Axiom of Choice

  • Horst Herrlich

Part of the Lecture Notes in Mathematics book series (LNM, volume 1876)

Table of contents

  1. Front Matter
    Pages I-XIV
  2. Horst Herrlich
    Pages 1-8
  3. Horst Herrlich
    Pages 9-20
  4. Horst Herrlich
    Pages 21-42
  5. Horst Herrlich
    Pages 43-116
  6. Horst Herrlich
    Pages 117-136
  7. Horst Herrlich
    Pages 137-141
  8. Horst Herrlich
    Pages 143-157
  9. Back Matter
    Pages 159-194

About this book


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.


Arithmetic Axiom of choice calculus coloring of graphs compactness game theory non-determinate games sets ultrafilters

Authors and affiliations

  • Horst Herrlich
    • 1
  1. 1.Department of MathematicsUniversity of BremenBremenGermany

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