Editing Hereditarily finite set (section)

===ZF===
[[File:Nested_set_V4.svg|thumb|400px|<math>~V_4~</math> represented with circles in place of [[Bracket (mathematics)#Sets and groups|curly brackets]]&nbsp;&nbsp;&nbsp;&nbsp;[[File:Loupe light.svg|15px|link=http://upload.wikimedia.org/wikipedia/commons/thumb/1/1b/Nested_set_V4.svg/1600px-Nested_set_V4.svg.png]] ]]

The hereditarily finite sets are a subclass of the [[Von Neumann universe]]. Here, the class of all well-founded hereditarily finite sets is denoted <math>V_{\omega}</math>. Note that this is also a set in this context.

If we denote by <math>\wp(S)</math> the [[power set]] of <math>S</math>, and by <math>V_0</math> the empty set, then <math>V_{\omega}</math> can be obtained by setting <math>V_{i+1}=\wp(V_i)</math> for each integer <math>i\ge 0</math>. Thus, <math>V_{\omega}</math> can be expressed as
{{bi|left=1.6|<math>\displaystyle V_\omega = \bigcup_{k=0}^\infty V_k</math>}}
and all its elements are finite.

This formulation shows, again, that there are only [[countably]] many hereditarily finite sets: <math>V_n</math> is finite for any finite <math>n</math>, its [[cardinality]] is <math>2\uparrow\uparrow (n-1)</math> in [[Knuth's up-arrow notation]] (a tower of <math>n-1</math> powers of two), and the union of countably many finite sets is countable.

Equivalently, a set is hereditarily finite if and only if its [[transitive set|transitive closure]] is finite.