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Axiom of Choice de Horst Herrlich

Axiom of Choice: Lecture Notes in Mathematics, cartea 1876

Autor Horst Herrlich
en Limba Engleză Paperback – 11 mai 2006
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.
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Specificații

ISBN-13: 9783540309895
ISBN-10: 3540309896
Pagini: 216
Ilustrații: XIV, 198 p. 1 illus.
Dimensiuni: 155 x 233 x 15 mm
Greutate: 0.31 kg
Ediția:2006
Editura: Springer Berlin, Heidelberg
Colecția Springer
Seria Lecture Notes in Mathematics

Locul publicării:Berlin, Heidelberg, Germany

Public țintă

Research

Cuprins

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).

Recenzii

From the reviews:
"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. … 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 reader’s appetite for studying the ZF universe." (Paul E. Howard, Mathematical Reviews, Issue 2007 f)

Textul de pe ultima copertă

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:
  1. Disasters happen without AC: Many fundamental mathematical results fail (being equivalent in ZF to AC or to some weak form of AC).
  2. Disasters happen with AC: Many undesirable mathematical monsters are being created (e.g., non measurable sets and undeterminate games).
  3. 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.