学习微积分60年有感（IV）
现在可以说，我这一辈子几乎都是在与微积分打交道，分不开。苦苦研习60年，人都快学傻了。现在，人老了，还整天唠叨不停，真烦人。
1957年秋，我进入南京大学数学天文系学习。当时，何旭初教授主讲一年级微积分课程，指定教学参考书是原苏联菲赫金哥尔茨的《微积分学教程》（共计三卷9大本），内容几乎涵盖了上世纪前半期世界数学分析的全部成就。
（读者请注意：在下面的英语句子中，有些空格被编辑器“吃”掉了。）
There is one fact of basic importance that we state now as a theorem.
THEOREM (Standard Part Principle)
For every finite hyperreal number b, there is exactly one realnumberr that is infinitely close to b.
PROOF We first show that there cannot be more than one realnumberinfinitely close to b.Suppose r and sare real numbers such that r ≈band s ≈ b. Then r ≈ s, and sincer and s are real, r must be equal to s .Thus there is at most one realnumber infinitely close to b.
We now show that there is a real number infinitely close to b. Let Abetheset of all real numbers less than b. Then any real number between twoelementsof Ais an element of A. By the Completeness Axiom for the realnumbers, Ais aninterval. Since the hyperreal number bis finite. Amust be aninterval of theform(－∝, r ) or (－∝, r ]for somereal number r. Every realnumber s belongs to A,so s < b. Also, every realnumber t > r does not belong to A, so t ≥ b. This showsthat ris infinitely close to b.
It was pointedoutearlier that the Completeness Axiom does not qualify as a real statement.Forthis reason, the Transfer Principle cannot be used to transfer theCompletenessAxiom to the hyperreal numbers. In fact, the Completeness Axiom isnot true forthe hyperreal numbers. By a closed hyperreal interval, we mean aset ofhyperreal numbers of the form [a, b], the set of all hyperreal numbersxforwhich a≤ x ≤ b, where aandbare hyperreal constants. Open and half-openhyperreal intervals are defined ina similar way. When we say that theCompleteness Axiom is not true for thehyperreal numbers, we mean that there actuallyare sets A of hyperreal numberssuch that:
(a) wheneverxandyare in A, any hyperreal number between xand yis in A.
(b) A is notahyperreal interval.
Here aretwoquitefamiliar examples.
EXAMPLE1 ThesetAof all infinitesimals has property (a ) above but is not a hyperrealinterval.It has property (a ) because any hyperreal number that is betweentwoinfinitesimals is itself infinitesimal. We show that Ais not ahyperrealinterval. Acannot be of the form [a, ∞] or (a,∞) becauseeveryinfinitesimal is less than 1. Acannot be of the form [a, b] or (a, b],becauseif bis positive infinitesimal, then 2·b is a largerinfinitesimal.Acannot be of the form [a,b) or (a,b), because if bis positive andnotinfinitesimal, then b/2 is less than b but still positive and notinfinitesimal.
The set B ofallfinite hyperreal numbers is another example of a set that has property (a)above but is not an interval.
Here aresomeexamples that may help to illustrate the nature of the hyperreal numbersystemand the use of the Transfer Axiom.
EXAMPLE2 let fbethe real function given by the equation.
Its graph istheunit semicircle with center at the origin. The following two realstatementshold for all real numbers x:
By theTransferAxiom, these real statements also hold for all hyperreal numbers x.Thereforethe natural extension f* of f is given by the sameequation
The domain off*isthe set of all hyperreal numbers between －1 and 1. Thehyperreal graph of f*, shownin Figure E.1, can be drawn on paper by drawingthe real graph of f(x)andtraining an infinitesimal microscope on certain keypoints.
EXAMPLE3 letfbethe identity function on the real numbers, f(x)=x. By the Transfer Axiom,theequation f(x)= x is true for all hyperreal x. Thus the natural extensionf*offis defined, and f*(x )=x for all hyperreal x. Figure E.2 shows thehyperrealgraph of f*. Under a microscope, it has a 45°slope.
Here is anexampleof a hyperreal function that is not the natural extension of a realfunction.
