Nonstandard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.
Nonstandard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:
Robinson argued that this principle of Leibniz's is a precursor of the transfer principle. Robinson continued:
Robinson continues:
A nonzero element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a standard natural number. Ordered fields that have infinitesimal elements are also called nonArchimedean. More generally, nonstandard analysis is any form of mathematics that relies on nonstandard models and the transfer principle. A field which satisfies the transfer principle for real numbers is a hyperreal field, and nonstandard real analysis uses these fields as nonstandard models of the real numbers.
Robinson's original approach was based on these nonstandard models of the field of real numbers. His classic foundational book on the subject Nonstandard Analysis was published in 1966 and is still in print^{[2]}. On page 88, Robinson writes:
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.
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There are at least three reasons to consider nonstandard analysis:
Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity. As noted in the article on hyperreal numbers, these formulations were widely criticized by George Berkeley and others. It was a challenge to develop a consistent theory of analysis using infinitesimals and the first person to do this in a satisfactory way was Abraham Robinson.^{[1]}
In 1958 Curt Schmieden and Detlef Laugwitz published an Article "Eine Erweiterung der Infinitesimalrechnung"^{[3]}  "An Extension of Infinitesimal Calculus", which proposed a construction of a ring containing infinitesimals. The ring was constructed from sequences of real numbers. Two sequences were considered equivalent if they differed only in a finite number of elements. Arithmetic operations were defined elementwise. However, the ring constructed in this way contains zero divisors and thus cannot be a field.
Some educators maintain that the use of infinitesimals is more intuitive and more easily grasped by students than the socalled "epsilondelta" approach to analytic concepts. See H. Jerome Keisler's book.^{[4]} This approach can sometimes provide easier proofs of results which are somewhat tedious in epsilondelta formulation of analysis. For example, proving the chain rule for differentiation is easier in a nonstandard setting. Much of the simplification comes from applying very easy rules of nonstandard arithmetic, viz:
together with the transfer principle mentioned below. Critics of nonstandard analysis maintain that these simplifications are really illusory since they merely mask use of elementary epsilondelta arguments. They also contend epsilondelta argument is not more challenging than understanding the axioms of the hyperreal numbers and their construction.
Another pedagogical application of nonstandard analysis is Edward Nelson's treatment of the theory of stochastic processes, presented in his monograph Radically Elementary Probability Theory.^{[5]}
Some recent work has been done in analysis using concepts from nonstandard analysis, particularly in investigating limiting processes of statistics and mathematical physics. Albeverio et al.^{[6]} discuss some of these applications.
There are two very different approaches to nonstandard analysis: the semantic or modeltheoretic approach and the syntactic approach. Both these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.
The semantic approach is by far the most popular approach to nonstandard analysis. Robinson's original formulation of nonstandard analysis falls into this category. As developed by him in his papers, it is based on studying models (in particular saturated models) of a theory. Since Robinson's work first appeared, a simpler semantic approach (due to Elias Zakon) has been developed using purely settheoretic objects called superstructures. In this approach a model of a theory is replaced by an object called a superstructure V(S) over a set S. Starting from a superstructure V(S) one constructs another object *V(S) using the ultrapower construction together with a mapping V(S) → *V(S) which satisfies the transfer principle. The map * relates formal properties of V(S) and *V(S). Moreover it is possible to consider a simpler form of saturation called countable saturation. This simplified approach is also more suitable for use by mathematicians who are not specialists in model theory or logic.
The syntactic approach requires much less logic and model theory to understand and use. This approach was developed in the mid1970s by the mathematician Edward Nelson. Nelson introduced an entirely axiomatic formulation of nonstandard analysis that he called Internal Set Theory or IST.^{[7]} IST is an extension of ZermeloFraenkel set theory in that alongside the basic binary membership relation , it introduces a new unary predicate standard which can be applied to elements of the mathematical universe together with some axioms for reasoning with this new predicate.
Syntactic nonstandard analysis requires a great deal of care in applying the principle of set formation (formally known as the axiom of comprehension) which mathematicians usually take for granted. As Nelson points out, a common fallacy in reasoning in IST is that of illegal set formation. For instance, there is no set in IST whose elements are precisely the standard integers. To avoid illegal set formation, one must only use predicates of ZFC to define subsets.^{[7]}
Another example of the syntactic approach is the Alternative Set Theory^{[8]} introduced by Vopěnka, trying to find settheory axioms more compatible with the nonstandard analysis than the axioms of the ZF set theory.
