PlanetMath: another definition of cofinality

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another definition of cofinality

(Definition)

Let $\kappa$ be a limit ordinal (e.g. a cardinal). The cofinality of $\kappa$ $\operatorname{cf}(\kappa)$ could also be defined as:

$\displaystyle \operatorname{cf}(\kappa)=\inf \{ \vert U\vert : U \subseteq \kappa$ s.t. $\displaystyle \sup \; U = \kappa \}$

( $\sup \; U$ is calculated using the natural order of the ordinals). The cofinality of a cardinal is always a regular cardinal and hence $\operatorname{cf}(\kappa) = \operatorname{cf}(\operatorname{cf}(\kappa))$ .

This definition is equivalent to the parent definition.

"another definition of cofinality" is owned by x_bas.

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Keywords:

supremum approximation cofinality

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Attachments:: $\operatorname{cf}(\operatorname{cf} \alpha) = \operatorname{cf} \alpha$ (Proof) by yesitis

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Cross-references: equivalent, regular cardinal, ordinals, order, cardinal, limit ordinal
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This is version 5 of another definition of cofinality, born on 2003-08-20, modified 2003-08-20.
Object id is 4626, canonical name is AnotherDefinitionOfCofinality.
Accessed 1680 times total.

Classification:

AMS MSC:

03E04 (Mathematical logic and foundations :: Set theory :: Ordered sets and their cofinalities; pcf theory)

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