Overview

AC, the axiom of choice, because of its non-constructive character, is the most controversial mathematical axiom, shunned by some, used indiscriminately by others. This treatise shows paradigmatically that: - Disasters happen without AC: Many fundamental mathematical results fail (being equivalent in ZF to AC or to some weak form of AC). - Disasters happen with AC: Many undesirable mathematical monsters are being created (e.g., non measurable sets and undeterminate games). - Some beautiful mathematical theorems hold only if AC is replaced by some alternative axiom, contradicting AC (e.g., by AD, the axiom of determinateness). Illuminating examples are drawn from diverse areas of mathematics, particularly from general topology, but also from algebra, order theory, elementary analysis, measure theory, game theory, and graph theory.

Full Product Details

Author:   Horst Herrlich
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2006 ed.
Volume:   1876
Dimensions:   Width: 15.50cm , Height: 1.10cm , Length: 23.30cm
Weight:   0.690kg
ISBN:  

9783540309895

ISBN 10:   3540309896
Pages:   198
Publication Date:   11 May 2006
Audience:   College/higher education ,  Undergraduate
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

Table of Contents

Origins: Hilbert's First Problem.- Choice Principles: Some Equivalents to the Axiom of Choice, Some Concepts Related to the Axiom of Choice.- Elementary Observations: Hidden Choice, Unnecessary Choice, Concepts Split Up: Compactness.- Disasters without Choice: Finiteness, Disasters in Cardinal Arithmetic, Disasters in Order Theory, Disasters in Algebra I: Vector Spaces, Disasters in Algebra II: Categories, Disasters in Elementary Analysis: The Reals and Continuity, Disasters in Topology I: Countable Sums, Disasters in Topology II: Products (The Tychonoff and the Cech-Stone Theorem), Disasters in Topology III: Function Spaces (The Ascoli Theorem), Disasters in Topology IV: The Baire Category Theorem, Disasters in Graph Theory: Coloring Problems.- Disasters with Choice: Disasters in Elementary Analysis, Disasters in Geometry: Paradoxical Decompositions.- Disasters either way: Disasters in Game Theory.- Beauty without Choice: Lindelöf = Compact, Measurability (The Axiom of Determinateness).

Reviews

From the reviews: <p> The book is an excellent introduction to the Axiom of Choice, its consequences and even its possible replacements. Although the author claims not to have written a textbook, compendium or history, the book might be used as any of these three. a ] there are many diagrams throughout the text and in the appendix detailing the relationships between the Axiom of Choice and its consequences. It certainly fulfills its stated purpose of whetting the readera (TM)s appetite for studying the ZF universe. (Paul E. Howard, Mathematical Reviews, Issue 2007 f)

Author Information

Tab Content 6

Author Website:  

Customer Reviews

Recent Reviews

No review item found!

Add your own review!

Countries Available

All regions