Hyperfinite set
Encyclopedia
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
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].
In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set.
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 
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a hyperfinite set or *-finite set is a type of internal set
Internal set
In mathematical logic, in particular in model theory and non-standard analysis, an internal set is a set that is a member of a model.Internal set is the key tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the...
. An internal set H of internal cardinality g ∈ *N (the hypernaturals) is hyperfinite if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
there exists an internal bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
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
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 infinitesimalInfinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
. 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
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
, considered as the set
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.
Ultrapower construction
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