ⓘ Cardinal characteristic of the continuum. In the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal numbe ..


ⓘ 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 ℵ 0 {\displaystyle \aleph _{0}}, and the cardinality of the continuum, that is, the cardinality of the set R {\displaystyle \mathbb {R} } of all real numbers. The latter cardinal is denoted 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} or c {\displaystyle {\mathfrak {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.


1. Background

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


2. Examples

As is standard in set theory, we denote by ω {\displaystyle \omega } the least infinite ordinal, which has cardinality ℵ 0 {\displaystyle \aleph _{0}} ; 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.


2.1. Examples nonN

The cardinal characteristic nonN {\displaystyle {\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.


2.2. Examples Bounding number b {\displaystyle {\mathfrak {b}}} and dominating number d {\displaystyle {\mathfrak {d}}}

We denote by ω {\displaystyle \omega ^{\omega }} the set of functions from ω {\displaystyle \omega } to ω {\displaystyle \omega }. For any two functions f: ω → ω {\displaystyle f:\omega \to \omega } and g: ω → ω {\displaystyle g:\omega \to \omega } we denote by f ≤ ∗ g {\displaystyle f\leq ^{*}g} the statement that for all but finitely many n ∈ ω, f n ≤ g n {\displaystyle n\in \omega,fn\leq gn}. The bounding number b {\displaystyle {\mathfrak {b}}} is the least cardinality of an unbounded set in this relation, that is, b = min { | F |: F ⊆ ω ∧ ∀ f: ω → ω ∃ g ∈ F g ≰ ∗ f }). {\displaystyle {\mathfrak {b}}=\min\{|F|:F\subseteq \omega ^{\omega }\land \forall f:\omega \to \omega \exists g\in Fg\nleq ^{*}f\}).}

The dominating number d {\displaystyle {\mathfrak {d}}} is the least cardinality of a set of functions from ω {\displaystyle \omega } to ω {\displaystyle \omega } such that every such function is dominated by that is, ≤ ∗ {\displaystyle \leq ^{*}} a member of that set, that is, d = min { | F |: F ⊆ ω ∧ ∀ f: ω → ω ∃ g ∈ f ≤ ∗ g }). {\displaystyle {\mathfrak {d}}=\min\{|F|:F\subseteq \omega ^{\omega }\land \forall f:\omega \to \omega \exists g\in Ff\leq ^{*}g\}).}

Clearly any such dominating set F {\displaystyle F} is unbounded, so b {\displaystyle {\mathfrak {b}}} is at most d {\displaystyle {\mathfrak {d}}}, and a diagonalisation argument shows that b > ℵ 0 {\displaystyle {\mathfrak {b}}> \aleph _{0}}. Of course if c = ℵ 1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} this implies that b = d = ℵ 1 {\displaystyle {\mathfrak {b}}={\mathfrak {d}}=\aleph _{1}}, but Hechler has shown that it is also consistent to have b {\displaystyle {\mathfrak {b}}} strictly less than d {\displaystyle {\mathfrak {d}}}.

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