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 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. 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 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 K = k 1 , k 2 , … , k n {displaystyle K={k_{1},k_{2},dots ,k_{n}}} with a hypernatural n. K is a near interval for if k1 = a and kn = b, and if the difference between successive elements of K is infinitesimal. Phrased otherwise, the requirement is that for every r ∈ there is a ki ∈ K such that ki ≈ r. This, for example, allows for an approximation to the unit circle, considered as the set e i θ {displaystyle e^{i heta }} for θ in the interval . In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set. In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences ⟨ u n , n = 1 , 2 , … ⟩ {displaystyle langle u_{n},n=1,2,ldots angle } of real numbers un. Namely, the equivalence class defines a hyperreal, denoted [ u n ] {displaystyle } in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *R is of the form [ A n ] {displaystyle } , and is defined by a sequence ⟨ A n ⟩ {displaystyle langle A_{n} angle } of finite sets A n ⊂ R , n = 1 , 2 , … {displaystyle A_{n}subset mathbb {R} ,n=1,2,ldots }