Hyperfinite set

In non-standard 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 most other 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 may be used to approximate other sets, where approximation in this context is considered to be a minimisation of symmetric difference. 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 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 $e^{i\theta }$ 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.

Relative to the ultrapower construction

The ultrapower construction of the hyperreals provides the context where hyperfinite sets can be described explicitly. An arbitrary hyperfinite set $[A_{n}]$ in *R is defined by a sequence $\langle A_{n}\rangle$ of finite sets $A_{n}\subset \mathbb {R} ,n=1,2,\ldots$, see Goldblatt (1998).

Reference

• Goldblatt, R. (1998) Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate Texts in Mathematics, 188. Springer-Verlag, New York.

External link

• M. Insall. "Hyperfinite Set". MathWorld.