landau published a little book, whose preface read "my daughters are chemistry majors and haven't a clue as to why ab=ba, or why that might need proving. Hence this book." Landau began with the Peano axioms. Landau's book is an aging classic, badly typeset; the same holds for the 1951 English translation.
Suppes's 1960 book on ZF set theory includes a fine derivation of the integers, rationals as equivalence classes, and the reals as Dedekind cuts. But the development of the natural numbers required the axiom of choice. This is the abstract equivalent of equipping the police with nuclear weapons. Something is amiss here. Quine's Set Theory and Its Logic tells this story from a much leaner set of axioms, one not including Choice. But his prose and Principia notation make for hard going, and his approach has no following.
In 2003, along come Little, Teo, and Van Brunt, who do a superb job of deriving the numbers from natural to complex, build some nice bridges to elementary analysis, and prove the fundamental theorem of algebra. The reals are developed as Cauchy sequences rather than Dedekind cuts. The presentation is elegant and well thought out. My only objection is that the authors are coy about having grounded their story in ZF; the axioms are there, but are not called axioms and are buried in narrative prose. Nevertheless, it can be deciphered that the minimalist ontological grounding for this marvelous exercise is as follows: the null set exists and seeds an inductive set. For better or worse, the names Dedekind, Peano, and von Neumann do not appear in the index.