- Hereditarily countable set
In

set theory , a set is called**hereditarily countable**if and only if it is acountable set of hereditarily countable sets. Thisinductive definition is in factwell-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 theaxiom 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 theaxiom of choice , and this set is designated $H\_\{aleph\_1\}$. The hereditarily countable sets form a model ofKripke–Platek set theory with theaxiom of infinity (KPI), if the axiom of countable choice is assumed in themetatheory .If $x\; in\; H\_\{aleph\_1\}$, then $L\_\{omega\_1\}(x)\; subset\; H\_\{aleph\_1\}$.

More generally, a set is

**hereditarily of cardinality less than κ**if and only it is ofcardinality 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 $H\_kappa\; !$. If the axiom of choice holds, then a set is hereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ.**ee also***

Hereditarily finite set

*Constructible universe **External links*** [

*http://www.jstor.org/pss/2273380 "On Hereditarily Countable Sets" by Thomas Jech*]

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