Non-standard analysis

In the most restricted sense, non-standard analysis is that 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 value is smaller than any element of F of the form 1/n, for n a natural number. 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.

Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. Robinson' original approach was based on so-called non-standard models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966. The book has been reissued in paperback by Princeton University Press (see reference below) and is widely available in popular bookstores.

There are a number of technical issues that must be addressed by a theory of analysis sufficiently powerful to allow development of infinitesimal calculus. For example, not every ordered field with infinitesimals is sufficiently rich to allow such a development. See the article on hyperreal numbers for a discussion of some of the relevant ideas.

 Contents

Motivation

There are at least three reasons to consider non-standard analysis:

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. As noted in the article on hyperreal numbers, these formulations were widely criticized by Bishop Berkeley and others. It was a challenge to develop a consistent theory of analysis using infinitesimals and it is arguable that the first person to solve this in a satisfactory way was Abraham Robinson, see reference below.

Pedagogical

Some authors maintain that use of infinitesimals is more intuitive and more easily grasped by students than the so-called "epsilon-delta" approach to analytic concepts. See Jerome Keisler's book referenced below. This approach can sometimes provide easier proofs of results which are somewhat tedious in epsilon-delta formulation of analysis. For example, proving the chain rule for differentiation is easier in a non-standard setting. Much of the simplification comes from applying very easy rules of nonstandard arithmetic, viz:

infinitesimal × bounded = infinitesimal
infinitesimal + infinitesimal = infinitesimal

together with the transfer principle mentioned below. Critics of non-standard analysis maintain that these simplifications are really illusory since they merely mask use of elementary epsilon-delta arguments. One stunning pedagogical application of non-standard analysis is Edward Nelson's treatment of the theory of stochastic processes, presented in his monograph Radically Elementary Probability Theory.

Though there is no scientific evidence either way on the pedagogical claim, the view that non-standard analysis simplifies teaching of calculus and is easier for students to grasp is still a minority view.

Technical

Some recent work has been done in analysis using concepts from non-standard analysis particularly in investigating limiting processes of statistics and mathematical physics. The Albeverio et-al reference below discusses some of these applications.

Approaches to non-standard analysis

There are two very different approaches to non-standard analysis: the semantic or model-theoretic approach and the syntactic approach. Note that 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 non-standard analysis. Robinson's original formulation of non-standard 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 Zakon) 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 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 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 non-standard analysis that he called Internal Set Theory or IST. IST is an extension of Zermelo-Fraenkel set theory in that alongside the basic binary membership relation [itex]\isin[itex], 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.

Despite its elegance and simplicity, syntactic non-standard 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.

Applications

Despite some initial hope in the mathematical community that non-standard 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. Upon reading a preprint of the Bernstein 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' proof reappeared many years later in Halmos' own work on quasi-triangular operators.

Other results are more along the line of reinterpreting or reproving previously known results. Of particular interest is Kamae's proof of the individual ergodic theorem or van den Dries and Wilkie's treatment of Gromov's theorem on groups of polynomial growth.

There are also applications of non-standard analysis to the theory of stochastic processes, particularly constructions of brownian motion as random walks. The Albeverio et-al reference below has an excellent introduction to this area of research.

Applications to calculus

As an application to mathematical education, H. Jerome Keisler has written a practical elementary text that develops differential and integral calculus using the hyperreal numbers, which, as we have seen has infinitesimal elements. These applications of non-standard analysis depend on the existence of the standard part of a limited hyperreal r. The standard part of r, denoted st r, is a standard real number infinitely close to r. Note that the standard part may not always be defined. In the following illustrative examples we will use the map * mentioned above which applies to sets, functions etc. Moreover, as is commonly the case, we assume that for real numbers r, *r is identical to r. This expresses the condition that R is considered to be embedded in *R. One of the expository devices Keisler uses is that of an imaginary infinite power microscope to distinguish points infinitely close together.

Criticisms

Despite the elegance and appeal of some aspects of non-standard analysis, there is a great deal of skepticism in the mathematical community about whether this machinery really adds anything that can not just as easily be achieved by standard methods. One noted critic of non-standard analysis is the Fields Medalist Alain Connes, as evinced by the following quote

The answer given by nonstandard analysis, a so-called nonstandard real, is equally deceiving. From every nonstandard real number one can construct canonically a subset of the interval [0, 1], which is not Lebesque measurable. No such set can be exhibited (Stern, 1985). This implies that not a single nonstandard real number can actually be exhibited.

A. Connes Noncommutative Geometry and Space-Time, Page 55 in The Geometric Universe, Huggett et al. The point of Connes' criticism is that nonstandard hyperreals are as fictitious as non-measurable sets. These sets can be shown to exist, assuming the axiom of choice of set theory, but are not constructible. Non-measurable sets are usually considered pathological, a sort of irritant that must be tolerated in order to have the axiom of choice available.

