Math 2.0

Mathematical logic and foundations Combinatorial set theory Problem

Suslin's problem

Builds on Total orderLinear continuumSeparable space

Problem

Let (L,<)(L, <) be a nonempty total order with these properties:

  1. LL has no least and no greatest element.
  2. LL is dense: between any two elements there is a third.
  3. LL is complete: each nonempty subset of LL that is bounded above has a least upper bound.
  4. Each family of pairwise disjoint nonempty open intervals of LL is countable.

Must LL be order isomorphic to the real line ℝ\R?

Remarks

The real line is characterised by clauses 1 to 3 together with separability: LL has a countable subset that meets each nonempty open interval. So the question asks whether the countable chain condition, clause 4, can replace separability. A counterexample is called a Suslin line, and Suslin's hypothesis says that no Suslin line exists. The problem is independent of ZFC, if ZFC is consistent. Jensen showed that the diamond principle, which holds in the constructible universe, implies that a Suslin line exists. Solovay and Tennenbaum showed that Suslin's hypothesis is consistent, and it follows from Martin's axiom together with the failure of the continuum hypothesis. A Suslin line exists if and only if a Suslin tree exists, that is, an Aronszajn tree in which each antichain is countable.

Reading path

To understand this page from the ground up, read these 45 pages in order. The foundations come first.

  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Supremum concept
  5. Total order concept
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Predicate definition
  11. Linear continuum definition
  12. Empty set object
  13. Family of sets definition
  14. Ordered pair definition
  15. Intersection definition
  16. Union definition
  17. Existential quantifier definition
  18. Universal quantifier definition
  19. Cartesian product definition
  20. Subset definition
  21. Intersection of a family of sets definition
  22. Union of a family of sets definition
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Topological space definition
  27. Antisymmetric relation definition
  28. Reflexive relation definition
  29. Transitive relation definition
  30. Complement definition
  31. Function definition
  32. Preorder definition
  33. Metric space definition
  34. Injective function definition
  35. Partial order definition
  36. Open ball definition
  37. Countable set definition
  38. Total order definition
  39. Metric topology definition
  40. Open set definition
  41. Closed set definition
  42. Closure definition
  43. Dense set definition
  44. Separable space definition
  45. Suslin's problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Suslin's_problem
About this page
TagD1T39KHPDN
URIm2://D1T39KHPDN
Kindproblem
WritingW0 stub
MSC03E50, 03E35, 03E05
StatusThis copy of the site has no build data. Run m2 status D1T39KHPDN.
AboutLinear continuum
UsesTotal order, Linear continuum, Separable space
Same as
Changed2 November 2026

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