Hereditarily countable set
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In set theory, a set is called hereditarily countable if and only if its transitive closure is a countable set. If the axiom of countable choice holds, then a set is hereditarily countable if and only if it is a countable set of hereditarily countable sets. The set of all hereditarily countable sets is symbolized by 
, meaning hereditarily of cardinality less than 
.
If 
, then 
.
More generally, a set is hereditarily of cardinality less than κ if and only if its transitive closure is of cardinality less than κ. The set of all such sets is symbolized by 
.
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