Non-standard calculus

Non-standard calculus
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Non-standard_calculus".

Gottfried Wilhelm Leibniz Inventor of infinitesimal calculus

In mathematics, non-standard calculus is the application of non-standard analysis techniques to differential and integral calculus. It provides a rigorous justification of purely formal calculations using infinitesimals (as envisioned by Leibniz) to derive facts about derivatives, integrals, and series. Such formal calculations with infinitesimals were widely used before alternative and rigorously justified methods, without infinitesimals, were introduced in the 19th century. See history of calculus. To quote H. Jerome Keisler:

"In '60, Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."

content

1 Motivation
2 Definition of derivative
3 Continuity
4 Uniform continuity
5 Heine-Cantor theorem
6 Why is the squaring function not uniformly continuous?
7 Dirichlet function
8 Limit
9 Limit of sequence
10 Basic theorems
11 Applications
12 Bishop vs Keisler
13 See also
14 References

Motivation

We illustrate this technique with the following naive calculation of the derivative of the function f(t)=t² at the value x:

$\frac{(x + \Delta x)^2 - x^2}{\Delta x} = 2 x + \Delta x \approx 2 x$

whenever the increment in x is "infinitesimal". Thus the derivative of f at x is 2x. The paradox involved in stripping away the term $Δ x$ is discussed at Ghosts of departed quantities.

This informal calculation is not itself an example of non-standard calculus, but it can be formalised and justified by the methods of non-standard analysis.

Definition of derivative

Naively speaking, non-standard analysis postulates the existence of positive numbers ε which are infinitely small, meaning that ε is smaller than any standard positive real, yet greater than zero. Every real number is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it. To define the derivative of f in this approach, one no longer needs an infinite limiting process as in standard calculus. Instead, one sets

$f'(x) = \mathrm{st} \left( \frac{f(x+\epsilon)-f(x)}{\epsilon} \right)$ ,

where st is the standard part function, yielding the standard real number infinitely close to the hyperreal argument of st. The addition of st to the formula resolves the centuries-old paradox already severely criticized by Bishop Berkeley (see Ghosts of departed quantities), and provides a rigorous basis to the approaches of both Isaac Newton and Gottfried Leibniz to infinitesimal calculus.

Continuity

A real function f is continuous at x if for every hyperreal x' infinitely close to x, the value f(x') is also infinitely close to f(x).

Here to be precise, f would have to be replaced by its natural hyperreal extension usually denoted f^* (see discussion of extension principle in main article at non-standard analysis).

Using the notation $\simeq$ for the relation of being infinitely close as above, the definition can be rewritten in an even shorter form as follows:

A function f is continuous at x if whenever $x'\simeq x$ , one has $f(x')\simeq f(x)$ .

The above requires fewer quantifiers than the (ε, δ)-definition familiar from standard elementary calculus:

f is continuous at x if for every ε>0, there exists a δ>0 such that whenever |x-x' | < δ, one has |ƒ(x) − ƒ(x' )| < ε.

Uniform continuity

A function f on an interval I is uniformly continuous if its natural extension f* in I* has the following property (see Keisler, Foundations of Infinitesimal Calculus ('07), p. 45):

for every pair of hyperreals x and y in I^*, if $x\simeq y$ then $f^*(x)\simeq f^*(y)$ .

This definition has a reduced quantifier complexity when compared with the standard (ε, δ)-definition, but has the same quantifier complexity as the definition of uniform continuity in terms of sequences in standard calculus.

Furthermore, the hyperreal definition as stated above is local in the sense that it only depends on the monad of each point in I*. Meanwhile, the standard (ε,δ)-definition is global in the sense that it is formulated in terms of pairs of points.

The localness of the hyperreal definition can be illustrated by the following three examples.

Example 1: a function f is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is continuous (in the sense of the formula above) at a positive infinitesimal, in addition to continuity at the standard points of the interval.

