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
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.
Cantor's diagonal argument shows that
is the well-known Continuum Hypothesis, which was shown to be consistent with the standard ZFC axioms for set theory by Kurt Gödel and to be independent of it by Paul Cohen.
If the Continuum Hypothesis fails and so
, natural questions arise about the cardinals strictly between
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
as cardinal characteristics of the continuum, so if the Continuum Hypothesis holds they are all equal to
As is standard in set theory, we denote by
the least infinite ordinal, which has cardinality
; it may be identified with the set of 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.
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.
is the least cardinality of an unbounded set in this relation, that is,
is the least cardinality of a set of functions from
, and a diagonalisation argument shows that
, but Hechler[1] has shown that it is also consistent to have
the set of all infinite subsets of
is the least cardinality of a subset
is defined to be the least cardinality of a filter base of a non-principal ultrafilter on
Kunen[2] gave a model of set theory in which
and using a countable support iteration of Sacks forcings, Baumgartner and Laver[3] constructed a model in which
is finite, and a family of subsets of
A maximal almost disjoint ("mad") family of subsets of
are not almost disjoint (that is, their intersection is infinite).
is the least cardinality of an infinite maximal almost disjoint family.
; Shelah[5] showed that it is consistent to have the strict inequality
A well-known diagram of cardinal characteristics is Cichoń's diagram, showing all pairwise relations provable in ZFC between 10 cardinal characteristics.