axiom of choice

Target entity: axiom of choice
Context triple: [von Neumann paradox in set theory, usesAxiom, axiom of choice]

• A. Zermelo–Fraenkel set theory
  Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
• B. axiom schema of separation
  The axiom schema of separation is a principle in set theory that guarantees the existence of subsets defined by properties or predicates, helping to avoid paradoxes by restricting unrestricted set formation.
• C. Zermelo set theory
  Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
• D. set theory
  Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
• E. Cantor’s theorem
  Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: axiom of choice
Target entity description: The axiom of choice is a fundamental principle in set theory asserting that one can select an element from each set in any collection of nonempty sets, with far-reaching consequences across mathematics.

• A. Zermelo–Fraenkel set theory
  Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
• B. axiom schema of separation
  The axiom schema of separation is a principle in set theory that guarantees the existence of subsets defined by properties or predicates, helping to avoid paradoxes by restricting unrestricted set formation.
• C. Zermelo set theory
  Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
• D. set theory
  Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
• E. Cantor’s theorem
  Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
• F. None of above. chosen