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 hyperinteger ''n'' is a

hyperreal number In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer

n

that is equal to its own

integer part In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or eq ...

. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

. An example of an infinite hyperinteger is given by the class of the

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...

in 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 of the hyperreals.

Discussion

The standard integer part function: :

\lfloor x \rfloor

is defined for all real ''x'' and equals the greatest integer not exceeding ''x''. By the

transfer principle In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the firs ...

of nonstandard analysis, there exists a natural extension: :

^*\! \lfloor \,\cdot\, \rfloor

defined for all hyperreal ''x'', and we say that ''x'' is a hyperinteger if

x = ^*\! \lfloor x \rfloor.

Thus, the hyperintegers are the

image An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...

of the integer part function on the hyperreals.

Internal sets

The set

^*\mathbb

of all hyperintegers is an internal subset of the hyperreal line

^*\mathbb

. The set of all finite hyperintegers (i.e.

\mathbb

itself) is not an internal subset. Elements of the complement

^*\mathbb\setminus\mathbb

are called, depending on the author, ''nonstandard'', ''unlimited'', or ''infinite'' hyperintegers. The reciprocal of an infinite hyperinteger is always an

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 " ...

. Nonnegative hyperintegers are sometimes called ''hypernatural'' numbers. Similar remarks apply to the sets

\mathbb

and

^*\mathbb

. Note that the latter gives a

non-standard model of arithmetic In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements o ...

in the sense of Skolem.

References

* Howard Jerome Keisler: '' Elementary Calculus: An Infinitesimal Approach''. First edition 1976; 2nd edition 1986. This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at http://www.math.wisc.edu/~keisler/calc.html {{Infinitesimal navbox Nonstandard analysis Infinity Numbers