Non-standard calculus


Non-standard calculus
Abraham Robinson

Contents

In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.

Calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter[citation needed], many mathematicians viewed infinitesimals as being naive and vague or meaningless.[1]

Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Jerome Keisler, "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."[2]

Motivation

To calculate the derivative f '\, of the function y = f(x) = x2 at x, both approaches agree on the algebraic manipulations:

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

This is a non-standard computation using the hyperreals if we interpret Δx as an infinitesimal and let the symbol "\approx" be the relation being infinitely close.

In order to make f ' a real-valued function, we must dispense with the final term Δx. In the standard approach using only real numbers, that is done by taking the limit as Δx tends to zero. In the non-standard approach using hyperreal numbers, the quantity Δx is taken to be an infinitesimal, a nonzero number that is closer to 0 than to any nonzero real. The manipulations displayed above then show that Δy / Δx is infinitely close to 2x, so the derivative of f at x is then 2x.

Discarding the "error term" is accomplished by an application of the standard part function. Dispensing with infinitesimal error terms was historically considered paradoxical by some writers, most notably George Berkeley.

Definition of derivative

The hyperreals can be constructed in the framework of ZFC, the standard axiomatisation of set theory used elsewhere in mathematics. To give an intuitive idea for the hyperreal approach, note that, 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 x is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it. To define the derivative of f at a standard real number x 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 real number infinitely close to the hyperreal argument of st, and f * is the natural extension of f to the hyperreals.

Continuity

A real function f is continuous at a standard real number x if for every hyperreal x' infinitely close to x, the value f(x' ) is also infinitely close to f(x). This captures Cauchy's definition of continuity.

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

Using the notation \approx 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'\approx x, one has f^*(x')\approx f^*(x)

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

f is continuous at x if for every ε > 0, there exists a δ > 0 such that for every x' , 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\approx y then f^*(x)\approx f^*(y).

This definition has a reduced quantifier complexity when compared with the standard (ε, δ)-definition. Namely, the epsilontic definition of uniform continuity requires four quantifiers, while the infinitesimal definition requires only two quantifiers. It has the same quantifier complexity as the definition of uniform continuity in terms of sequences in standard calculus, which however is not expressible in the first-order language of the real numbers.

Furthermore, the hyperreal definition as stated above is local[citation needed] 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.[citation needed]

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 every 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:[3]

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.

In fact, it is the alternation of the \forall and \exists that causes the complexity.

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)\approx f(c)\approx 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 above.

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

Example: Dirichlet function

Consider the Dirichlet function

I_Q(x)=\begin{cases} 1 & \text{ if }x \text{ is rational}, \\
0 & \text{ if } x \text{ 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 an of π. Now let the index n be an infinite hypernatural number. By the transfer principle, the natural extension of the Dirichlet function takes the value 1 at an. Note that the hyperrational point an 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  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 infinite hypernatural n, we have st(xn)=L (here the extension principle is used to define xn 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.\;

Extreme value theorem

To show that a real continuous function f on [0,1] has a maximum, let N be an infinite hyperinteger. The interval [0, 1] has a natural hyperreal extension. The function ƒ is also naturally extended to hyperreals between 0 and 1. Consider the partition of the hyperreal interval [0,1] into N subintervals of equal infinitesimal length 1/N, with partition points xi = i /N as i "runs" from 0 to N. In the standard setting (when N is finite), a point with the maximal value of ƒ can always be chosen among the N+1 points xi, by induction. Hence, by the transfer principle, there is a hyperinteger i0 such that 0 ≤ i0 ≤ N and f(x_{i_0})\geq f(x_i) for all i = 0, …, N (an alternative explanation is that every hyperfinite set admits a maximum). Consider the real point

c= {\rm st}(x_{i_0})

where st is the standard part function. An arbitrary real point x lies in a suitable sub-interval of the partition, namely x\in [x_i,x_{i+1}], so that st(xi) = x. Applying st to the inequality f(x_{i_0})\geq f(x_i), we obtain {\rm st}(f(x_{i_0}))\geq {\rm st}(f(x_i)). By continuity of ƒ we have

{\rm st}(f(x_{i_0}))= f({\rm st} (x_{i_0}))=f(c).

Hence ƒ(c) ≥ ƒ(x), for all x, proving c to be a maximum of the real function ƒ. See Keisler (1986, p. 164).

Intermediate value theorem

As another illustration of the power of Robinson's approach, we present a short proof of the intermediate value theorem (Bolzano's theorem) using infinitesimals.

Let f be a continuous function on [a,b] such that f(a)<0 while f(b)>0. Then there exists a point c in [a,b] such that f(c)=0.

The proof proceeds as follows. Let N be an infinite hyperinteger. Consider a partition of [a,b] into N intervals of equal length, with partition points xi as i runs from 0 to N. Consider the collection I of indices such that f(xi)>0. Let i0 be the least element in I (such an element exists by the transfer principle, as I is a hyperfinite set; see non-standard analysis). Then the real number

c=\mathrm{st}(x_{i_0})

is the desired zero of f. Such a proof reduces the quantifier complexity of a standard proof of the IVT.

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 Riemann integral. This integral is defined as the limit, if it exists, of a directed family of Riemann 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 partition or mesh and

sup k(xk + 1xk)

the width of the mesh. In the definition of the Riemann integral, the limit of the Riemann 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 Riemann-integrable on [a, b] if and only if for every internal mesh of infinitesimal width, the quantity

 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 Riemann 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 ba 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 εk'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.

See also

Notes

  1. ^ For example, Richard Courant described infinitesimals on page 81 of Differential and Integral Calculus, Vol I, as "devoid of any clear meaning" and "naive befogging". Similarly on page 101, Courant described them as "incompatible with the clarity of ideas demanded in mathematics", "entirely meaningless", "fog which hung round the foundations", and a "hazy idea".
  2. ^ Elementary Calculus: An Infinitesimal Approach
  3. ^ Kevin Houston, How to Think Like a Mathematician, ISBN 9780521719780

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)
  • The Origins of Cauchy's Rigorous Calculus, Judith Grabiner, Dover Publications (February 4, 2005), ISBN 0486438155

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