Category: Nonstandard analysis

Infinitely near point
In algebraic geometry, an infinitely near point of an algebraic surface S is a point on a surface obtained from S by repeatedly blowing up points. Infinitely near points of algebraic surfaces were int
Elementary Calculus: An Infinitesimal Approach
Elementary Calculus: An Infinitesimal approach is a textbook by H. Jerome Keisler. The subtitle alludes to the infinitesimal numbers of the hyperreal number system of Abraham Robinson and is sometimes
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 op
Ultrafilter (set theory)
In the mathematical field of set theory, an ultrafilter is a maximal proper filter: it is a filter on a given non-empty set which is a certain type of non-empty family of subsets of that is not equal
In nonstandard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") is a certain subset of namely a maximal filter on that is, a proper filter on that cannot be enl
Saturated model
In mathematical logic, and particularly in its subfield model theory, a saturated model M is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ul
Influence of nonstandard analysis
Abraham Robinson's theory of nonstandard analysis has been applied in a number of fields.
Loeb space
In mathematics, a Loeb space is a type of measure space introduced by Loeb using nonstandard analysis.
Standard part function
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal t
Hyperfinite set
In nonstandard 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 an
Law of continuity
The law of continuity is a heuristic principle introduced by Gottfried Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite
Hyperreal number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension o
Internal set theory
Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the nonstandard analysis introduced by Abraham Robinson. Instead
In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or S-continuity) of an internal function f at a point a is defined as follows: for all x infinitely close to a, the
Monad (nonstandard analysis)
In nonstandard analysis, a monad (also called halo) is the set of points infinitesimally close to a given point. Given a hyperreal number x in R∗, the monad of x is the set If x is finite (limited), t
Increment theorem
In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then for some infinitesima
Criticism of nonstandard analysis
Nonstandard analysis and its offshoot, nonstandard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.
Leibniz's notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinit
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
Smooth infinitesimal analysis
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all funct
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 direc
In nonstandard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard nat
Non-Archimedean ordered field
In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Examples are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the
Constructive nonstandard analysis
In mathematics, constructive nonstandard analysis is a version of Abraham Robinson's nonstandard analysis, developed by Moerdijk (1995), Palmgren (1998), Ruokolainen (2004). Ruokolainen wrote: The pos
Internal set
In mathematical logic, in particular in model theory and nonstandard analysis, an internal set is a set that is a member of a model. The concept of internal sets is a tool in formulating the transfer
Surreal number
In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value
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
Nonstandard calculus
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some argumen