Figure E.1
Figure E.2
EXAMPLE4 One hyperreal function, which we have already studied insomedetail, is the standard part function st(x). For real numbers the standardpartfunction has the same values as the identity function,
st(x) = x forall real x.
However,thehyperreal graph of st(x), shown in Figure E.3, is very different fromthehyperreal graph of the identity function f*. The domain of the standardpartfunction is the set of all finite numbers, while f* has domain R*. Thusforinfinite x, f*(x) = x, but st(x)is undefined. If xis finite but not real,f*(x)=x but st(x)≠ x. Under the microscope, aninfinitesimal piece of the graphof the standard part function is horizontal,while the identity function has a45° slope.
The standardpartfunction is not the natural extension of the identity function, and henceisnot the natural extension of any real function.
Figure E.3
The standardpartfunction is the only hyperreal function used in this course except for naturalextensionsof real functions.
We conclude withafew words about the construction of the real and the hyperreal numbers.BeforeWeierstrass, the rational numbers were on solid ground but the realnumbersweresomething new. Before one could safely use the axioms for the realnumbers, ithad to be shown that the axioms did not lead to a contradiction.This was doneby starting with the rational numbers and constructing astructure whichsatisfied all the axioms for the real numbers. Since anythingproved from theaxioms is true in this structure, the axioms cannot lead to acontradiction.
The idea istoconstruct real numbers out of Cauchy sequences of rational numbers.
DEFINITION
A Cauchy Sequence is a sequence <________> of numberssuchthat for every real ε > 0 there is an integer ____ such that thenumbers
are all within ε of each other.
Two Cauchy sequences
of rational numbers are called Cauchy equivalent, in symbols<_____>=<____>, if the differencesequence
converges to zero. (Intuitively this means that the two sequences havethesame limit.)
PROPERTIES OF CAUCHY EQUIVALENCE
(1)If <_______> =<______> and<______> = <_______>
then the sum sequences are equivalent.
(2)Under the same hypotheses, the product sequences are equivalent,
(3)if__ = __ for all but finitely many n, then
The set of real numbers is then defined as the set of all equivalenceclassesof Cauchy sequences of rational numbers. A rational number rcorresponds to theequivalence class of the constant sequence < r, r,r,…>. The sum of theequivalence class of <___ > and theequivalence class of <___ > is defined as the equivalence class of thesum sequence
The product is defined in a similar way. It can be shown that alltheaxioms for the real numbers hold for this structure.
Today the real numbers are on solid ground and the hyperreal numbersare anew idea. Robinson used the ultraproduct construction of Skolem to showthat theaxioms for the hyperreal numbers (for example, as used in this book)do not leadto a contradiction. The method is much like the construction of thereal numbersfrom the rationals. But this time the real number system is thestarting point.We construct hyperrealnumbers out of arbitrary (not justCauchy) sequences ofreal numbers.
By an ultraproduct equivalence we mean an equivalence relation= on thesetof all sequences of real numbers which have the properties of Cauchyequivalence(1)-(3) and also
(4)If each __belongs to the set {0,1} the <__>isequivalent to exactly one of the constant sequences <0,0,0,…>or<1,1,1,…>.
Given an ultraproduct equivalence relation, the set of hyperrealnumbers isdefined as the set of all equivalence classes of sequences of realnumbers. Areal number rcorresponds to the equivalence class of the constantsequence<r, r,r…>. Sums and products are defined as forCauchysequences. The natural extension f*of a real function f(x)is defined sothatthe image of the equivalence class of <__> istheequivalence class of <___>. It can be provedthatultraproduct equivalence relations exist, and that all the axioms for therealand hyperreal numbers hold for the structure defined in this way.
When hyperreal numbers are constructed as equivalence classes ofsequencesof real numbers, we can give specific examples of infinite hyperrealnumbers.The equivalence class of
<1,2,3,……n,……>
is a positive infinite hyperreal number. The equivalence class of
<1,4,9……n2……>
is larger, and the equivalence class of
<1,2,4……__……>
is a still larger infinite hyperreal number.
We can also give examples of nonzero infinitesimals. Theequivalenceclasses of
<1, 1/2,1/3……1/n, ……>,
<1,1/4,1/9……n–2,……>,
and <1,1/2,1/4……___________,……>,
are progressively smaller positive infinitesimals.
