Cardinal function

In mathematics, a cardinal function is a function which returns cardinal numbers.

Cardinal functions in set theory

• The most frequently used cardinal function is a function which assigns to a set its cardinality.

• Cardinal characteristics of an ideal of subsets of X are

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 add(I)≥$\aleph _{1}$.

• ${\rm {cov}}(I)=\min\{|{\mathcal {A}}|:{\mathcal {A}}\subseteq I\wedge \bigcup {\mathcal {A}}=X{\big \}}$.

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).

• ${\rm {non}}(I)=\min\{|A|:A\subseteq X\ \wedge \ A\notin I{\big \}}$,

The "uniformity number" of I (sometimes also written ${\rm {unif}}(I)$) is the size of the smallest set not in I. By our assumption on I, add(I) ≤ non(I).

• ${\rm {cof}}(I)=\min\{|{\mathcal {B}}|:{\mathcal {B}}\subseteq I\wedge (\forall A\in I)(\exists B\in {\mathcal {B}})(A\subseteq B){\big \}}.$

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).

See also

Cichoń's diagram