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THE AXIOM
OF CHOICE

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For an overview of the Axiom of Choice see the Relevance of the Axiom of Choice (FAQ Launcher).
  1. Introduction
    • Notation and Format
    • Philosophy
    • Overview
  2. Axiom of Choice Quiz
  3. Foundations
    • Formal systems, axiomatic theories, and metatheory
    • Euclidean geometry and the parallel postulate
    • ZFC
  4. Quiz Solutions and Weak Forms of AC
    • Explanation of Quiz Answers
    • Weak forms of AC
    • Some theorems whose proofs within ZFC require Countable Choice or Dependent Choice
  5. Common Equivalents of AC
    • Choice Principles, Ordering Principles, and Maximality Principles
  6. Less Well Known Equivalents of AC
    • Cardinality Theorems
    • Tychonoff's Theorem
    • Every vectory space has a basis
  7. Theorems Whose Proofs Within ZFC Require AC
    • Equivalents of the Ultrafilter Theorem
    • A discontinuous additive function
    • "Bad" sets of reals
  8. Conclusion
    • Axioms that imply AC
    • The axiom of determinacy: An axiom which contradicts AC
    • Category Theory: A new foundation of mathematics?
    • Accept AC?
  9. Appendix
    • ZFC Axioms
    • Notation Index
    • Glossary
    • Bibliography

 
 

 

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