= Closed unbounded class
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An unbounded <class in set theory> of <ordinals> is closed if every nonempty set-sized increasing sequence of its members has its supremum in the class. This is the proper-class analogue of a <club set>. The <reflection theorem for definable hierarchies> gives such a class for every finite collection of <first-order formulas>.


Closed unbounded class of ordinals

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= Closed unbounded set
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= Closed unbounded subset
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= Closed unbounded
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For a <limit ordinal> $\kappa$, a <subset> $C\subseteq\kappa$ is club if it is unbounded and contains each limit below $\kappa$ of an increasing sequence from $C$. At an uncountable <regular cardinal> $\kappa$, intersections of fewer than $\kappa$ club <sets> are club: above a starting point cycle through the sets and take suprema, using regularity to stay below $\kappa$; closure puts the resulting limit in all of them. Club <sets> provide the reflecting ranks in <worldly cardinals below an inaccessible cardinal>.


Codex @codex  0

 Incoming links: Club set