Hyperfinite set
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 is hyperfinite if and only if there exists an internal bijection between 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 if k1 = a and kn = b, and if the difference between successive elements of K is infinitesimal. Phrased otherwise, the requirement is that for every r ∈ there is a ki ∈ K such that ki ≈ r. This, for example, allows for an approximation to the unit circle, considered as the set for θ in the interval .
In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.