Definitions

# Non-standard analysis

Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.

Non-standard analysis was introduced 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, G. W. 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. 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:

It is shown in this book that Leibniz's ideas can be fully vindicated and that 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.

A non-zero 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. 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.

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.

## 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.

In 1958 Curt Schmieden and Detlef Laugwitz published an Article "Eine Erweiterung der Infinitesimalrechnung" - "An Extension of Infinitesimal Calculus" (see reference below), 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

Some educators maintain that the use of infinitesimals is more intuitive and more easily grasped by students than the so-called "epsilon-delta" approach to analytic concepts. See H. 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 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.

### 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. 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 Elias 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 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 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 $isin$, 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. NSA was used by Larry Manevitz and Shmuel Weinberger to prove a result in algebraic topology (see below).

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 on non-standard calculus 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.

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.

## Connes' critique

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 cannot just as easily be achieved by standard methods. One noted critic of non-standard analysis is the Fields Medalist Alain Connes, as evidenced 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 Lebesgue 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", Huggett et al., The Geometric Universe, p. 55

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. 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. Moreover, Robinson notes on page 48 of the 1966 edition of his book that his theory does not require the axiom of choice but can also be based upon the ultrafilter lemma. Robinson infinitesimals can also be obtained using a free ultrafilter over the natural numbers.

These criticisms notwithstanding, however, there is absolutely no controversy about the mathematical validity of the approach and the results of non-standard analysis. In particular, the following two points should be kept in mind:

*IST is a conservative extension of ZFC. This is shown in Edward Nelson's AMS 1977 paper in an appendix written by William Powell;

* Model theoretic non-standard analysis. for example based on superstructures, which is now a commonly used approach, does not need any new set-theoretic axioms beyond those of ZFC.

## Bishop's critique

Bishop's critique of Keisler's book based on Robinson's theory is documented at Errett Bishop.

## Logical framework

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

$V_0\left(mathbf\left\{S\right\}\right) = mathbf\left\{S\right\}$

$V_\left\{n+1\right\}\left(mathbf\left\{S\right\}\right) =V_\left\{n\right\}\left(mathbf\left\{S\right\}\right) cup$
2^{V_{n}(mathbf{S})}

$V\left(mathbf\left\{S\right\}\right) = bigcup_\left\{n in mathbb\left\{N\right\}\right\} V_\left\{n\right\}\left(mathbf\left\{S\right\}\right).$

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

$*: V\left(mathbb\left\{R\right\}\right) rightarrow V\left(*mathbb\left\{R\right\}\right)$

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:

$forall x in A, Phi\left(x, alpha_1, ldots, alpha_n\right)$
$exists x in A, Phi\left(x, alpha_1, ldots, alpha_n\right)$

For example, the formula

$forall x in A, exists y in 2^B, x in y$
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,
$forall x in A, exists y, x in y$
does not have bounded quantification because the quantification of y is unrestricted.

## Internal sets

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:

• 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:

$P\left(A_1, ldots, A_n\right) iff P\left(*A_1, ldots, *A_n\right)$

• Countable saturation: If {Ak}k is a decreasing sequence of nonempty internal sets, with k ranging over the natural numbers, then

$bigcap_k A_k neq emptyset$

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

$A_k = \left\{k in *mathbb\left\{N\right\}: k geq n\right\}$

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

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

$r cong s iff forall theta in mathbb\left\{R\right\}^+, |r - s| leq theta$

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) 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 a hyperinteger, i.e. an element of *NN, 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 y-axis and having infinitesimal slope is parallel to the x-axis.

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 f on the interval [a,b] is continuous if and only if for every hyperreal x in the interval *[a,b],

$\left[*f\right]\left(x\right) cong \left[*f\right]\left(operatorname\left\{st\right\}\left(x\right)\right).,$

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

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

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:

## References

• Sergio Albeverio, Jans Erik Fenstad, Raphael Hoegh-Krohn, Tom Lindstrøm: Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press 1986.
• P. Halmos, Invariant subspaces for Polynomially Compact Operators, Pacific Journal of Mathematics, 16:3 (1966) 433-437.
• 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. A chapter on Internal Set Theory is available at http://www.math.princeton.edu/~nelson/books/1.pdf
• Edward Nelson: Radically Elementary Probability Theory, Princeton University Press, 1987, available as a pdf at http://www.math.princeton.edu/~nelson/books/rept.pdf
• Robinson, Abraham (1996). Non-standard analysis. Revised edition, Princeton University Press. ISBN 0-691-04490-2.
• Allen Bernstein and Abraham Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific Journal of Mathematics 16:3 (1966) 421-431
• Curt Schmieden and Detlef Laugwitz: Eine Erweiterung der Infinitesimalrechnung, Mathematische Zeitschrift 69 (1958), 1-39
• 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.
• Robert, Alain: Nonstandard analysis, Wiley, New York 1988. ISBN 0-471-91703-6
• Manevitz, Larry M.; Weinberger, Shmuel: Discrete circle actions: a note using non-standard analysis. Israel J. Math. 94 (1996), 147--155.

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