Editing Hereditarily finite set (section)

==Formal definition==
A [[recursive]] definition of [[well-foundedness|well-founded]] hereditarily finite sets is as follows:
: ''Base case'': The empty set is a hereditarily finite set.
: ''Recursion rule'': If <math>a_1,\dots a_k</math> are hereditarily finite, then so is <math>\{a_1,\dots a_k\}</math>.
Only sets that can be built by a finite number of applications of these two rules are hereditarily finite.

===Representation===
This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets:

* <math>\{\}</math> (i.e. <math>\emptyset</math>, the Neumann ordinal "0")
* <math>\{\{\}\}</math> (i.e. <math>\{\emptyset\}</math> or <math>\{0\}</math>, the Neumann ordinal "1")
* <math>\{\{\{\}\}\}</math>
* <math>\{\{\{\{\}\}\}\}</math> and then also <math>\{\{\},\{\{\}\}\}</math> (i.e. <math>\{0,1\}</math>, the Neumann ordinal "2"),
* <math>\{\{\{\{\{\}\}\}\}\}</math>, <math>\{\{\{\},\{\{\}\}\}\}</math> as well as <math>\{\{\},\{\{\{\}\}\}\}</math>,
* ... sets represented with <math>6</math> bracket pairs, e.g. <math>\{\{\{\{\{\{\}\}\}\}\}\}</math>. There are six such sets
* ... sets represented with <math>7</math> bracket pairs, e.g. <math>\{\{\{\{\{\{\{\}\}\}\}\}\}\}</math>. There are twelve such sets
* ... sets represented with <math>8</math> bracket pairs, e.g. <math>\{\{\{\{\{\{\{\{\}\}\}\}\}\}\}\}</math> or <math>\{\{\}, \{\{\}\}, \{\{\},\{\{\}\}\}\}</math> (i.e. <math>\{0,1,2\}</math>, the Neumann ordinal "3")
* ... etc.
In this way, the number of sets with <math>n</math> bracket pairs is<ref>{{Cite OEIS|A004111}}</ref>
{{bi|left=1.6|1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 247, 548, 1226, 2770, 6299, 14426, ...}}