Cardinal characteristic of the continuum

Examples

As is standard in set theory, we denote by ${\displaystyle \omega }$ the least infinite ordinal, which has cardinality ${\displaystyle {\mathrm{\aleph }}_0}$; it may be identified with the set of natural numbers.

A number of cardinal characteristics naturally arise as cardinal invariants for ideals that are closely connected with the structure of the reals, such as the ideal of Lebesgue null sets and the ideal of meagre sets.

non(N)

The cardinal characteristic ${\displaystyle \text{non}(\mathcal{N})}$ is the least cardinality of a non-measurable set; equivalently, it is the least cardinality of a set that is not a Lebesgue null set.

Bounding number and dominating number

We denote by ${\displaystyle {\omega }^{\omega }}$ the set of functions from ${\displaystyle \omega }$ to ${\displaystyle \omega }$. For any two functions ${\displaystyle f:\omega \to \omega }$ and ${\displaystyle g:\omega \to \omega }$ we denote by ${\displaystyle f{\le }^{\ast }g}$ the statement that for all but finitely many ${\displaystyle n\in \omega ,f(n)\le g(n)}$. The bounding number ${\displaystyle \mathfrak{b}}$ is the least cardinality of an unbounded set in this relation, that is, ${\displaystyle \mathfrak{b}=min(\{|F|:F\subseteq {\omega }^{\omega }\wedge \mathrm{\forall }g:\omega \to \omega \mathrm{\exists }f\in F(f{\nleqq }^{\ast }g)\}).}$

The dominating number ${\displaystyle \mathfrak{d}}$ is the least cardinality of a set of functions from ${\displaystyle \omega }$ to ${\displaystyle \omega }$ such that every such function is dominated by (that is, ${\displaystyle {\le }^{\ast }}$) a member of that set, that is, ${\displaystyle \mathfrak{d}=min(\{|F|:F\subseteq {\omega }^{\omega }\wedge \mathrm{\forall }g:\omega \to \omega \mathrm{\exists }f\in F(g{\le }^{\ast }f)\}).}$

Clearly any such dominating set ${\displaystyle F}$ is unbounded, so ${\displaystyle \mathfrak{b}}$ is at most ${\displaystyle \mathfrak{d}}$, and a diagonalisation argument shows that ${\displaystyle \mathfrak{b}>{\mathrm{\aleph }}_0}$. Of course if ${\displaystyle \mathfrak{c}={\mathrm{\aleph }}_1}$ this implies that ${\displaystyle \mathfrak{b}=\mathfrak{d}={\mathrm{\aleph }}_1}$, but Hechler has shown that it is also consistent to have ${\displaystyle \mathfrak{b}}$ strictly less than ${\displaystyle \mathfrak{d}.}$

Splitting number and reaping number

We denote by ${\displaystyle [\omega {]}^{\omega }}$ the set of all infinite subsets of ${\displaystyle \omega }$. For any ${\displaystyle a,b\in [\omega {]}^{\omega }}$, we say that ${\displaystyle a}$ splits ${\displaystyle b}$ if both ${\displaystyle b\cap a}$ and ${\displaystyle b\setminus a}$ are infinite. The splitting number ${\displaystyle \mathfrak{s}}$ is the least cardinality of a subset ${\displaystyle S}$ of ${\displaystyle [\omega {]}^{\omega }}$ such that for all ${\displaystyle b\in [\omega {]}^{\omega }}$, there is some ${\displaystyle a\in S}$ such that ${\displaystyle a}$ splits ${\displaystyle b}$. That is, ${\displaystyle \mathfrak{s}=min(\{|S|:S\subseteq [\omega {]}^{\omega }\wedge \mathrm{\forall }b\in [\omega {]}^{\omega }\mathrm{\exists }a\in S(|b\cap a|={\mathrm{\aleph }}_0\wedge |b\setminus a|={\mathrm{\aleph }}_0)\}).}$

The reaping number ${\displaystyle \mathfrak{r}}$ is the least cardinality of a subset ${\displaystyle R}$ of ${\displaystyle [\omega {]}^{\omega }}$ such that no element ${\displaystyle a}$ of ${\displaystyle [\omega {]}^{\omega }}$ splits every element of ${\displaystyle R}$. That is, ${\displaystyle \mathfrak{r}=min(\{|R|:R\subseteq [\omega {]}^{\omega }\wedge \mathrm{\forall }a\in [\omega {]}^{\omega }\mathrm{\exists }b\in R(|b\cap a|<{\mathrm{\aleph }}_0\vee |b\setminus a|<{\mathrm{\aleph }}_0)\}).}$

Ultrafilter number

The ultrafilter number ${\displaystyle \mathfrak{u}}$ is defined to be the least cardinality of a filter base of a non-principal ultrafilter on ${\displaystyle \omega }$. Kunen gave a model of set theory in which ${\displaystyle \mathfrak{u}={\mathrm{\aleph }}_1}$ but ${\displaystyle \mathfrak{c}={\mathrm{\aleph }}_{{\omega }_1},}$ and using a countable support iteration of Sacks forcings, Baumgartner and Laver constructed a model in which ${\displaystyle \mathfrak{u}={\mathrm{\aleph }}_1}$ and ${\displaystyle \mathfrak{c}={\mathrm{\aleph }}_2}$.

Almost disjointness number

Two subsets ${\displaystyle A}$ and ${\displaystyle B}$ of ${\displaystyle \omega }$ are said to be almost disjoint if ${\displaystyle |A\cap B|}$ is finite, and a family of subsets of ${\displaystyle \omega }$ is said to be almost disjoint if its members are pairwise almost disjoint. A maximal almost disjoint ("mad") family of subsets of ${\displaystyle \omega }$ is thus an almost disjoint family ${\displaystyle \mathcal{A}}$ such that for every subset ${\displaystyle X}$ of ${\displaystyle \omega }$ not in ${\displaystyle \mathcal{A}}$, there is a set ${\displaystyle A\in \mathcal{A}}$ such that ${\displaystyle A}$ and ${\displaystyle X}$ are not almost disjoint (that is, their intersection is infinite). The almost disjointness number ${\displaystyle \mathfrak{a}}$ is the least cardinality of an infinite maximal almost disjoint family. A basic result is that ${\displaystyle \mathfrak{b}\le \mathfrak{a}}$; Shelah showed that it is consistent to have the strict inequality ${\displaystyle \mathfrak{b}<\mathfrak{a}}$.

Cichoń's diagram

A well-known diagram of cardinal characteristics is Cichoń's diagram, showing all pairwise relations provable in ZFC between 10 cardinal characteristics.