Hyperfinite set

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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.^[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 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].^[2]

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 $A_{n}\subset {\mathbb {R}},n=1,2,\ldots$ ^[4]

Notes

↑ J. E. Rubio (1994). Optimization and nonstandard analysis. Marcel Dekker. p. 110. ISBN 0-8247-9281-5.
↑ 2.0 2.1 2.2 R. Chuaqui (1991). Truth, possibility, and probability: new logical foundations of probability and statistical inference. Elsevier. pp. 182–3. ISBN 0-444-88840-3.
↑ L. Ambrosio et al. (2000). Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory. Springer. p. 203. ISBN 3-540-64803-8.
↑ R. Goldblatt (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Springer. p. 188. ISBN 0-387-98464-X.

External links

M. Insall, "Hyperfinite Set", MathWorld.

v t e Infinitesimals

History	Adequality Infinitesimal calculus Leibniz's notation Integral symbol Criticism of non-standard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle

Related branches	Non-standard analysis Non-standard calculus Internal set theory Synthetic differential geometry Constructive non-standard analysis

Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers

Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of Continuity Overspill Microcontinuity Transcendental law of homogeneity

Scientists	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler

Physics Engineering	Infinitesimal strain theory Infinitesimal oscillations of a pendulum

Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

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