The mistake of Leibniz and his contemporaries was to identify alltheinfinitesimals with zero. This leads to an immediate contradictionbecausedy/dxbecomes 0/0. In the present treatment the equivalence classes of
<1, 1/2,1/3, ……, 1/n, ……>
and <0, 0, ……0, ……>
are different hyperreal numbers. They are not equal but merely havethesame standard part, zero. This avoids the contradiction and once againmakesinfinitesimals a mathematically sound method.
For more information about the ideas touched on in this epilogue, see theinstructor’s supplement, Foundations of Infinitesimal Calculus,which has as elf-containedtreatment of ultraproducts and the hyperreal numbers.
FOR FURTHER READING ON THE HISTORY OF THE CALCULUS SEE:
The History of the Calculus and its Conceptual Development: Carl c.Boyer,Dover, New York, 1959.
Mathematical Thought from Ancient to Modern Times; Morris Kline,OxfordUniv.
Press, New York, 1972.
No-standard Analysis: Abraham Robinson, North-Holland, Amsterdam,London,1966.
FOR ADVANCED READING ON INFINITESIMAL ANALYSIS SEE NO-STANDARD ANALYSISBYABRAHAM ROBINSON AND:
Lectures on Non-standard Analysis; M. Machover and J.Hirschfeld,Springer-Verlag, Berlin, Heidelberg, New York, 1969.
Victoria Symposium on Nonstandard Analysis; A. Hurd and p.Loeb,Springer-Verlag, Berlin, Heidelberg, New York, 1973.
Studies in Model Theory; M. Morley, Editor, Mathematical AssociationofAmerica, Providence, 1973.
Applied Nonstandard Analysis: M. Davis, Wiley, New York, 1977.
Introduction to the Theory of Infinitesimals: K.D Stroyan and W.A.J.Luxemburg, Academic Press, New York and London, 1976.
Foundations ofInfinitesimal Stochastic Analysis: K.D.（全文完）
1957年秋，我进入南京大学数学天文系学习。当时，何旭初教授主讲一年级微积分课程，指定教学参考书是原苏联菲赫金哥尔茨的《微积分学教程》（共计三卷9大本），内容几乎涵盖了上世纪前半期世界数学分析的全部成就。
在此期间，对基于（ε，δ）方法的极限论，顶礼膜拜，视为神圣，从不怀疑。
在20年之后，1978年，全国改革开放，轰轰烈烈。我看准了Keisler的无穷小微积分，观点全变了。普及无穷小微积分，而且坚持不断，矢志不移。这样，又一路走过了人生的40年。
袁萌 12月31日
（读者请注意：在下面的英语句子中，有些空格被编辑器“吃”掉了。）
There is one fact of basic importance that we state now as a theorem.
THEOREM (Standard Part Principle)
For every finite hyperreal number b, there is exactly one realnumberr that is infinitely close to b.
PROOF We first show that there cannot be more than one realnumberinfinitely close to b.Suppose r and sare real numbers such that r ≈band s ≈ b. Then r ≈ s, and sincer and s are real, r must be equal to s .Thus there is at most one realnumber infinitely close to b.
We now show that there is a real number infinitely close to b. Let Abetheset of all real numbers less than b. Then any real number between twoelementsof Ais an element of A. By the Completeness Axiom for the realnumbers, Ais aninterval. Since the hyperreal number bis finite. Amust be aninterval of theform(－∝, r ) or (－∝, r ]for somereal number r. Every realnumber s belongs to A,so s < b. Also, every realnumber t > r does not belong to A, so t ≥ b. This showsthat ris infinitely close to b.
It was pointedoutearlier that the Completeness Axiom does not qualify as a real statement.Forthis reason, the Transfer Principle cannot be used to transfer theCompletenessAxiom to the hyperreal numbers. In fact, the Completeness Axiom isnot true forthe hyperreal numbers. By a closed hyperreal interval, we mean aset ofhyperreal numbers of the form [a, b], the set of all hyperreal numbersxforwhich a≤ x ≤ b, where aandbare hyperreal constants. Open and half-openhyperreal intervals are defined ina similar way. When we say that theCompleteness Axiom is not true for thehyperreal numbers, we mean that there actuallyare sets A of hyperreal numberssuch that:
(a) wheneverxandyare in A, any hyperreal number between xand yis in A.
