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 = {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 with a hypernatural n. K is a near interval for [a,b] 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 ∈ [a,b] there is a ki ∈ K such that ki ≈ r. This, for example, allows for an approximation to the unit circle, considered as the set 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 constructionedit
In terms of the ultrapower construction, the hyperreal line *R is defined as the collection of equivalence classes of sequences of real numbers un. Namely, the equivalence class defines a hyperreal, denoted in Goldblatt's notation. Similarly, an arbitrary hyperfinite set in *R is of the form , and is defined by a sequence of finite sets [4]
Referencesedit
^J. E. Rubio (1994). Optimization and nonstandard analysis. Marcel Dekker. p. 110. ISBN0-8247-9281-5.
^ abcR. Chuaqui (1991). Truth, possibility, and probability: new logical foundations of probability and statistical inference. Elsevier. pp. 182–3. ISBN0-444-88840-3.
^L. Ambrosio; et al. (2000). Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory. Springer. p. 203. ISBN3-540-64803-8.
^Rob Goldblatt (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Springer. p. 188. ISBN0-387-98464-X.
hyperfinite, nonstandard, analysis, branch, mathematics, hyperfinite, finite, type, internal, internal, internal, cardinality, hypernaturals, hyperfinite, only, there, exists, internal, bijection, between, share, properties, finite, sets, hyperfinite, minimal,. 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 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 k1 k2 kn displaystyle K k 1 k 2 dots k n with a hypernatural n K is a near interval for a b 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 a b there is a ki K such that ki r This for example allows for an approximation to the unit circle considered as the set ei8 displaystyle e i theta for 8 in the interval 0 2p 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 editIn terms of the ultrapower construction the hyperreal line R is defined as the collection of equivalence classes of sequences un n 1 2 displaystyle langle u n n 1 2 ldots rangle nbsp of real numbers un Namely the equivalence class defines a hyperreal denoted un displaystyle u n nbsp in Goldblatt s notation Similarly an arbitrary hyperfinite set in R is of the form An displaystyle A n nbsp and is defined by a sequence An displaystyle langle A n rangle nbsp of finite sets An R n 1 2 displaystyle A n subseteq mathbb R n 1 2 ldots nbsp 4 References edit J E Rubio 1994 Optimization and nonstandard analysis Marcel Dekker p 110 ISBN 0 8247 9281 5 a b c 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 Rob Goldblatt 1998 Lectures on the hyperreals An introduction to nonstandard analysis Springer p 188 ISBN 0 387 98464 X External links editM Insall Hyperfinite Set MathWorld Retrieved from https en wikipedia org w index php title Hyperfinite set amp oldid 1125097958, wikipedia, wiki, book, books, library,