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 â *N (the hypernaturals) is hyperfinite if and only if there exists an internal bijection 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 integration.
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<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 â [a,b] there is a k<sub>i</sub> â K such that k<sub>i</sub> â r. This, for example, allows for an approximation to the unit circle, considered as the set for ø in the interval [0,2ÃÂ].
In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.
In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences of real numbers u<sub>n</sub>. 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