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In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set. Formal definition A recursive definition of well-founded hereditarily finite sets is as follows: Base case: The empty set is a hereditarily finite set. Recursion rule: If a 1 , … a k {\displaystyle a_{1},\dots a_{k}} are hereditarily finite, then so is { a 1 , … a k } {\displaystyle \{a_{1},\dots a_{k}\}} . 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: { } {\displaystyle \{\}} (i.e. ∅ {\displaystyle \emptyset } , the Neumann ordinal "0") { { } } {\displaystyle \{\{\}\}} (i.e. { ∅ } {\displaystyle \{\emptyset \}} or { 0 } {\displaystyle \{0\}} , the Neumann ordinal "1") { { { } } } {\displaystyle \{\{\{\}\}\}} { { { { } } } } {\displaystyle \{\{\{\{\}\}\}\}} and then also { { } , { { } } } {\displaystyle \{\{\},\{\{\}\}\}} (i.e. { 0 , 1 } {\displaystyle \{0,1\}} , the Neumann ordinal "2"), { { { { { } } } } } {\displaystyle \{\{\{\{\{\}\}\}\}\}} , { { { } , { { } } } } {\displaystyle \{\{\{\},\{\{\}\}\}\}} as well as { { } , { { { } } } } {\displaystyle \{\{\},\{\{\{\}\}\}\}} , ... sets represented with 6 {\displaystyle 6} bracket pairs, e.g. { { { { { { } } } } } } {\displaystyle \{\{\{\{\{\{\}\}\}\}\}\}} . There are six such sets ... sets represented with 7 {\displaystyle 7} bracket pairs, e.g. { { { { { { { } } } } } } } {\displaystyle \{\{\{\{\{\{\{\}\}\}\}\}\}\}} . There are twelve such sets ... sets represented with 8 {\displaystyle 8} bracket pairs, e.g. { { { { { { { { } } } } } } } } {\displaystyle \{\{\{\{\{\{\{\{\}\}\}\}\}\}\}\}} or { { } , { { } } , { { } , { { } } } } {\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}} (i.e. { 0 , 1 , 2 } {\displaystyle \{0,1,2\}} , the Neumann ordinal "3") ... etc. In this way, the number of sets with n {\displaystyle n} bracket pairs is Discussion The set { { } , { { { } } } } {\displaystyle \{\{\},\{\{\{\}\}\}\}} is an example for such a hereditarily finite set and so is the empty set { } {\displaystyle \{\}} , as noted. On the other hand, the sets { 7 , N , π } {\displaystyle \{7,{\mathbb {N} },\pi \}} or { 3 , { N } } {\displaystyle \{3,\{{\mathbb {N} }\}\}} are examples of finite sets that are not hereditarily finite. For example, the first cannot be hereditarily finite since it contains at least one infinite set as an element, when N = { 0 , 1 , 2 , … } {\displaystyle {\mathbb {N} }=\{0,1,2,\dots \}} . The class of all hereditarily finite sets is denoted by H ℵ 0 {\displaystyle H_{\aleph _{0}}} , meaning that the cardinality of each member is smaller than ℵ 0 {\displaystyle \aleph _{0}} . (Analogously, the class of hereditarily countable sets is denoted by H ℵ 1 {\displaystyle H_{\aleph _{1}}} .) H ℵ 0 {\displaystyle H_{\aleph _{0}}} is in bijective correspondence with ℵ 0 {\displaystyle \aleph _{0}} . It can also be denoted by V ω {\displaystyle V_{\omega }} , which denotes the ω {\displaystyle \omega } th sta.... Discover the Eugenio G Omodeo Alberto Policriti popular books. Find the top 100 most popular Eugenio G Omodeo Alberto Policriti books.