• The most frequently used cardinal function is the function that assigns to a
set A its
cardinality, denoted by |
A|. •
Aleph numbers and
beth numbers can both be seen as cardinal functions defined on
ordinal numbers. •
Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers. • Cardinal characteristics of a (proper)
ideal I of
subsets of
X are: :{\rm add}(I) = \min\{|\mathcal{A}| : \mathcal{A}\subseteq I \wedge \bigcup \mathcal{A} \notin I\}. ::The "additivity" of
I is the smallest number of sets from
I whose
union is not in
I any more. As any ideal is closed under finite unions, this number is always at least \aleph_0; if
I is a
σ-ideal, then \operatorname{add}(I) \ge \aleph_1. :\operatorname{cov}(I) = \min\{|\mathcal{A}| : \mathcal{A} \subseteq I \wedge \bigcup \mathcal{A} = X\}. :: The "covering number" of
I is the smallest number of sets from
I whose union is all of
X. As
X itself is not in
I, we must have add(
I) ≤ cov(
I). :\operatorname{non}(I) = \min\{|A| : A \subseteq X\ \wedge\ A \notin I\}, :: The "uniformity number" of
I (sometimes also written {\rm unif}(I)) is the size of the smallest set not in
I. Assuming
I contains all
singletons, add(
I) ≤ non(
I). :{\rm cof}(I) = \min\{|\mathcal{B}| : \mathcal{B} \subseteq I \wedge \forall A \in I(\exists B \in \mathcal{B})(A\subseteq B)\}. :: The "cofinality" of
I is the
cofinality of the
partial order (
I, ⊆). It is easy to see that we must have non(
I) ≤ cof(
I) and cov(
I) ≤ cof(
I). :In the case that I is an ideal closely related to the structure of the
reals, such as the ideal of
Lebesgue null sets or the ideal of
meagre sets, these cardinal invariants are referred to as
cardinal characteristics of the continuum. • For a
preordered set (\mathbb{P},\sqsubseteq) the
bounding number {\mathfrak b}(\mathbb{P}) and
dominating number {\mathfrak d}(\mathbb{P}) are defined as ::{\mathfrak b}(\mathbb{P}) = \min\big\{|Y| : Y \subseteq \mathbb{P}\ \wedge\ (\forall x\in \mathbb{P})(\exists y\in Y)(y\not\sqsubseteq x)\big\}, ::{\mathfrak d}(\mathbb{P}) = \min\big\{|Y| : Y \subseteq \mathbb{P}\ \wedge\ (\forall x\in \mathbb{P})(\exists y\in Y)(x\sqsubseteq y)\big\}.\exists^\infty n\in\mathbb{N}" means: "there are infinitely many natural numbers
n such that...", and "\forall^\infty n\in\mathbb{N}" means "for all except finitely many natural numbers
n we have...". --> • In
PCF theory the cardinal function pp_\kappa(\lambda) is used. == Cardinal functions in topology ==