Example 2: a function f is uniformly continuous on the semi-open interval [0,∞) if and only if it is continuous at the standard points of the interval, and in addition, the natural extension f* is continuous at every positive infinite hyperreal point.

Example 3: similarly, the failure of uniform continuity for the squaring function

$x^2\,$

is due to the absence of continuity at a single infinite hyperreal point, see below.

Concerning quantifier complexity, the following remarks were made by Kevin Houston:

"The number of quantifiers in a mathematical statement gives a rough measure of the statement’s complexity. Statements involving three or more quantifiers can be difficult to understand. This is the main reason why it is hard to understand the rigorous definitions of limit, convergence, continuity and differentiability in analysis as they have many quantifiers. [new paragraph] In fact, it is the alternation of the \forall and \exists that causes the complexity." Kevin Houston, How to Think Like a Mathematician, ISBN 9780521719780

Heine-Cantor theorem

The fact that a continuous function on a compact interval I is necessarily uniformly continuous (the Heine–Cantor theorem) admits a succinct non-standard proof. Let x, y be hyperreals in the (natural extension of) I. Since I is bounded, both x and y admit standard parts. Since I is closed, st(x) and st(y) belong to I. If x and y are infinitely close, then by the triangle inequality, they have the same standard part

$c = \operatorname{st}(x) = \operatorname{st}(y).\,$

Since the function is assumed continuous at c, we have

$f(x)\simeq f(c)\simeq f(y),\,$

and therefore f(x) and f(y) are infinitely close.

Why is the squaring function not uniformly continuous?

Let f(x) = x² defined on $\mathbb{R}$ . Let $N\in \mathbb{R}^*$ be an infinite hyperreal. The hyperreal number $N + \tfrac{1}{N}$ is infinitely close to N. Meanwhile, the difference

$f(N+\tfrac{1}{N}) - f(N) = N^2 + 2 + \tfrac{1}{N^2} - N^2 = 2 + \tfrac{1}{N^2}$

is not infinitesimal. Therefore the squaring function is not uniformly continuous, according to the definition in uniform continuity.

A similar proof may be giving in the standard setting (Fitzpatrick 2006, Example 3.15).

Dirichlet function

Consider the Dirichlet function

$I_Q(x)=\begin{cases}1\, \mbox{ if }\, x\, \mbox{ is rational} \\0\, \mbox{ if }\, x\, \mbox{ is irrational} \end{cases}$ .

It is well-known that the function is discontinuous at every point. Let us check this in terms of the non-standard definition of continuity above, for instance let us show that the Dirichlet function is not continuous at π. Consider the continued fraction approximation a_n of π. Now let the index n be an infinite hyperinteger. By the transfer principle, the natural extension of the Dirichlet function takes the value 1 at a_n. Note that the hyperrational point a_n is infinitely close to π. Thus the natural extension of the Dirichlet function takes different values at these two infinitely close points, and therefore the Dirichlet function is not continuous at π.

Limit

While the thrust of Robinson's approach is that one can dispense with the limit-theoretic approach using multiple quantifiers, the notion of limit can be easily recaptured in terms of the standard part function st, namely

$\lim_{x\to a} f(x) = L\,$

if and only if whenever the difference x − a is infinitesimal, the difference ƒ(x) − L is infinitesimal, as well, or in formulas:

if st(x − a) = 0 then st(ƒ(x)) = L,

cf. (ε, δ)-definition of limit.

Limit of sequence

Given a sequence of real numbers $\{x_n|n\in \mathbb{N}\}\;$ , if $L\in \mathbb{R}\;$ we say L is the limit of the sequence and write

$L = \lim_{n \to \infty} x_n$

if for every nonstandard hyperinteger n, we have st(x_n)=L (here the extension principle is used to define x_n for every hyperinteger n).

