Math 2.0

Mathematical logic and foundations Set theory: axioms and basics Theorem

Zorn's lemma

Builds on ChainUpper boundMaximal elementAxiom of choice

Statement

Assume the axiom of choice. Let (P,≤)(P, \le) be a nonempty partially ordered set in which each chain has an upper bound in PP. Then PP has a maximal element.

Remarks

Over ZF, Zorn's lemma is equivalent to the axiom of choice. If one requires an upper bound also for the empty chain, the hypothesis "PP is nonempty" is not needed. Applications: each vector space has a basis, and each nonzero ring with unity has a maximal ideal.

History

Kazimierz Kuratowski proved the lemma in 1922, and Max Zorn proved it independently in 1935. The Hausdorff maximal principle of 1914 is an earlier statement of the same kind.

Reading path

To understand this page from the ground up, read these 29 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 relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionChoice function, definition, no evidence levelChoice functiondefinitionPartial order, definition, no evidence levelPartial orderdefinitionAxiom of choice, statement, no evidence levelAxiom of choicestatementMaximal element, definition, no evidence levelMaximal elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionUpper bound, definition, no evidence levelUpper bounddefinitionChain, definition, no evidence levelChaindefinitionZorn's lemma, statement, no evidence levelZorn's lemmastatement
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. Antisymmetric relation definition
  18. Reflexive relation definition
  19. Transitive relation definition
  20. Function definition
  21. Preorder definition
  22. Choice function definition
  23. Partial order definition
  24. Axiom of choice statement
  25. Maximal element definition
  26. Total order definition
  27. Upper bound definition
  28. Chain definition
  29. Zorn's lemma statement

Where this leads

13 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Zorn's_lemma, Statement of the lemma
About this page
Tag04RXTQA5ZA
URIm2://04RXTQA5ZA
Aliaseszorn-lemma, kuratowski-zorn-lemma
Kindstatement, theorem
WritingW1 stated
MSC03E25
StatusThis copy of the site has no build data. Run m2 status 04RXTQA5ZA.
UsesChain, Upper bound, Maximal element, Axiom of choice
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