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| In [[non-standard analysis]], a branch of [[mathematics]], a '''hyperfinite set''' or '''*-finite set''' is a type of [[internal set]]. An internal set ''H'' of internal cardinality ''g'' ∈ *'''N''' (the [[hypernatural]]s) is hyperfinite [[if and only if]] there exists an internal [[bijection]] between ''G'' = {1,2,3,...,''g''} and ''H''.<ref>{{cite book|title=Optimization and nonstandard analysis|author=J. E. Rubio|publisher=Marcel Dekker|year=1994|isbn=0-8247-9281-5|page=110}}</ref><ref name=Chuaqui /> Hyperfinite sets share the properties of finite sets: A hyperfinite set has minimal and maximal elements, and a hyperfinite union of a hyperfinite collection of hyperfinite sets may be derived. The sum of the elements of any hyperfinite subset of *'''R''' always exists, leading to the possibility of well-defined [[integration (mathematics)|integration]].<ref name=Chuaqui>{{cite book|title=Truth, possibility, and probability: new logical foundations of probability and statistical inference|author=R. Chuaqui|publisher=Elsevier|year=1991|isbn=0-444-88840-3|pages=182–3}}</ref>
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| Hyperfinite sets can be used to approximate other sets. If a hyperfinite set approximates an interval, it is called a ''near interval'' with respect to that interval. Consider a hyperfinite set <math>K = {k_1,k_2, \dots ,k_n}</math> with a hypernatural ''n''. ''K'' is a near interval for [''a'',''b''] if ''k''<sub>1</sub> = ''a'' and ''k''<sub>''n''</sub> = ''b'', and if the difference between successive elements of ''K'' is [[infinitesimal]]. Phrased otherwise, the requirement is that for every ''r'' ∈ [''a'',''b''] there is a ''k''<sub>''i''</sub> ∈ ''K'' such that ''k''<sub>''i''</sub> ≈ ''r''. This, for example, allows for an approximation to the [[unit circle]], considered as the set <math>e^{i\theta}</math> for θ in the interval [0,2π].<ref name=Chuaqui />
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| In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.<ref>{{cite book|title=Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory|author=[[Luigi Ambrosio|L. Ambrosio]] et al.|publisher=Springer|year=2000|isbn=3-540-64803-8|page=203}}</ref>
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| == Ultrapower construction ==
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| In terms of the [[ultrapower]] construction, the hyperreal line *'''R''' is defined as the collection of [[equivalence class]]es of sequences <math>\langle u_n, n=1,2,\ldots \rangle</math> of real numbers ''u''<sub>''n''</sub>. Namely, the equivalence class defines a hyperreal, denoted <math>[u_n]</math> in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *'''R''' is of the form <math>[A_n]</math>, and is defined by a sequence <math>\langle A_n \rangle</math> of finite sets <math>A_n \subset \mathbb{R}, n=1,2,\ldots</math><ref>{{cite book|author=R. Goldblatt|year=1998|title=Lectures on the hyperreals. An introduction to nonstandard analysis|page=188|publisher=Springer|isbn=0-387-98464-X}}</ref>
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| == Notes ==
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| <references/>
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| == External links ==
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| *{{mathworld |urlname=HyperfiniteSet |title=Hyperfinite Set |author=M. Insall}}
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| {{Infinitesimals}}
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| {{DEFAULTSORT:Hyperfinite Set}}
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| [[Category:Non-standard analysis]]
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