Cardinal characteristic of the continuum
In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between  (the cardinality of the set of natural numbers), and the cardinality of the continuum, that is, the cardinality of the set  of all real numbers. The latter cardinal is denoted  or . A variety of such cardinal characteristics arise naturally, and much work has been done in determining what relations between them are provable, and constructing models of set theory for various consistent configurations of them. Background 

Cantor's diagonal argument shows that  is strictly greater than , but it does not specify whether it is the least cardinal greater than  (that is, ). Indeed the assumption that  is the well-known Continuum Hypothesis, which was shown to be consistent with the standard ZFC axioms for set theory by Kurt Godel and to be independent of it by Paul Cohen. If the Continuum Hypothesis fails and so  is at least , natural questions arise about the cardinals strictly between  and , for example regarding Lebesgue measurability. By considering the least cardinal with some property, one may get a definition for an uncountable cardinal that is consistently less than . Generally one only considers definitions for cardinals that  are provably greater than  and at most  as cardinal characteristics of the continuum, so if the Continuum Hypothesis holds they are all equal to . Examples 

As is standard in set theory, we denote by  the least infinite ordinal, which has cardinality ; it may be identified with the set of all natural numbers. A number of cardinal characteristics naturally arise as cardinal invariants for ideals which 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 non() 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  the set of functions from  to . For any two functions  and  we denote by  the statement that for all but finitely many . The bounding number  is the least cardinality of an unbounded set in this relation, that is, 

The dominating number  is the least cardinality of a set of functions from  to  such that every such function is dominated by (that is, ) a member of that set, that is,

Clearly 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