Hyperfinite set explained

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 = and H.^[1] 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,} ... ,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.^[3]

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 R , n =1 , 2 ,\ldots ^[4]

Notes and References

1. Book: Optimization and nonstandard analysis. J. E. Rubio. Marcel Dekker. 1994. 0-8247-9281-5. 110.
2. Book: Truth, possibility, and probability: new logical foundations of probability and statistical inference. limited. R. Chuaqui. Rolando Chuaqui. Elsevier. 1991. 0-444-88840-3. 182–3.
3. Book: Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory. limited. L. Ambrosio. Luigi Ambrosio. Springer. 2000. 3-540-64803-8. 203. etal.
4. Book: Robert Goldblatt

   . Rob Goldblatt. Robert Goldblatt. 1998. Lectures on the hyperreals. An introduction to nonstandard analysis. limited. 188. Springer. 0-387-98464-X.