Mathematical logic and foundations Definition
Cardinal characteristics of the continuum
Builds on Cardinal numberCardinality of the continuumMeagre setFunction
Definition
Let be the set of functions from to . For , write if for all but finitely many . Let be the family of meagre subsets of . A cardinal characteristic of the continuum is an uncountable cardinal 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:
- The bounding number is the least size of a set with no such that for all .
- The dominating number is the least size of a set such that each has for some .
- The additivity is the least size of a subfamily of whose union is not meagre.
- The covering number is the least size of a subfamily of whose union is .
- The uniformity is the least size of a subset of that is not meagre.
Remarks
Example: , where is the cardinality of the continuum. Check: for countably many , the function bounds them all, so . Each dominating family is unbounded, which gives , and itself dominates, so . Non-example: the least size of a family of meagre sets whose union is meagre is . 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 give three more cardinals. Together with , 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 . Martin's axiom implies that they all equal .
Reading path
To understand this page from the ground up, read these 56 pages in order. The foundations come first.
- Element definition
- Proposition concept
- Real numbers object
- Set definition
- Conjunction definition
- Disjunction definition
- Implication definition
- Predicate definition
- Empty set object
- Family of sets definition
- Ordered pair definition
- Intersection definition
- Union definition
- Existential quantifier definition
- Universal quantifier definition
- Cartesian product definition
- Subset definition
- Intersection of a family of sets definition
- Union of a family of sets definition
- Uniqueness quantifier definition
- Binary relation definition
- Universal set definition
- Power set definition
- Topological space definition
- Antisymmetric relation definition
- Reflexive relation definition
- Transitive relation definition
- Complement definition
- Function definition
- Preorder definition
- Metric space definition
- Codomain definition
- Injective function definition
- Partial order definition
- Open ball definition
- Countable set definition
- Surjective function definition
- Least element definition
- Total order definition
- Metric topology definition
- Bijective function definition
- Well-order definition
- Open set definition
- Countably infinite set definition
- Equinumerous sets definition
- Ordinal number definition
- Closed set definition
- Interior definition
- Cardinality definition
- Cardinal number definition
- Closure definition
- Aleph-null object
- Nowhere dense set definition
- Cardinality of the continuum object
- Meagre set definition
- Cardinal characteristics of the continuum definition
Further reading
About this page
| Tag | R8S00S3Z94 |
|---|---|
| URI | m2://R8S00S3Z94 |
| Kind | definition |
| Writing | W0 stub |
| MSC | 03E17 |
| Defines | Cardinal characteristics of the continuum |
| Uses | Cardinal number, Cardinality of the continuum, Meagre set, Function |
| Changed | 2 November 2026 |
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