In his now famous book Non Commutative Geometry, Connes offers an alternative approach to infinitesimals based on ideals of compact operators on a Hilbert space. Note that in this treatment, the Dixmier trace plays a central role, but its definition is itself dependent on the choice of a free ultrafilter on the natural numbers, which is certainly nonconstructive.

These criticisms notwithstanding, however, there is absolutely no controversy about the mathematical validity of the approach and the results of non-standard analysis.

Logical framework

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

[itex]V_0(\mathbf{S}) = \mathbf{S} [itex]
[itex]V_{n+1}(\mathbf{S}) =V_{n}(\mathbf{S}) \cup

2^{V_{n}(\mathbf{S})}[itex]

[itex]V(\mathbf{S}) = \bigcup_{n \in \mathbb{N}} V_{n}(\mathbf{S})[itex]

Thus the superstructure over S is obtained by starting from S and iterating the operation of adjoining the powerset 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

[itex] *: V(\mathbb{R}) \rightarrow V(*\mathbb{R}) [itex]

which satisfies some additional properties.

Transfer map. Note V(R) and V(*R) are not actually disjoint

To formulate these principles we state first 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:

[itex] \forall x \in A, \Phi(x, \alpha_1, \ldots, \alpha_n) [itex]
[itex] \exists x \in A, \Phi(x, \alpha_1, \ldots, \alpha_n) [itex]

For example, the formula

[itex] \forall x \in A, \ \exists y \in 2^B, \ x \in y [itex]

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

[itex] \forall x \in A, \ \exists y, \ x \in y [itex]

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

A set x is internal iff is an element of *A for some element A of V(R). Note that *A itself is internal if A belongs to V(R).

We now formulate the basic logical framework of nonstandard analysis:

• Extension principle: The mapping * is the identity on R.
• Transfer principle: For any formula P(x1, ..., xn) with bounded quantification and with free variables x1, ..., xn, and for any elements A1, ..., An of V(R), the following equivalence holds:
[itex]P(A_1, \ldots, A_n) \iff P(*A_1, \ldots, *A_n) [itex]
• Countable saturation: If {Ak}k is a nondecreasing sequence of nonempty internal sets, then
[itex]\bigcap_k A_k \neq \emptyset [itex]

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

First consequences

The symbol *N denotes the nonstandard natural numbers. By the extension principle, this is a superset of N. The set *NN is nonempty. To see this, apply countable saturation to the sequence of internal sets

[itex] A_k = \{k \in *\mathbb{N}: k \geq n\} [itex]

The sequence {Ak}kN has a nonempty intersection, proving the result.

We begin with some definitions: Hyperreals r, s are infinitely close iff

[itex] r \cong s \iff \forall \theta \in \mathbb{R}^+, \ | r -s| \leq \theta[itex]

A hyperreal r is infinitesimal iff it is infinitely close to 0. r is limited or bounded iff 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 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.

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 real-valued function on the interval [a,b] is continuous iff for every hyperreal x in the interval *[a,b],

[itex] [*f](x) \cong [*f](\operatorname{st}x).\,[itex]

Similarly,

Theorem. A real-valued function f is differentiable at the real value x if and only if for every infinitesimal hyperreal number h, the value

[itex] f'(x)= \operatorname{st} \left(h^{-1}([*f](x+h) - [*f](x))\right) [itex]

exists and is independent of h. In this case f'(x) is a real number and is the derivative of f at x.

Related topics

The following topics are of central importance and are discussed in the articles below.

References

• Sergio Albeverio, Jans Erik Fenstad, Raphael Hoegh-Krohn, Tom Lindtrom:Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press 1986.
• P. Halmos,Invariant subspaces for Polynomially Compact Operators, Pacific Journal of Mathematics, Vol. 16, 1966.
• T. Kamae: A simple proof of the ergodic theorem using nonstandard analysis, Israel Journal of Mathematics vol. 42, Number 4, 1982.
• H. Jerome Keisler: An Infinitesimal Approach to Stochastic Analysis, vol. 297 of Memoirs of the American Mathematical Society, 1984.
• H. Jerome Keisler: Elementary Calculus: An Approach Using Infinitesimals. 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
• Edward Nelson: Internal Set Theory: A New Approach to Nonstandard Analysis, Bulletin of the American Mathematical Society, Vol. 83, Number 6, November 1977.
• Edward Nelson: Radically Elementary Probability Theory, Princeton University Press, 1987.
• Abraham Robinson: Non-standard Analysis, Princeton University Press, 1996.
• Allen Bernstein and Abraham Robinson: Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific Journal of Mathematics, Vol. 16, 1966
• L. van den Dries and A. J. Wilkie: Gromov's Theorem on Groups of Polynomial Growth and Elementary Logic, Journal of Algebra, Vol 89, 1984.

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