nonstandard analysis The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...

, a branch of

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

there exists an internal

bijection In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

between ''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

K = \

with a hypernatural ''n''. ''K'' is a near interval for 'a'',''b''if ''k''₁ = ''a'' and ''k''_''n'' = ''b'', and if the difference between successive elements of ''K'' is

infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...

. 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 In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

, considered as the set

e^

for θ in the interval ,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 The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All fact ...

construction, the hyperreal line *R is defined as the collection of

equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...

es of sequences

\langle u_n, n=1,2,\ldots \rangle

of real numbers ''u''_''n''. Namely, the equivalence class defines a hyperreal, denoted

_n /math> in Goldblatt's notation.  Similarly, an arbitrary hyperfinite set in *R is of the form_n /math>, and is defined by a sequence \langle A_n \rangle of finite sets A_n \subseteq \mathbb, n=1,2,\ldots

References

External links

* {{DEFAULTSORT:Hyperfinite Set Nonstandard analysis