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 limits rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.
Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. He wrote:
... the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latter.

Robinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:
However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.

Robinson continues:
... Leibniz's ideas can be fully vindicated and... they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.

In 1973, intuitionist Arend Heyting praised nonstandard analysis as "a standard model of important mathematical research".

Introduction

A non-zero element of an ordered field is infinitesimal if and only if its absolute value is smaller than any element of that is of the form, for a standard natural number. Ordered fields that have infinitesimal elements are also called non-Archimedean. More generally, nonstandard analysis is any form of mathematics that relies on nonstandard models and the transfer principle. A field that satisfies the transfer principle for real numbers is called a real closed 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. On page 88, Robinson writes:
The existence of nonstandard models of arithmetic was discovered by Thoralf Skolem. Skolem's method foreshadows the ultrapower construction

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 Hyperreal number for a discussion of some of the relevant ideas.

Basic definitions

In this section we outline one of the simplest approaches to defining a hyperreal field. Let be the field of real numbers, and let be the semiring of natural numbers. Denote by the set of sequences of real numbers. A field is defined as a suitable quotient of, as follows. Take a nonprincipal ultrafilter. In particular, contains the Fréchet filter. Consider a pair of sequences
We say that and are equivalent if they coincide on a set of indices that is a member of the ultrafilter, or in formulas:
The quotient of by the resulting equivalence relation is a hyperreal field, a situation summarized by the formula.

Motivation

There are at least three reasons to consider nonstandard analysis: historical, pedagogical, and technical.

Historical

Much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity. These formulations were widely criticized by George Berkeley and others. The challenge of developing a consistent and satisfactory theory of analysis using infinitesimals was first met by Abraham Robinson.
In 1958 Curt Schmieden and Detlef Laugwitz published an article "Eine Erweiterung der Infinitesimalrechnung" 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.

Pedagogical

, David Tall, and other educators maintain that the use of infinitesimals is more intuitive and more easily grasped by students than the "epsilon–delta" approach to analytic concepts. This approach can sometimes provide easier proofs of results than the corresponding epsilon–delta formulation of the proof. Much of the simplification comes from applying very easy rules of nonstandard arithmetic, as follows:
together with the transfer principle.
Another pedagogical application of nonstandard analysis is Edward Nelson's treatment of the theory of stochastic processes.

Technical

Some recent work has been done in analysis using concepts from nonstandard analysis, particularly in investigating limiting processes of statistics and mathematical physics. Sergio Albeverio et al. discuss some of these applications.

Approaches

There are two, main, different approaches to nonstandard analysis: the semantic or model-theoretic approach and the syntactic approach. Both of these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.
Robinson's original formulation of nonstandard analysis falls into the category of the semantic approach. As developed by him in his papers, it is based on studying models of a theory. Since Robinson's work first appeared, a simpler semantic approach has been developed using purely set-theoretic objects called superstructures. In this approach a model of a theory is replaced by an object called a superstructure over a set. Starting from a superstructure one constructs another object using the ultrapower construction together with a mapping that satisfies the transfer principle. The map * relates formal properties of and. 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 mid-1970s by the mathematician Edward Nelson. Nelson introduced an entirely axiomatic formulation of nonstandard analysis that he called internal set theory. IST is an extension of Zermelo–Fraenkel 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, which mathematicians usually take for granted. As Nelson points out, a 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.
Another example of the syntactic approach is the Vopěnka's alternative set theory, which tries to find set-theory axioms more compatible with the nonstandard analysis than the axioms of ZF.

Robinson's book

Abraham Robinson's book Non-standard Analysis was published in 1966. Some of the topics developed in the book were already present in his 1961 article by the same title. In addition to containing the first full treatment of nonstandard analysis, the book contains a detailed historical section where Robinson challenges some of the received opinions on the history of mathematics based on the pre–nonstandard analysis perception of infinitesimals as inconsistent entities. Thus, Robinson challenges the idea that Augustin-Louis Cauchy's "sum theorem" in Cours d'Analyse concerning the convergence of a series of continuous functions was incorrect, and proposes an infinitesimal-based interpretation of its hypothesis that results in a correct theorem.

Invariant subspace problem

Abraham Robinson and Allen Bernstein used nonstandard analysis to prove that every polynomially compact linear operator on a Hilbert space has an invariant subspace.
Given an operator on Hilbert space, consider the orbit of a point in under the iterates of. Applying Gram–Schmidt one obtains an orthonormal basis for. Let be the corresponding nested sequence of "coordinate" subspaces of. The matrix expressing with respect to is almost upper triangular, in the sense that the coefficients are the only nonzero sub-diagonal coefficients. Bernstein and Robinson show that if is polynomially compact, then there is a hyperfinite index such that the matrix coefficient is infinitesimal. Next, consider the subspace of. If in has finite norm, then is infinitely close to.
Now let be the operator acting on, where is the orthogonal projection to. Denote by the polynomial such that is compact. The subspace is internal of hyperfinite dimension. By transferring upper triangularisation of operators of finite-dimensional complex vector space, there is an internal orthonormal Hilbert space basis for where runs from to, such that each of the corresponding -dimensional subspaces is -invariant. Denote by the projection to the subspace. For a nonzero vector of finite norm in, one can assume that is nonzero, or to fix ideas. Since is a compact operator, is infinitely close to and therefore one has also. Now let be the greatest index such that. Then the space of all standard elements infinitely close to is the desired invariant subspace.
Upon reading a preprint of the Bernstein and Robinson paper, Paul Halmos reinterpreted their proof using standard techniques. Both papers appeared back-to-back in the same issue of the Pacific Journal of Mathematics. Some of the ideas used in Halmos's proof reappeared many years later in Halmos's own work on quasi-triangular operators.