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 formula_1 (the cardinality of the set of natural numbers), and the cardinality of the continuum, that is, the cardinality of the set formula_2 of all real numbers. The latter cardinal is denoted formula_3 or formula_4. 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 formula_4 is strictly greater than formula_1, but it does not specify whether it is the "least" cardinal greater than formula_1 (that is, formula_8). Indeed the assumption that formula_9 is the well-known Continuum Hypothesis, which was shown to be independent of the standard ZFC axioms for set theory by Paul Cohen. If the Continuum Hypothesis fails and so formula_4 is at least formula_11, natural questions arise about the cardinals strictly between formula_1 and formula_4, 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 formula_4. Generally one only considers definitions for cardinals that are provably greater than formula_1 and at most formula_4 as cardinal characteristics of the continuum, so if the Continuum Hypothesis holds they are all equal to formula_8. Examples. As is standard in set theory, we denote by formula_18 the least infinite ordinal, which has cardinality formula_1; 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(formula_20) 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 formula_21 and dominating number formula_22. We denote by formula_23 the set of functions from formula_18 to formula_18. For any two functions formula_26 and formula_27 we denote by formula_28 the statement that for all but finitely many formula_29. The "bounding number" formula_21 is the least cardinality of an unbounded set in this relation, that is, formula_31 The "dominating number" formula_22 is  the least cardinality of a set of functions from formula_18 to formula_18 such that every such function is dominated by (that is, formula_35) a member of that set, that is, formula_36 Clearly any such dominating set formula_37 is unbounded, so formula_21 is at most formula_22, and a diagonalisation argument shows that formula_40. Of course if formula_9 this implies that formula_42, but Hechler has shown that it is also consistent to have formula_21 strictly less than formula_22. Splitting number formula_45 and reaping number formula_46. We denote by formula_47 the set of all infinite subsets of formula_18. For any formula_49, we say  that formula_50 "splits" formula_51 if both formula_52 and formula_53 are infinite. The "splitting number" formula_45 is the least cardinality of a subset formula_55 of formula_47 such that for all formula_57, there is some formula_58 such that formula_50 splits formula_51. That is, formula_61 The "reaping number" formula_46 is the least cardinality of a subset formula_63 of formula_47 such that no element formula_50 of formula_47 splits every element of formula_63. That is, formula_68 Ultrafilter number formula_69. The ultrafilter number formula_69 is defined to be the least cardinality of a filter base of a non-principal ultrafilter on formula_18. Kunen gave a model  of set theory in which formula_72 but formula_73, and using a countable support iteration of Sacks forcings, Baumgartner and Laver constructed a model in which formula_72 and formula_75. Almost disjointness number formula_76. Two subsets formula_77 and formula_78 of formula_18 are said to be "almost disjoint" if formula_80 is finite, and a family of subsets of formula_18 is said to be almost disjoint if its members are pairwise almost disjoint. A "maximal almost disjoint" (mad) family of subsets of formula_18 is thus an almost disjoint family formula_83 such that for every subset formula_84 of formula_18 not in formula_83, there is a  set formula_87 such that formula_77 and formula_84 are not almost disjoint (that is, their intersection is infinite). The almost disjointness number formula_76 is the least cardinality of an infinite maximal almost disjoint family. A basic result is that formula_91; Shelah showed that it is consistent to have the strict inequality formula_92. 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.