# Hyperfinite set

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{{Short description|Type of internal set in nonstandard analysis}}
In [nonstandard analysis](/source/nonstandard_analysis), a branch of [mathematics](/source/mathematics), a '''hyperfinite set''' or '''*-finite set''' is a type of [internal set](/source/internal_set). An internal set ''H'' of internal cardinality ''g'' ∈ *'''N''' (the [hypernatural](/source/hypernatural)s) is hyperfinite [if and only if](/source/if_and_only_if) there exists an internal [bijection](/source/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](/source/integration_(mathematics)).<ref name=Chuaqui>{{cite book|title=Truth, possibility, and probability: new logical foundations of probability and statistical inference|url=https://archive.org/details/truthpossibility00chua_120|url-access=limited|author=R. Chuaqui|author-link= Rolando Chuaqui|publisher=Elsevier|year=1991|isbn=0-444-88840-3|pages=[https://archive.org/details/truthpossibility00chua_120/page/n202 182]–3}}</ref>

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](/source/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](/source/unit_circle), considered as the set <math>e^{i\theta}</math> for θ in the interval [0,2π].<ref name=Chuaqui />

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|url=https://archive.org/details/calculusvariatio00ambr_557|url-access=limited|author=L. Ambrosio|author-link=Luigi Ambrosio|publisher=Springer|year=2000|isbn=3-540-64803-8|page=[https://archive.org/details/calculusvariatio00ambr_557/page/n205 203]|display-authors=etal}}</ref>

==Ultrapower construction==
In terms of the [ultrapower](/source/ultrapower) construction, the hyperreal line *'''R''' is defined as the collection of [equivalence class](/source/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 \subseteq \mathbb{R}, n=1,2,\ldots</math><ref>{{cite book|author=Rob Goldblatt|authorlink = Robert Goldblatt|year=1998|title=Lectures on the hyperreals. An introduction to nonstandard analysis|url=https://archive.org/details/lecturesonhyperr00gold_525|url-access=limited|page=[https://archive.org/details/lecturesonhyperr00gold_525/page/n202 188]|publisher=Springer|isbn=0-387-98464-X}}</ref>

==References==
{{Reflist}}

== External links ==
*{{mathworld |urlname=HyperfiniteSet |title=Hyperfinite Set |author=M. Insall}}

{{Infinitesimals}}

{{DEFAULTSORT:Hyperfinite Set}}
Category:Nonstandard analysis

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Adapted from the Wikipedia article [Hyperfinite set](https://en.wikipedia.org/wiki/Hyperfinite_set) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Hyperfinite_set?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.