Hyperfinite set

{{Short description|Type of internal set in nonstandard analysis}}

In nonstandard 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 hypernaturals) is hyperfinite if and only if there exists an internal bijection between G = {1,2,3,...,g} and H.{{cite book|title=Optimization and nonstandard analysis|author=J. E. Rubio|publisher=Marcel Dekker|year=1994|isbn=0-8247-9281-5|page=110}} 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.{{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}}

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 $K = \{k_1,k_2, \dots ,k_n\}$ with a hypernatural n. K is a near interval for [a,b] if k₁ = a and k_n = 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_i ∈ K such that k_i ≈ r. This, for example, allows for an approximation to the unit circle, considered as the set $e^{i\theta}$ 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.{{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}}

Ultrapower construction

In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences $\langle u_n, n=1,2,\ldots \rangle$ of real numbers u_n. Namely, the equivalence class defines a hyperreal, denoted $[u_n]$ in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *R is of the form $[A_n]$ , and is defined by a sequence $\langle A_n \rangle$ of finite sets $A_n \subseteq \mathbb{R}, n=1,2,\ldots$ {{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}}

References

External links

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

Category:Nonstandard analysis