Cardinal characteristic of the continuum

The cardinal characteristic non(${\displaystyle {\mathcal {N}}}$) 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.

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 ${\displaystyle f\leq ^{*}g}$ the statement that for all but finitely many ${\displaystyle n\in \omega ,f(n)\leq g(n)}$. The bounding number ${\displaystyle {\mathfrak {b}}}$ is the least cardinality of an unbounded set in this relation, that is, ${\displaystyle {\mathfrak {b}}=\min(\{|F|:F\subseteq \omega ^{\omega }\land \forall f:\omega \to \omega \exists g\in F(g\nleq ^{*}f)\}).}$

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, ${\displaystyle \leq ^{*}}$) a member of that set, that is, ${\displaystyle {\mathfrak {d}}=\min(\{|F|:F\subseteq \omega ^{\omega }\land \forall f:\omega \to \omega \exists g\in F(f\leq ^{*}g)\}).}$

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 ${\displaystyle {\mathfrak {b}}>\aleph _{0}}$. Of course if ${\displaystyle {\mathfrak {c}}=\aleph _{1}}$ this implies that ${\displaystyle {\mathfrak {b}}={\mathfrak {d}}=\aleph _{1}}$, but Hechler[1] has shown that it is also consistent to have ${\displaystyle {\mathfrak {b}}}$ strictly less than ${\displaystyle {\mathfrak {d}}}$.

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, ${\displaystyle {\mathfrak {s}}=\min(\{|S|:S\subseteq [\omega ]^{\omega }\land \forall b\in [\omega ]^{\omega }\exists a\in S(|b\cap a|=\aleph _{0}\land |b\setminus a|=\aleph _{0})\}).}$

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, ${\displaystyle {\mathfrak {r}}=\min(\{|R|:R\subseteq [\omega ]^{\omega }\land \forall a\in [\omega ]^{\omega }\exists b\in R(|b\cap a|<\aleph _{0}\lor |b\setminus a|<\aleph _{0})\}).}$

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[2] gave a model of set theory in which ${\displaystyle {\mathfrak {u}}=\aleph _{1}}$ but ${\displaystyle {\mathfrak {c}}=\aleph _{\aleph _{1}}}$, and using a countable support iteration of Sacks forcings, Baumgartner and Laver[3] constructed a model in which ${\displaystyle {\mathfrak {u}}=\aleph _{1}}$ and ${\displaystyle {\mathfrak {c}}=\aleph _{2}}$.

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[4] is that ${\displaystyle {\mathfrak {b}}\leq {\mathfrak {a}}}$; Shelah[5] showed that it is consistent to have the strict inequality ${\displaystyle {\mathfrak {b}}<{\mathfrak {a}}}$.