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.[1][2] Hyperfinite sets share the properties of most other 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 may be used to approximate other sets, where approximation in this context is considered to be a minimisation of symmetric difference. 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.[2]
In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.[3]
Notes
- ^ J. E. Rubio (1994). Optimization and nonstandard analysis. Marcel Dekker. p. 110. ISBN 0824792815.
- ^ a b c R. Chuaqui (1991). Truth, possibility, and probability: new logical foundations of probability and statistical inference. Elsevier. pp. 182–3. ISBN 0444888403.
- ^ L. Ambrosio; et al. (2000). Calculus of variations and partial differential equations: topics on geometrical evolution problems and degree theory. Springer. p. 203. ISBN 3540648038.
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External link
- M. Insall. "Hyperfinite Set". MathWorld.