Math 2.0

Mathematical logic and foundations Axiom

Axiom of choice

Builds on Choice functionFunction

Statement

Let XX be a set of nonempty sets. Then there is a choice function on XX, that is, a function c ⁣:X→⋃Xc \colon X \to \bigcup X such that c(S)∈Sc(S) \in S for each S∈XS \in X.

Remarks

An equivalent form: for each family (Ai)i∈I(A_i)_{i \in I} of nonempty sets, the product ∏i∈IAi\prod_{i \in I} A_i is nonempty. Over ZF, it is equivalent to Zorn's lemma, to the well-ordering theorem and to the comparability of cardinals. It is independent of ZF, if ZF is consistent. Kurt Gödel showed in 1938 that ZF does not refute it, and Paul Cohen showed in 1963 that ZF does not prove it.

History

Ernst Zermelo formulated the axiom in 1904 for his proof of the well-ordering theorem.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionFunction, definition, no evidence levelFunctiondefinitionChoice function, definition, no evidence levelChoice functiondefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Set definition
  4. Disjunction definition
  5. Implication definition
  6. Predicate definition
  7. Family of sets definition
  8. Ordered pair definition
  9. Union definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Union of a family of sets definition
  15. Uniqueness quantifier definition
  16. Binary relation definition
  17. Function definition
  18. Choice function definition
  19. Axiom of choice statement

Where this leads

31 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Axiom_of_choice, lead section
About this page
Tag0HTGB13FE2
URIm2://0HTGB13FE2
Aliasesac, axiom-of-choice
Kindstatement, axiom
WritingW1 stated
MSC03E25
StatusThis copy of the site has no build data. Run m2 status 0HTGB13FE2.
UsesChoice function, Function
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