Cofinality and Regular Cardinals

Cofinality is an intrinsic measure of the "width" of an ordinal, capturing how quickly one can approach it from below. Regular and singular cardinals represent a fundamental partition of the infinite cardinals.

Cofinality

Definition5.5Cofinality

The cofinality of a limit ordinal $\alpha$ , denoted $\mathrm{cf}(\alpha)$ , is the least order type of a cofinal subset of $\alpha$ . A subset $S \subseteq \alpha$ is cofinal if $\sup S = \alpha$ . Equivalently, $\mathrm{cf}(\alpha)$ is the least ordinal $\beta$ such that there exists a strictly increasing function $f: \beta \to \alpha$ with $\sup f[\beta] = \alpha$ .

ExampleCofinality computations

$\mathrm{cf}(\omega) = \omega$ (no finite set is cofinal in $\omega$ , but $\omega$ is cofinal in itself).
$\mathrm{cf}(\omega_1) = \omega_1$ (no countable set is cofinal in $\omega_1$ ; this is a key property).
$\mathrm{cf}(\omega + \omega) = \omega$ (the sequence $\omega, \omega+1, \omega+2, \ldots$ is cofinal).
$\mathrm{cf}(\aleph_\omega) = \omega$ (the sequence $\aleph_0, \aleph_1, \aleph_2, \ldots$ is cofinal with order type $\omega$ ).

Regular and Singular Cardinals

Definition5.6Regular and singular

An infinite cardinal $\kappa$ is regular if $\mathrm{cf}(\kappa) = \kappa$ , and singular if $\mathrm{cf}(\kappa) < \kappa$ .

Every successor cardinal $\aleph_{\alpha+1}$ is regular. A limit cardinal is singular if it can be expressed as a supremum of fewer than $\kappa$ cardinals each less than $\kappa$ .

Theorem5.4Konig's theorem

If $\kappa_i < \lambda_i$ for all $i \in I$ , then:

$\sum_{i \in I} \kappa_i < \prod_{i \in I} \lambda_i.$

In particular, $\mathrm{cf}(2^\kappa) > \kappa$ for every infinite cardinal $\kappa$ .

Consequences

ExampleApplications of Konig's theorem

$2^{\aleph_0} \neq \aleph_\omega$ : since $\mathrm{cf}(\aleph_\omega) = \omega = \aleph_0$ , Konig gives $\mathrm{cf}(2^{\aleph_0}) > \aleph_0$ , but $\mathrm{cf}(\aleph_\omega) = \omega = \aleph_0$ , contradiction.
More generally, $2^{\aleph_0}$ cannot be any cardinal with countable cofinality: not $\aleph_\omega$ , not $\aleph_{\omega+\omega}$ , not $\aleph_{\omega_1 \cdot \omega}$ , etc.
Konig's theorem is one of the few restrictions on the exponential function in cardinal arithmetic.

RemarkInaccessible cardinals

A cardinal $\kappa > \aleph_0$ is strongly inaccessible if it is regular and a strong limit ( $2^\lambda < \kappa$ for all $\lambda < \kappa$ ). The existence of inaccessible cardinals cannot be proved in ZFC (if ZFC is consistent). If $\kappa$ is inaccessible, then $V_\kappa$ is a model of ZFC, so the existence of inaccessibles implies the consistency of ZFC. This is the starting point of the large cardinal hierarchy.