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[parent] 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.



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Keywords:  supremum approximation cofinality

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$\operatorname{cf}(\operatorname{cf} \alpha) = \operatorname{cf} \alpha$ (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 MSC03E04 (Mathematical logic and foundations :: Set theory :: Ordered sets and their cofinalities; pcf theory)

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