# Hyperfinite set

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 hypernaturals) is hyperfinite if and only if there exists an internal bijection between *G* = {1,2,3,...,*g*} and *H*.^{[1]}^{[2]} 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.^{[2]}

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 with a hypernatural *n*. *K* is a near interval for [*a*,*b*] if *k*_{1} = *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 for θ in the interval [0,2π].^{[2]}

In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.^{[3]}

## Ultrapower construction[edit]

In terms of the ultrapower construction, the hyperreal line ***R** is defined as the collection of equivalence classes of sequences of real numbers *u*_{n}. Namely, the equivalence class defines a hyperreal, denoted in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in ***R** is of the form , and is defined by a sequence of finite sets ^{[4]}

## Notes[edit]

**^**J. E. Rubio (1994).*Optimization and nonstandard analysis*. Marcel Dekker. p. 110. ISBN 0-8247-9281-5.- ^
^{a}^{b}^{c}R. Chuaqui (1991).*Truth, possibility, and probability: new logical foundations of probability and statistical inference*. Elsevier. pp. 182–3. ISBN 0-444-88840-3. **^**L. Ambrosio; et al. (2000).*Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory*. Springer. p. 203. ISBN 3-540-64803-8.**^**R. Goldblatt (1998).*Lectures on the hyperreals. An introduction to nonstandard analysis*. Springer. p. 188. ISBN 0-387-98464-X.

## External links[edit]

- M. Insall. "Hyperfinite Set".
*MathWorld*.