This definition has no quantifier alternations.The standard (ε, δ)-style definition on the other hand does have quantifier alternations:

$L = \lim_{n \to \infty} x_n\Longleftrightarrow \forall \epsilon>0\;, \exists N \in \mathbb{N}\;, \forall n \in \mathbb{N} : n >N \rightarrow d(x_n,L)<\epsilon.\;$

Basic theorems

If f is a real valued function defined on an interval a, b, then the transfer operator applied to f, denoted by *f, is an internal, hyperreal-valued function defined on the hyperreal interval [*a, *b.

Theorem. Let f be a real-valued function defined on an interval a, b. Then f is differentiable at a < x < b if and only if for every non-zero infinitesimal h, the value

$\Delta_h f := \operatorname{st} \frac{[*f](x+h)-[*f](x)}{h}$

is independent of h. In that case, the common value is the derivative of f at x.

This fact follows from the transfer principle of non-standard analysis and overspill.

Note that a similar result holds for differentiability at the endpoints a, b provided the sign of the infinitesimal h is suitably restricted.

For the second theorem, we consider the Cauchy integral. This integral is defined as the limit, if it exists, of a directed family of Cauchy sums; these are sums of the form

$\sum_{k=0}^{n-1} f(\xi_k) (x_{k+1} - x_k)$

where

$a = x_0 \leq \xi_0 \leq x_1 \leq \ldots x_{n-1} \leq \xi_{n-1} \leq x_n = b.$

We will call such a sequence of values a Cauchy integral mesh and

$\sup_k (x_{k+1} - x_k)$

the width of the mesh. In the definition of the Cauchy integral, the limit of the Cauchy sums is taken as the width of the mesh goes to 0.

Theorem. Let f be a real-valued function defined on an interval a, b. Then f is Cauchy-integrable on a, b if and only if for every internal Cauchy integral mesh of infinitesimal width

$S_M = \operatorname{st} \sum_{k=0}^{n-1} [*f](\xi_k) (x_{k+1} - x_k)$

is independent of the mesh. In this case, the common value is the Cauchy integral of f over a, b.

Applications

One immediate application is an extension of the standard definitions of differentiation and integration to internal functions on intervals of hyperreal numbers.

An internal hyperreal-valued function f on a, b is S-differentiable at x, provided

$\Delta_h f = \operatorname{st} \frac{f(x+h)-f(x)}{h}$

exists and is independent of the infinitesimal h. The value is the S derivative at x.

Theorem. Suppose f is S-differentiable at every point of a, b where b − a is a bounded hyperreal. Suppose furthermore that

$|f'(x)| \leq M \quad a \leq x \leq b.$

Then for some infinitesimal ε

$|f(b) - f(a)| \leq M (b-a) + \epsilon.$

To prove this, let N be a non-standard natural number. Divide the interval a, b into N subintervals by placing N − 1 equally spaced intermediate points:

$a = x_0 < x_1< \cdots < x_{N-1} < x_N = b$

Then

$|f(b) - f(a)| \leq \sum_{k=1}^{N-1} |f(x_{k+1}) - f(x_{k})| \leq \sum_{k=1}^{N-1} \left\{|f'(x_k)| + \epsilon_k\right\}|x_{k+1} - x_{k}|.$

Now the maximum of any internal set of infinitesimals is infinitesimal. Thus all the ε_k's are dominated by an infinitesimal ε. Therefore,

$|f(b) - f(a)| \leq \sum_{k=1}^{N-1} (M + \epsilon)(x_{k+1} - x_{k}) = M(b-a) + \epsilon (b-a)$

from which the result follows.

Bishop vs Keisler

Bishop's critique of non-standard calculus is detailed at Bishop-Keisler controversy.

References

Fitzpatrick, Patrick (2006), Advanced Calculus, Brooks/Cole

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.)

H. Jerome Keisler: Foundations of Infinitesimal Calculus, available for downloading at http://www.math.wisc.edu/~keisler/foundations.html (10 jan '07)