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Mathematical logic and foundations Definition

Cardinal characteristics of the continuum

Builds on Cardinal numberCardinality of the continuumMeagre setFunction

Definition

Let ωω\omega^\omega be the set of functions from ℕ\N to ℕ\N. For f,g∈ωωf, g \in \omega^\omega, write f≤∗gf \le^* g if f(n)≤g(n)f(n) \le g(n) for all but finitely many nn. Let ℳ\mathcal{M} be the family of meagre subsets of ℝ\R. A cardinal characteristic of the continuum is an uncountable cardinal κ≤2ℵ0\kappa \le 2^{\aleph_0} of the following kind. It is the least size of a family of reals, or of sets of reals, with a given property. The standard examples include:

  1. The bounding number 𝔟\mathfrak{b} is the least size of a set F⊆ωωF \subseteq \omega^\omega with no gg such that f≤∗gf \le^* g for all f∈Ff \in F.
  2. The dominating number 𝔡\mathfrak{d} is the least size of a set F⊆ωωF \subseteq \omega^\omega such that each g∈ωωg \in \omega^\omega has g≤∗fg \le^* f for some f∈Ff \in F.
  3. The additivity add⁡(ℳ)\operatorname{add}(\mathcal{M}) is the least size of a subfamily of ℳ\mathcal{M} whose union is not meagre.
  4. The covering number cov⁡(ℳ)\operatorname{cov}(\mathcal{M}) is the least size of a subfamily of ℳ\mathcal{M} whose union is ℝ\R.
  5. The uniformity non⁡(ℳ)\operatorname{non}(\mathcal{M}) is the least size of a subset of ℝ\R that is not meagre.

Remarks

Example: ℵ1≤𝔟≤𝔡≤𝔠\aleph_1 \le \mathfrak{b} \le \mathfrak{d} \le \mathfrak{c}, where 𝔠=2ℵ0\mathfrak{c} = 2^{\aleph_0} is the cardinality of the continuum. Check: for countably many fkf_k, the function g(n)=max⁡k≤nfk(n)g(n) = \max_{k \le n} f_k(n) bounds them all, so 𝔟>ℵ0\mathfrak{b} > \aleph_0. Each dominating family is unbounded, which gives 𝔟≤𝔡\mathfrak{b} \le \mathfrak{d}, and ωω\omega^\omega itself dominates, so 𝔡≤𝔠\mathfrak{d} \le \mathfrak{c}. Non-example: the least size of a family of meagre sets whose union is meagre is 11. It is defined in the same style, but it is not uncountable, so it is not a cardinal characteristic. The same three definitions with the Lebesgue null sets in place of ℳ\mathcal{M} give three more cardinals. Together with 𝔟\mathfrak{b}, 𝔡\mathfrak{d} and the cofinalities of the two ideals, they make up the ten cardinals of Cichoń's diagram. Under the continuum hypothesis all these cardinals equal ℵ1\aleph_1. Martin's axiom implies that they all equal 𝔠\mathfrak{c}.

Reading path

To understand this page from the ground up, read these 56 pages in order. The foundations come first.

  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Intersection of a family of sets definition
  19. Union of a family of sets definition
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Universal set definition
  23. Power set definition
  24. Topological space definition
  25. Antisymmetric relation definition
  26. Reflexive relation definition
  27. Transitive relation definition
  28. Complement definition
  29. Function definition
  30. Preorder definition
  31. Metric space definition
  32. Codomain definition
  33. Injective function definition
  34. Partial order definition
  35. Open ball definition
  36. Countable set definition
  37. Surjective function definition
  38. Least element definition
  39. Total order definition
  40. Metric topology definition
  41. Bijective function definition
  42. Well-order definition
  43. Open set definition
  44. Countably infinite set definition
  45. Equinumerous sets definition
  46. Ordinal number definition
  47. Closed set definition
  48. Interior definition
  49. Cardinality definition
  50. Cardinal number definition
  51. Closure definition
  52. Aleph-null object
  53. Nowhere dense set definition
  54. Cardinality of the continuum object
  55. Meagre set definition
  56. Cardinal characteristics of the continuum definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Cardinal_characteristic_of_the_continuum
About this page
TagR8S00S3Z94
URIm2://R8S00S3Z94
Kinddefinition
WritingW0 stub
MSC03E17
DefinesCardinal characteristics of the continuum
UsesCardinal number, Cardinality of the continuum, Meagre set, Function
Changed2 November 2026

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