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- In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. This inductive definition is 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. If , then . More generally, a set is hereditarily of cardinality less than κ if 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. (en)
- 在集合论中，一个集合被称为继承可数的，当且仅当它的传递闭包是可数集合。如果可数选择公理成立，则一个集合是继承可数的，当且仅当它是继承可数集合的可数集合。所有继承有限集合的集合符号化为 ，意味着势小于 的继承。 如果 ，则 。 更一般的说，一个集合是势小于κ的继承，当且仅当它的传递闭包有着小于κ的势。所有这样的集合的集合符号化为 。 (zh)
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- 1828 (xsd:nonNegativeInteger)
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- 在集合论中，一个集合被称为继承可数的，当且仅当它的传递闭包是可数集合。如果可数选择公理成立，则一个集合是继承可数的，当且仅当它是继承可数集合的可数集合。所有继承有限集合的集合符号化为 ，意味着势小于 的继承。 如果 ，则 。 更一般的说，一个集合是势小于κ的继承，当且仅当它的传递闭包有着小于κ的势。所有这样的集合的集合符号化为 。 (zh)
- In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets. This inductive definition is 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. If , then . (en)
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- Hereditarily countable set (en)
- 继承可数集合 (zh)
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