Math 2.0

Mathematical logic and foundations Theorem

Hausdorff maximal principle

Builds on ChainMaximal elementAxiom of choice

Statement

Assume the axiom of choice. Let (P,≤)(P, \le) be a partially ordered set. Then each chain of PP is a subset of a maximal chain. A maximal chain is a chain that is not a proper subset of another chain of PP.

Remarks

Over ZF, the principle is equivalent to the axiom of choice and to Zorn's lemma. With the empty chain, it says that each partially ordered set has a maximal chain. Felix Hausdorff proved it in 1914.

Reading path

To understand this page from the ground up, read these 28 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 orderdefinitionChain, definition, no evidence levelChaindefinitionHausdorff maximal principle, statement, no evidence levelHausdorff maximal pri…statement
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. Chain definition
  28. Hausdorff maximal principle statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Hausdorff_maximal_principle, Statement
About this page
Tag0KM4KVR9MV
URIm2://0KM4KVR9MV
Aliasesmaximal-chain-principle
Kindstatement, theorem
WritingW1 stated
MSC03E25
StatusThis copy of the site has no build data. Run m2 status 0KM4KVR9MV.
UsesChain, Maximal element, Axiom of choice
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