(b) A is notahyperreal interval.
Here aretwoquitefamiliar examples.
EXAMPLE1 ThesetAof all infinitesimals has property (a ) above but is not a hyperrealinterval.It has property (a ) because any hyperreal number that is betweentwoinfinitesimals is itself infinitesimal. We show that Ais not ahyperrealinterval. Acannot be of the form [a, ∞] or (a,∞) becauseeveryinfinitesimal is less than 1. Acannot be of the form [a, b] or (a, b],becauseif bis positive infinitesimal, then 2·b is a largerinfinitesimal.Acannot be of the form [a,b) or (a,b), because if bis positive andnotinfinitesimal, then b/2 is less than b but still positive and notinfinitesimal.
The set B ofallfinite hyperreal numbers is another example of a set that has property (a)above but is not an interval.
Here aresomeexamples that may help to illustrate the nature of the hyperreal numbersystemand the use of the Transfer Axiom.
EXAMPLE2 let fbethe real function given by the equation.
Its graph istheunit semicircle with center at the origin. The following two realstatementshold for all real numbers x:
By theTransferAxiom, these real statements also hold for all hyperreal numbers x.Thereforethe natural extension f* of f is given by the sameequation
The domain off*isthe set of all hyperreal numbers between －1 and 1. Thehyperreal graph of f*, shownin Figure E.1, can be drawn on paper by drawingthe real graph of f(x)andtraining an infinitesimal microscope on certain keypoints.
EXAMPLE3 letfbethe identity function on the real numbers, f(x)=x. By the Transfer Axiom,theequation f(x)= x is true for all hyperreal x. Thus the natural extensionf*offis defined, and f*(x )=x for all hyperreal x. Figure E.2 shows thehyperrealgraph of f*. Under a microscope, it has a 45°slope.
Here is anexampleof a hyperreal function that is not the natural extension of a realfunction.
Figure E.1
Figure E.2
EXAMPLE4 One hyperreal function, which we have already studied insomedetail, is the standard part function st(x). For real numbers the standardpartfunction has the same values as the identity function,
st(x) = x forall real x.
However,thehyperreal graph of st(x), shown in Figure E.3, is very different fromthehyperreal graph of the identity function f*. The domain of the standardpartfunction is the set of all finite numbers, while f* has domain R*. Thusforinfinite x, f*(x) = x, but st(x)is undefined. If xis finite but not real,f*(x)=x but st(x)≠ x. Under the microscope, aninfinitesimal piece of the graphof the standard part function is horizontal,while the identity function has a45° slope.
The standardpartfunction is not the natural extension of the identity function, and henceisnot the natural extension of any real function.
Figure E.3
The standardpartfunction is the only hyperreal function used in this course except for naturalextensionsof real functions.
We conclude withafew words about the construction of the real and the hyperreal numbers.BeforeWeierstrass, the rational numbers were on solid ground but the realnumbersweresomething new. Before one could safely use the axioms for the realnumbers, ithad to be shown that the axioms did not lead to a contradiction.This was doneby starting with the rational numbers and constructing astructure whichsatisfied all the axioms for the real numbers. Since anythingproved from theaxioms is true in this structure, the axioms cannot lead to acontradiction.
The idea istoconstruct real numbers out of Cauchy sequences of rational numbers.
DEFINITION
A Cauchy Sequence is a sequence <________> of numberssuchthat for every real ε > 0 there is an integer ____ such that thenumbers
are all within ε of each other.
Two Cauchy sequences
of rational numbers are called Cauchy equivalent, in symbols<_____>=<____>, if the differencesequence
converges to zero. (Intuitively this means that the two sequences havethesame limit.)
PROPERTIES OF CAUCHY EQUIVALENCE
(1)If <_______> =<______> and<______> = <_______>
then the sum sequences are equivalent.
