Math 2.0

Mathematical logic and foundations Set theory: axioms and basics Theorem

Well-ordering theorem

Builds on Well-orderAxiom of choice

Statement

Assume the axiom of choice. Then each set AA has a well-order.

Remarks

Another name is Zermelo's theorem. Over ZF, it is equivalent to the axiom of choice and to Zorn's lemma. A well-order of ℝ\mathbb{R} is hard or impossible to visualize, because its construction needs the axiom of choice. With a well-order, each set allows proofs by transfinite induction.

History

Ernst Zermelo proved the theorem in 1904 with the axiom of choice, which he formulated for this purpose.

Reading path

To understand this page from the ground up, read these 28 pages in order. The foundations come first.

Show the path as a graph
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 choicestatementLeast element, definition, no evidence levelLeast elementdefinitionTotal order, definition, no evidence levelTotal orderdefinitionWell-order, definition, no evidence levelWell-orderdefinitionWell-ordering theorem, statement, no evidence levelWell-ordering theoremstatement
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. Least element definition
  26. Total order definition
  27. Well-order definition
  28. Well-ordering theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Well-ordering_theorem, lead section
About this page
Tag0N109NG5ZX
URIm2://0N109NG5ZX
Aliaseszermelo-theorem, well-ordering-principle
Kindstatement, theorem
WritingW1 stated
MSC03E25
StatusThis copy of the site has no build data. Run m2 status 0N109NG5ZX.
UsesWell-order, Axiom of choice
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.