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| In [[set theory]], a set is called '''hereditarily countable''' if it is a [[countable set]] of [[hereditary property|hereditarily]] countable sets. This [[inductive definition]] is in fact [[well-founded]] and can be expressed in the language of [[first-order logic|first-order]] set theory. A set is hereditarily countable if and only if it is countable, and every element of its [[transitive set|transitive closure]] is countable. If the [[axiom of countable choice]] holds, then a set is hereditarily countable if and only if its transitive closure is countable.
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| The [[class (set theory)|class]] of all hereditarily countable sets can be proven to be a set from the axioms of [[Zermelo–Fraenkel set theory]] (ZF) without any form of the [[axiom of choice]], and this set is designated <math>H_{\aleph_1}</math>. The hereditarily countable sets form a model of [[Kripke–Platek set theory]] with the [[axiom of infinity]] (KPI), if the axiom of countable choice is assumed in the [[metatheory]].
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| If <math>x \in H_{\aleph_1}</math>, then <math>L_{\omega_1}(x) \subset H_{\aleph_1}</math>.
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| More generally, a set is '''hereditarily of cardinality less than κ''' if and only it is of [[cardinality]] less than κ, and all its elements are hereditarily of cardinality less than κ; the class of all such sets can also be proven to be a set from the axioms of ZF, and is designated <math>H_\kappa \!</math>. If the axiom of choice holds and the cardinal κ is regular, then a set is hereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ.
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| ==See also==
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| *[[Hereditarily finite set]]
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| *[[Constructible universe]]
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| ==External links==
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| *[http://www.jstor.org/pss/2273380 "On Hereditarily Countable Sets"] by [[Thomas Jech]]
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| [[Category:Set theory]]
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| [[Category:Large cardinals]]
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| {{settheory-stub}}
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Latest revision as of 13:46, 9 January 2015
The author is called Irwin Wunder but it's not the most masucline title out there. Doing ceramics is what her family and her appreciate. California is exactly where I've always been living and I love each day living right here. Bookkeeping is what I do.
my site ... home std test