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 (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 ${\displaystyle K=\{k_{1},k_{2},\dots ,k_{n}\}}$ 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 ${\displaystyle e^{i\theta }}$ 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

In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences ${\displaystyle \langle u_{n},n=1,2,\ldots \rangle }$ of real numbers u_n. Namely, the equivalence class defines a hyperreal, denoted ${\displaystyle [u_{n}]}$ in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *R is of the form ${\displaystyle [A_{n}]}$, and is defined by a sequence ${\displaystyle \langle A_{n}\rangle }$ of finite sets ${\displaystyle A_{n}\subseteq \mathbb {R} ,n=1,2,\ldots }$^[4]