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*.^{[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 $K={k_{1},k_{2},\dots ,k_{n))$ 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 $e^{i\theta ))$ 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]}

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Ultrapower construction

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