Hyperfinite set

In <a href="/facts/Nonstandard_analysis/lEpNpWMB">nonstandard analysis</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a hyperfinite set or *-finite set is a type of <a href="/facts/Internal_set/IHp7mBaK">internal set</a>. An internal set H of internal cardinality g ∈ *N (the <a href="/facts/Hypernatural/TYasI1IP">hypernaturals</a>) is hyperfinite <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> there exists an internal <a href="/facts/Bijection/j4gTuTmW">bijection</a> between G = {1,2,3,...,g} and H. 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 <a href="/facts/Integration_(mathematics)/V7lyV997">integration</a>.
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 
 
 
 
 K
 =
 {
 
 k
 
 1
 
 
 ,
 
 k
 
 2
 
 
 ,
 …
 ,
 
 k
 
 n
 
 
 }
 
 
 {\displaystyle K=\{k_{1},k_{2},\dots ,k_{n}\}}
 
 with a hypernatural n. K is a near interval for [a,b] if k1 = a and kn = b, and if the difference between successive elements of K is <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a>. Phrased otherwise, the requirement is that for every r ∈ [a,b] there is a ki ∈ K such that ki ≈ r. This, for example, allows for an approximation to the <a href="/facts/Unit_circle/VsTffjPK">unit circle</a>, considered as the set 
 
 
 
 
 e
 
 i
 θ
 
 
 
 
 {\displaystyle e^{i\theta }}
 
 for θ in the interval [0,2π].
In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.

Hyperfinite set open-in-new

Hyperfinite set