Specht module: Difference between revisions
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A tabloid is an equivalence class of labelings of the Young diagram that are not necessarily tableaux; there is a unique tableau in each equivalence class. |
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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. | |||
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]]. | |||
If <math>x \in H_{\aleph_1}</math>, then <math>L_{\omega_1}(x) \subset H_{\aleph_1}</math>. | |||
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. | |||
==See also== | |||
*[[Hereditarily finite set]] | |||
*[[Constructible universe]] | |||
==External links== | |||
*[http://www.jstor.org/pss/2273380 "On Hereditarily Countable Sets"] by [[Thomas Jech]] | |||
[[Category:Set theory]] | |||
[[Category:Large cardinals]] | |||
{{settheory-stub}} | |||
Revision as of 01:32, 2 December 2013
In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. This inductive definition is in fact well-founded and can be expressed in the language of first-order set theory. A set is hereditarily countable if and only if it is countable, and every element of its 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.
The 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 . 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.
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 . 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.