Gabriel's Horn: Difference between revisions

Revision as of 05:08, 23 February 2014

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-{{Redirect-acronym|Nested set|the [[Nested set model]] in relational databases}}
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-[[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]] ]]
-In [[mathematics]] and [[set theory]], '''hereditarily finite sets''' are defined [[recursion|recursively]] as [[finite set]]s consisting of 0 or more hereditarily finite sets.
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-==Formal definition==
-A recursive definition of a hereditarily finite set goes as follows:
-: ''Base case'': The empty set is a hereditarily finite set.
-: ''Recursion rule'': If ''a''<sub>1</sub>,...,''a''<sub>''k''</sub> are hereditarily finite, then so is {''a''<sub>1</sub>,...,''a''<sub>''k''</sub>}.
-The set of all hereditarily finite sets is denoted ''V''<sub>ω</sub>. If we denote ''P''(''S'') for the [[power set]] of ''S'', ''V''<sub>ω</sub> can also be constructed by first taking the empty set written ''V''<sub>0</sub>, then ''V''<sub>1</sub> = ''P''(''V''<sub>0</sub>), ''V''<sub>2</sub> = ''P''(''V''<sub>1</sub>),..., ''V''<sub>''k''</sub> = ''P''(''V''<sub>''k''&minus;1</sub>),... Then
-:<math>\bigcup_{k=0}^\infty V_k = V_\omega.</math>
-==Discussion==
-The hereditarily finite sets are a subclass of the [[Von Neumann universe]]. They are a [[model theory|model]] of the axioms consisting of the [[Zermelo–Fraenkel axioms|axioms of set theory]] with the [[axiom of infinity]] replaced by its negation, thus proving that the axiom of infinity is not a consequence of the other axioms of set theory.
-Notice that there are [[countably]] many hereditarily finite sets, since ''V<sub>n</sub>'' is finite for any finite ''n'' (its [[cardinality]] is <sup>''n''&minus;1</sup>2, see [[tetration]]), 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.  V<sub>ω</sub> is also symbolized by <math>H_{\aleph_0}</math>, meaning hereditarily of cardinality less than <math>\aleph_0</math>.
-==See also==
-*[[Hereditarily countable set]]
-{{Set theory}}
-[[Category:Set theory]]

Gabriel's Horn: Difference between revisions

Revision as of 05:08, 23 February 2014

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