Cardinal characteristic of the continuum explained

In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between

\aleph _{0} (the cardinality of the set of natural numbers), and the cardinality of the continuum, that is, the cardinality of the set

R of all real numbers. The latter cardinal is denoted 2 ^{\aleph _{0}} or ak c . 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

ak c is strictly greater than \aleph _{0} , but it does not specify whether it is the least cardinal greater than \aleph _{0} (that is, \aleph _{1} ). Indeed the assumption that ak c = \aleph _{1} 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 ak c is at least \aleph _{2} , natural questions arise about the cardinals strictly between \aleph _{0} and ak c , 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 ak c . Generally one only considers definitions for cardinals that are provably greater than \aleph _{0} and at most ak c as cardinal characteristics of the continuum, so if the Continuum Hypothesis holds they are all equal to \aleph _{1} .

Examples

As is standard in set theory, we denote by

\omega the least infinite ordinal, which has cardinality \aleph _{0} ; 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.

non(N)

The cardinal characteristic

non ( l{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.

Bounding number and dominating number

We denote by

\omega ^{\omega} the set of functions from \omega to \omega . For any two functions f:\omega\to\omega and g:\omega\to\omega we denote by f \leq ^{*} g the statement that for all but finitely many n\in\omega, f(n) \leq g(n) . The bounding number ak b is the least cardinality of an unbounded set in this relation, that is, ak b = min (\{|F| : F\subseteq\omega ^{\omega} \land \forall f:\omega\to\omega \exists g \in F (g \nleq ^{*} f)\}).

The dominating number

ak d is the least cardinality of a set of functions from \omega to \omega such that every such function is dominated by (that is, \leq ^{*} ) a member of that set, that is,ak 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

F is unbounded, so ak b is at most ak d , and a diagonalisation argument shows that ak b>\aleph _{0} . Of course if ak c = \aleph _{1} this implies that ak b = ak d = \aleph _{1} , but Hechler^[1] has shown that it is also consistent to have ak b strictly less than ak d .

Splitting number and reaping number

We denote by

[\omega] ^{\omega} the set of all infinite subsets of \omega . For any a,b\in[\omega] ^{\omega} , we say that a splits b if both b \cap a and b \setminus a are infinite. The splitting number ak s is the least cardinality of a subset S of [\omega] ^{\omega} such that for all b\in[\omega] ^{\omega} , there is some a\in S such that a splits b . That is, ak 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

ak r is the least cardinality of a subset R of [\omega] ^{\omega} such that no element a of [\omega] ^{\omega} splits every element of R . That is, ak 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)\}).}

Ultrafilter number

The ultrafilter number

ak u is defined to be the least cardinality of a filter base of a non-principal ultrafilter on \omega . Kunen^[2] gave a model of set theoryin which ak u = \aleph _{1} but ak c = \aleph _{\aleph _{1}} , and using a countable support iteration of Sacks forcings, Baumgartner and Laver^[3] constructed a model in which ak u = \aleph _{1} and ak c = \aleph _{2} .

Almost disjointness number

Two subsets

A and B of \omega are said to be almost disjoint if |A\cap B| is finite, and a family of subsets of \omega is said to be almost disjoint if its members are pairwise almost disjoint. A maximal almost disjoint ("mad") family of subsets of \omega is thus an almost disjoint family l{A} such that for every subset X of \omega not in l{A} , there is a set A\in l{A} such that A and X are not almost disjoint(that is, their intersection is infinite). The almost disjointness number ak{a} is the least cardinality of an infinite maximal almost disjoint family.A basic result^[4] is thatak{b}\leq ak{a}

Shelah^[5] showed that it is consistent to have the strict inequality

ak{b}< ak{a} .

Cichoń's diagram

A well-known diagram of cardinal characteristics is Cichoń's diagram, showing all pairwise relations provable in ZFC between 10 cardinal characteristics.

Further reading

• Tomek Bartoszyński and Haim Judah. Set Theory On the Structure of the Real Line. A K Peters, 1995.
• Book: Vaughan, Jerry E. . Chapter 11: Small uncountable cardinals and topology . van Mill . Jan . Reed . George M. . Open Problems in Topology . 196–218 . 1990 . . Amsterdam . 0-444-88768-7 . https://archive.org/details/openproblemsinto0000unse/page/196 . PDF . December 5, 2011 .
• Book: Blass, Andreas . Andreas Blass

. Andreas Blass . Chapter 6 : Combinatorial Cardinal Characteristics of the Continuum . Foreman . Matthew . Matthew Foreman . Kanamori . Akihiro . Akihiro Kanamori . Handbook of Set Theory . 1 . 395–490 . January 12, 2010 . . 978-1-4020-4843-2 . December 5, 2011.

• Book: Bartoszyński, Tomek . Tomek Bartoszyński

. Tomek Bartoszyński . Chapter 7 : Invariants of Measure and Category . Foreman . Matthew . Kanamori . Akihiro . Handbook of Set Theory . 1 . 491–556 . January 12, 2010 . Springer . 978-1-4020-4843-2 . math.LO/9910015.

• Book: Jech . Thomas . Thomas Jech . Set Theory . Third Millennium . . Berlin, New York . Springer Monographs in Mathematics . 978-3-540-44085-7 . 2003 . 1007.03002 .
• Book: Halbeisen, Lorenz J. . Combinatorial Set Theory: With a Gentle Introduction to Forcing . Springer-Verlag. London . 2012 . Springer Monographs in Mathematics . Springer Monographs in Mathematics . 978-1-4471-2172-5 . 10.1007/978-1-4471-2173-2.

Notes and References

1. Stephen Hechler. On the existence of certain cofinal subsets of {} ^{\omega\omega} . In T. Jech (ed), Axiomatic Set Theory, Part II. Volume 13(2) of Proc. Symp. Pure Math., pp 155–173. American Mathematical Society, 1974
2. [Kenneth Kunen]
3. [James Earl Baumgartner]
4. [Eric van Douwen]
5. [Saharon Shelah]