(2)Under the same hypotheses, the product sequences are equivalent,
(3)if__ = __ for all but finitely many n, then
The set of real numbers is then defined as the set of all equivalenceclassesof Cauchy sequences of rational numbers. A rational number rcorresponds to theequivalence class of the constant sequence < r, r,r,…>. The sum of theequivalence class of <___ > and theequivalence class of <___ > is defined as the equivalence class of thesum sequence
The product is defined in a similar way. It can be shown that alltheaxioms for the real numbers hold for this structure.
Today the real numbers are on solid ground and the hyperreal numbersare anew idea. Robinson used the ultraproduct construction of Skolem to showthat theaxioms for the hyperreal numbers (for example, as used in this book)do not leadto a contradiction. The method is much like the construction of thereal numbersfrom the rationals. But this time the real number system is thestarting point.We construct hyperrealnumbers out of arbitrary (not justCauchy) sequences ofreal numbers.
By an ultraproduct equivalence we mean an equivalence relation= on thesetof all sequences of real numbers which have the properties of Cauchyequivalence(1)-(3) and also
(4)If each __belongs to the set {0,1} the <__>isequivalent to exactly one of the constant sequences <0,0,0,…>or<1,1,1,…>.
Given an ultraproduct equivalence relation, the set of hyperrealnumbers isdefined as the set of all equivalence classes of sequences of realnumbers. Areal number rcorresponds to the equivalence class of the constantsequence<r, r,r…>. Sums and products are defined as forCauchysequences. The natural extension f*of a real function f(x)is defined sothatthe image of the equivalence class of <__> istheequivalence class of <___>. It can be provedthatultraproduct equivalence relations exist, and that all the axioms for therealand hyperreal numbers hold for the structure defined in this way.
When hyperreal numbers are constructed as equivalence classes ofsequencesof real numbers, we can give specific examples of infinite hyperrealnumbers.The equivalence class of
<1,2,3,……n,……>
is a positive infinite hyperreal number. The equivalence class of
<1,4,9……n2……>
is larger, and the equivalence class of
<1,2,4……__……>
is a still larger infinite hyperreal number.
We can also give examples of nonzero infinitesimals. Theequivalenceclasses of
<1, 1/2,1/3……1/n, ……>,
<1,1/4,1/9……n–2,……>,
and <1,1/2,1/4……___________,……>,
are progressively smaller positive infinitesimals.
The mistake of Leibniz and his contemporaries was to identify alltheinfinitesimals with zero. This leads to an immediate contradictionbecausedy/dxbecomes 0/0. In the present treatment the equivalence classes of
<1, 1/2,1/3, ……, 1/n, ……>
and <0, 0, ……0, ……>
are different hyperreal numbers. They are not equal but merely havethesame standard part, zero. This avoids the contradiction and once againmakesinfinitesimals a mathematically sound method.
For more information about the ideas touched on in this epilogue, see theinstructor’s supplement, Foundations of Infinitesimal Calculus,which has as elf-containedtreatment of ultraproducts and the hyperreal numbers.
FOR FURTHER READING ON THE HISTORY OF THE CALCULUS SEE:
The History of the Calculus and its Conceptual Development: Carl c.Boyer,Dover, New York, 1959.
Mathematical Thought from Ancient to Modern Times; Morris Kline,OxfordUniv.
Press, New York, 1972.
No-standard Analysis: Abraham Robinson, North-Holland, Amsterdam,London,1966.
FOR ADVANCED READING ON INFINITESIMAL ANALYSIS SEE NO-STANDARD ANALYSISBYABRAHAM ROBINSON AND:
Lectures on Non-standard Analysis; M. Machover and J.Hirschfeld,Springer-Verlag, Berlin, Heidelberg, New York, 1969.
Victoria Symposium on Nonstandard Analysis; A. Hurd and p.Loeb,Springer-Verlag, Berlin, Heidelberg, New York, 1973.
Studies in Model Theory; M. Morley, Editor, Mathematical AssociationofAmerica, Providence, 1973.
Applied Nonstandard Analysis: M. Davis, Wiley, New York, 1977.
Introduction to the Theory of Infinitesimals: K.D Stroyan and W.A.J.Luxemburg, Academic Press, New York and London, 1976.
Foundations ofInfinitesimal Stochastic Analysis: K.D.（全文完）