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". 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 formula_1 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 formula_2 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.
Ultrapower construction.
In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences formula_3 of real numbers "u""n". Namely, the equivalence class defines a hyperreal, denoted formula_4 in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *R is of the form formula_5, and is defined by a sequence formula_6 of finite sets formula_7
References.
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