Hereditarily countable set

In set theory, a set is called hereditarily countable if it is a countable set of hereditarily countable sets.

Results

The inductive definition above is well-founded and can be expressed in the language of first-order set theory.

Equivalent properties

A set is hereditarily countable if and only if it is countable, and every element of its transitive closure is countable.

The collection of all hereditarily countable sets

The class of all hereditarily countable sets can be proven to be a set from the axioms of Zermelo–Fraenkel set theory (ZF) and is set is designated $H_{\aleph _{1}}$. In particular, the existence does not require any form of the axiom of choice. Constructive Zermelo–Fraenkel (CZF) does not prove the class to be a set.

The set $H_{\aleph _{1}}$ is included in the set $V_{\omega _{1}}$ from the von Neumann hierarchy, that is, $H_{\aleph _{1}}\subseteq V_{\omega _{1}}$. Every hereditarily finite set is hereditarily countable, so $H_{\aleph _{1}}\supseteq \mathrm {HF} =V_{\omega }$. Since $V_{\omega }$ is countable, we in fact have $H_{\aleph _{1}}\supseteq V_{\omega +1}$.

An ordinal is hereditarily countable if and only if it is countable.

Model theory

This class is 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 $x\in H_{\aleph _{1}}$, then $L_{\omega _{1}}(x)\subseteq H_{\aleph _{1}}$.

Generalizations

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 $H_{\kappa }\!$. 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