any such dominating set is unbounded, so is at most , and a diagonalisation argument shows that . Of course if this implies that , but Hechler has shown that it is also consistent to have strictly less than . Splitting number and reaping number We denote by the set of all infinite subsets of . For any , we say that splits if both and are infinite. The splitting number is the least cardinality of a subset of such that for all , there is some such that splits . That is, The reaping number is the least cardinality of a subset of such that no element of splits every element of . That is, Ultrafilter number The ultrafilter number is defined to be the least cardinality of a filter base of a non-principal ultrafilter on . Kunen gave a model of set theory in which but , and using a countable support iteration of Sacks forcings, Baumgartner and Laver constructed a model in which and . Almost disjointness number Two subsets and of are said to be almost disjoint if is finite, and a family of subsets of is said to be almost disjoint if its members are pairwise almost disjoint. A maximal almost disjoint (mad) family of subsets of is thus an almost disjoint family such that for every subset of not in , there is a set such that and are not almost disjoint (that is, their intersection is infinite). The almost disjointness number is the least cardinality of an infinite maximal almost disjoint family. A basic result<ref>Eric van Douwen. The Integers and Topology. In K. Kunen and J.E. Vaughan (eds) Handbook of Set-Theoretic Topology. North-Holland, Amsterdam, 1984.</ref> is that ; Shelah showed that it is consistent to have the strict inequality . Cichoń's diagram A well known diagram of cardinal characteristics is Cichoń's diagram, showing all pair-wise relations provable in ZFC between 10 cardinal characteristics. References Further reading Tomek Bartoszyński and Haim Judah. Set Theory On the Structure of the Real Line''. A K Peters, 1995. Cardinal numbers