Cardinal characteristic of the continuum

Background

Cantor's diagonal argument shows that \mathfrak 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 \mathfrak 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 them by Paul Cohen. If the continuum hypothesis fails and so \mathfrak c is at least \aleph_2, natural questions arise about the cardinals strictly between \aleph_0 and \mathfrak 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 \mathfrak c. Generally one only considers definitions for cardinals that are provably greater than \aleph_0 and at most \mathfrak 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 that 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 \text{non}(\mathcal{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 \mathfrak b is the least cardinality of an unbounded set in this relation, that is, \mathfrak b = \min({|F| : F\subseteq\omega^\omega \land \forall g:\omega\to\omega ,\exists f \in F (f \nleq^* g)}).

The dominating number \mathfrak 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, \mathfrak d = \min({|F| : F\subseteq\omega^\omega \land \forall g:\omega\to\omega ,\exists f \in F (g \leq^ f)}).

Clearly any such dominating set F is unbounded, so \mathfrak b is at most \mathfrak d, and a diagonalisation argument shows that \mathfrak b\aleph_0. Of course if \mathfrak c=\aleph_1 this implies that \mathfrak b=\mathfrak d=\aleph_1, but Hechler has shown that it is also consistent to have \mathfrak b strictly less than \mathfrak 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 \mathfrak 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, \mathfrak 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 \mathfrak 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, \mathfrak r = \min({|R| : R\subseteq[\omega]^\omega \land \forall a\in[\omega]^\omega \exists b \in R (|b \cap a|

Ultrafilter number

The ultrafilter number \mathfrak u is defined to be the least cardinality of a filter base of a non-principal ultrafilter on \omega. Kunen gave a model of set theory in which \mathfrak u = \aleph_1 but \mathfrak c = \aleph_{\omega_1}, and using a countable support iteration of Sacks forcings, Baumgartner and Laver constructed a model in which \mathfrak u = \aleph_1 and \mathfrak 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 \mathcal{A} such that for every subset X of \omega not in \mathcal{A}, there is a set A\in\mathcal{A} such that A and X are not almost disjoint (that is, their intersection is infinite). The almost disjointness number \mathfrak{a} is the least cardinality of an infinite maximal almost disjoint family. A basic result is that \mathfrak{b}\leq\mathfrak{a}; Shelah showed that it is consistent to have the strict inequality \mathfrak{b}.

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.

References

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]]. ''Set Theory An Introduction to Independence Proofs''. Studies in Logic and the Foundations of Mathematics vol. 102, Elsevier, 1980
3. [[James Earl Baumgartner]] and [[Richard Laver]]. Iterated perfect-set forcing. ''Annals of Mathematical Logic'' '''17''' (1979) pp 271–288.
4. [[Eric van Douwen]]. The Integers and Topology. In K. Kunen and J.E. Vaughan (eds) ''Handbook of Set-Theoretic Topology''. North-Holland, Amsterdam, 1984.
5. [[Saharon Shelah]]. On cardinal invariants of the continuum. In J. Baumgartner, D. Martin and S. Shelah (eds) ''Axiomatic Set Theory'', Contemporary Mathematics 31, American Mathematical Society, 1984, pp 183-207.