In
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 an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
there exists an internal
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
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
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 quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
. 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
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 factor ...
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 a ...
es of sequences
of real numbers ''u''
''n''. Namely, the equivalence class defines a hyperreal, denoted