Despite some initial hope in the mathematical community that nonstandard analysis would alter the way mathematicians thought about and reasoned with real numbers, this expectation never materialized. Moreover the list of new applications in mathematics is still very small. One of these results is the theorem proven by Abraham Robinson and Allen Bernstein that every polynomially compact linear operator on a Hilbert space has an invariant subspace^{[9]}. Upon reading a preprint of the BernsteinRobinson paper, Paul Halmos reinterpreted their proof using standard techniques^{[10]}. Both papers appeared backtoback in the same issue of the Pacific Journal of Mathematics. Some of the ideas used in Halmos' proof reappeared many years later in Halmos' own work on quasitriangular operators.
Other results are more along the line of reinterpreting or reproving previously known results. Of particular interest is Kamae's proof^{[11]} of the individual ergodic theorem or van den Dries and Wilkie's treatment^{[12]} of Gromov's theorem on groups of polynomial growth. NSA was used by Larry Manevitz and Shmuel Weinberger to prove a result in algebraic topology^{[13]}.
There are also applications of nonstandard analysis to the theory of stochastic processes, particularly constructions of Brownian motion as random walks. Albeverio etal^{[6]} have an excellent introduction to this area of research.
As an application to mathematical education, H. Jerome Keisler has written an elementary text^{[4]} on nonstandard calculus that develops differential and integral calculus using the hyperreal numbers, which include infinitesimal elements. These applications of nonstandard analysis depend on the existence of the standard part of a finite hyperreal r. The standard part of r, denoted st(r), is a standard real number infinitely close to r. One of the visualization devices Keisler uses is that of an imaginary infiniteresolution microscope to distinguish points infinitely close together. Keisler's book is now out of print, but is freely available from his website, see references below. Keisler's approach was harshly criticized by Errett Bishop.
Despite the elegance and appeal of some aspects of nonstandard analysis, there has been skepticism in the mathematical community about whether the nonstandard machinery adds anything that cannot easily be achieved by standard methods. These criticisms notwithstanding, however, there is no controversy about the mathematical validity of the approach and the results of nonstandard analysis. It is known that IST is a conservative extension of ZFC. This is shown in Edward Nelson's 1977 AMS Bulletin paper in an appendix written by William Powell.
Bishop's critique of NSA and of Keisler's elementary calculus book based on Robinson's theory is documented at Criticism of nonstandard analysis#Bishop's criticism.
Given any set S, the superstructure over a set S is the set V(S) defined by the conditions
Thus the superstructure over S is obtained by starting from S and iterating the operation of adjoining the power set of S and taking the union of the resulting sequence. The superstructure over the real numbers includes a wealth of mathematical structures: For instance, it contains isomorphic copies of all separable metric spaces and metrizable topological vector spaces. 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
which satisfies some additional properties.
To formulate these principles we first state some definitions: A formula has bounded quantification if and only if the only quantifiers which occur in the formula have range restricted over sets, that is are all of the form:
For example, the formula
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,
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:
One can show using ultraproducts 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 superset of N. The set *N − N is nonempty. To see this, apply countable saturation to the sequence of internal sets
The sequence {A_{n}}_{n ∈ N} has a nonempty intersection, proving the result.
We begin with some definitions: Hyperreals r, s are infinitely close if and only if
A hyperreal r is infinitesimal if and only if it is infinitely close to 0. r is limited or bounded if and only if its absolute value is dominated by (less than) 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 a hyperinteger, i.e. an element of *N − N, then 1/n 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.
Example: The plane (x,y) with x and y ranging over *R is internal, and is a model of plane Euclidean geometry. The plane with x and y restricted to bounded values (analogous to the Dehn plane) is external, and in this bounded plane the parallel postulate is violated. For example, any line passing through the point (0,1) on the yaxis and having infinitesimal slope is parallel to the xaxis.
Theorem. For any bounded hyperreal r there is a unique standard real denoted st(r) infinitely close to r. The mapping st is a ring homomorphism from the ring of bounded hyperreals to R.
The mapping st is also external.
One way of thinking of the standard part of a hyperreal, is in terms of Dedekind cuts; any bounded hyperreal s defines a cut by considering the pair of sets (L,U) where L is the set of standard rationals a less than s and U is the set of standard rationals b greater than s. The real number corresponding to (L,U) can be seen to satisfy the condition of being the standard part of s.
One intuitive characterization of continuity is as follows:
Theorem. A realvalued function f on the interval [a,b] is continuous if and only if for every hyperreal x in the interval *[a,b],
Similarly,
Theorem. A realvalued function f is differentiable at the real value x if and only if for every infinitesimal hyperreal number h, the value
exists and is independent of h. In this case f'(x) is a real number and is the derivative of f at x.
The following topics are of central importance and are discussed in the articles below.
The following articles are related:
