Hyperfinite set

In non-standard analysis, a hyperfinite set or *-finite set is a type of internal set. An internal set H is hyperfinite if and only if there exists an internal bijection $f:G\rightarrow H$ with G = {1,2,3,...,g} for g ∈ *N. Hyperfinite sets share the properties of most other sets: A hyperfinite set H has minimal and maximal elements, and a hyperfinite union of a hyperfinite collection of hyperfinite sets may be derived. The sum of any hyperfinite subset of *R always exists.

Hyperfinite sets may be used to approximate other sets, where approximation in this context is considered to be a minimisation of symmetric difference. Such hyperfinite sets are called near intervals. 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.