non-standard analysis


Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute valueMathworldPlanetmathPlanetmathPlanetmathPlanetmath is smaller than any element of F of the form 1n, for n a natural numberMathworldPlanetmath. Ordered fields that have infinitesimal elements are also called non-Archimedean. More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the transfer principle for real numbers is a hyperreal field, and non-standard real analysis uses these fields as non-standard models of the real numbers.

Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. Robinson’s original approach was based on these non-standard models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print.

Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article on hyperreal numbers for a discussion of some of the relevant ideas.

Given any set S, the superstructure over a set S is the set V⁒(S) defined by the conditions

V0⁒(𝐒)=𝐒
Vn+1⁒(𝐒)=Vn⁒(𝐒)βˆͺ2Vn⁒(𝐒)
V⁒(𝐒)=⋃nβˆˆβ„•Vn⁒(𝐒)

Thus the superstructure over S is obtained by starting from S and iterating the operationMathworldPlanetmath of adjoining the power setMathworldPlanetmath of S and taking the union of the resulting sequenceMathworldPlanetmath. The superstructure over the real numbers includes a wealth of mathematical structures: For instance, it contains isomorphicPlanetmathPlanetmathPlanetmath copies of all separablePlanetmathPlanetmath metric spaces and metrizable topological vector spacesMathworldPlanetmath. Virtually all of mathematics that interests an analyst goes on within V⁒(R).

The working view of nonstandard analysis is a set *R and a mapping

*:V(ℝ)β†’V(*ℝ)

which satisfies some additional properties. *ℝ is of course embedded in ℝ.

To formulate these principles we state first some definitions: A formulaMathworldPlanetmathPlanetmath has bounded quantification if and only if the only quantifiersMathworldPlanetmath which occur in the formula have range restricted over sets, that is are all of the form:

βˆ€x∈A,Φ⁒(x,Ξ±1,…,Ξ±n)
βˆƒx∈A,Φ⁒(x,Ξ±1,…,Ξ±n)

For example, the formula

βˆ€x∈A,βˆƒy∈2B,x∈y

has bounded quantification, the universally quantified variable x ranges over A, the existentially quantified variable y ranges over the powerset of B. On the other hand,

βˆ€x∈A,βˆƒy,x∈y

does not have bounded quantification because the quantification of y is unrestricted.

A set x is internal if and only if x is an element of *A for some element A of V⁒(R). *A itself is internal if A belongs to V⁒(R).

We now formulate the basic logical framework of nonstandard analysis: ExtensionPlanetmathPlanetmathPlanetmathPlanetmath principle: The mapping * is the identityPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on R.

Transfer principle: For any formula P⁒(x1,…,xn) with bounded quantification and with free variablesMathworldPlanetmathPlanetmath x1,…,xn, and for any elements A1,…,An of V⁒(R), the following equivalence holds: :

P(A1,…,An)⇔P(*A1,…,*An)

CountableMathworldPlanetmath saturation: If Akk is a decreasing sequence of nonempty internal sets, with k ranging over the natural numbers, then :

β‹‚kAkβ‰ βˆ…

One can show using ultraproductsMathworldPlanetmath that such a map * exists. Elements of V⁒(R) are called standard. Elements of *R are called hyperreal numbers.

The symbol *N denotes the nonstandard natural numbers. By the extension principle, this is a supersetMathworldPlanetmath of N. The set *N-N is not empty. To see this, apply countable saturation to the sequence of internal sets

Ak={k∈*β„•:kβ‰₯n}

The sequence Akk is in N has a non-empty intersectionMathworldPlanetmath, proving the result.

We begin with some definitions: Hyperreals r, s are infinitely close if and only if

rβ‰…sβ‡”βˆ€ΞΈβˆˆβ„+,|r-s|≀θ

A hyperreal r is infinitesimal if and only if it is infinitely close to 0. r is limited or boundedPlanetmathPlanetmath if and only if its absolute value is dominated by a standard integer. The bounded hyperreals form a subring of *R containing the reals. In this ring, the infinitesimal hyperreals are an ideal. For example, if n is an element of *N-N, then 1n is an infinitesimal.

The set of bounded hyperreals or the set of infinitesimal hyperreals are external subsets of V(*R); what this means in practice is that bounded quantification, where the bound is an internal set, never ranges over these sets.

This entry was adapted from the Wikipedia article http://en.wikipedia.org/wiki/Nonstandard_analysisNon-standard analysis as of December 19, 2006.

Title non-standard analysis
Canonical name NonstandardAnalysis
Date of creation 2013-03-22 16:28:26
Last modified on 2013-03-22 16:28:26
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 13
Author PrimeFan (13766)
Entry type Definition
Classification msc 03H05
Synonym nonstandard analysis