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.
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 independent of the standard ZFC axioms for set theory 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 .
As is standard, we denote by the least infinite ordinal, which has cardinality and 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.
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,
The ultrafilter number is defined to be the least cardinality of a filter base of an ultrafilter on . Kunen gave a model of set theory in which but , and using a countable support iteration of Sacks forcings, Baumgartner and Laver demonstrated 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 , 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 is that ; Shelah showed that it is consistent to have the strict inequality .
A well known diagram of cardinal characteristics is Cichoń's diagram, showing all relations provable in ZFC between 10 cardinal characteristics.
- ↑ Stephen Hechler. On the existence of certain cofinal subsets of . In T. Jech (ed), Axiomatic Set Theory, Part II. Volume 13(2) of Proc. Symp. Pure Math., pp 155–173. American Mathematical Society, 1974
- ↑ Kenneth Kunen. Set Theory An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics vol. 102, Elsevier, 1980
- ↑ James Earl Baumgartner and Richard Laver. Iterated perfect-set forcing. Annals of Mathematical Logic 17 (1979) pp 271–288.
- ↑ Eric van Douwen. The Integers and Topology. In K. Kunen and J.E. Vaughan (eds) Handbook of Set-Theoretic Topology. North-Holland, Amsterdam, 1984.
- ↑ 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.
- Tomek Bartoszyński and Haim Judah. Set Theory On the Structure of the Real Line. A K Peters, 1995.