微积分手机版，核心理论是什么？

当前，微积分手机版即将大规模投递出去，微积分手机版的核心理论是否正确？柯西序列等价类理论是关键。

实际上，承认实数是柯西序列等价类的定义，那么，在数学的逻辑上，就必须接受超实数的柯西序列等价类的定义。1978年，袁萌的有关论文说明了这个“数学真理”。

我与陈启清深信，新一届国家教育指导委员会，在理性思考上，不会拒绝微积分手机版。

袁萌   陈启清   8月15日

附：微积分手机版的核心理论（摘抄自该手机版结束语的相关段落原文）

DEFINITION

A Cauchy Sequence is a sequence (a1, a2 , •• • ) of numbers such that for every real B > 0 there is an integer n, such that the numbers  are all within B of each other.

Two Cauchy sequences

(a1, az, .. . ), of rational numbers are called Cauchy equivalent, in symbols (a 1 , a2 , .• . ) = (b 1 , b2 , •.. ), if the difference sequence (a1 - b1 ,a2 - b2 , ... )

converges to zero. (Intuitively this means that the two sequences have the same limit.)

PROPERTIES OF CAUCHY EQUIVALENCE

(1) If(a1 ,a2 , ... ) = (a'1 ,a~, ... ) and (b1 ,b2 , •.. ) = (b~,b~, ... ) then the sum sequences are equivalent,

912 EPILOGUE （结束语页码）

(a1 + b1 ,a2 + b2 , •.. ) = (a'1 + b'1 ,a~ + b~, ... ).

(2) Under the same hypotheses, the product sequences are equivalent,

(at•b 1 .a2·b2, ... ) = (a'1 ·b'1 ,a~·b~, ... ).

(3) If a .. = b .. for all but finitely many n, then

(a1 ,a2 , ... ) = (b1 ,b2 , ... ).

The set of real numbers is then defined as the set of all equivalence classes of Cauchy sequences of rational numbers. （实数的柯西定义）A rational number r corresponds to the equivalence class of the constant sequence (r, r, r, .. . ). The sum of the equivalence class of (a 1 , a2 , •. • ) and the equivalence class of (b1 , b2 , .• • ) is defined as the equivalence class of the sum sequence (a1 + b1 ,a2 + b2 , ..• ).

The product is defined in a similar way. It can be shown that all the axioms for the real numbers hold for this structure. （今天）Today the real numbers are on solid ground and the hyperreal numbers are a new idea.（新思想） Robinson used the ultraproduct（超乘机） construction of Skolem to show that the axioms for the hyperreal numbers (for example, as used in this book) do not lead to a contradiction.

The method is much like the construction of the real numbers from the rationals.（见袁萌论文） But this time the real number system is the starting point. We construct hype1-real numbers out of arbitrary (not just Cauchy) sequences of real numbers. By an ultraproduct equivalence we mean an equivalence relation = on the set of all sequences of real numbers which have the properties of Cauchy equivalence (1 }-(3) and also

(4) If each a .. belongs to the set {0, 1} then (a1 ,a2 , ... ) is equivalent to exactly one of the constant sequences (0, 0, 0, ... ) or (1, 1, 1, ... ).

Given an ultraproduct equivalence relation, the set of hyperreal numbers is defined as the set of all equivalence classes of sequences of real numbers.（超实数的等价类定义） A real number r corresponds to the equivalence class of the constant sequence (r, r, r, .. . ). Sums and products are defined as for Cauchy sequences. The natural extension f* of a real function f(x) is defined so that the image of the equivalence class of (a1 , a2 , .. • ) is the equivalence class of (f(a1),f(a2), .. • ). It can be proved that ultraproduct equivalence relations exist, and that all the axioms for the real and hyperreal numbers hold for the structure defined in this way. When hyperreal numbers are constructed as equivalence classes of sequences of real numbers, we can give specific examples of infinite hyperreal numbers. The equivalence class of  (1, 2, 3, .... 11, ... )  is a positive infinite hyperreal number. The equivalence class of  (1, 4, 9, ... , 172, ... )

is larger, and the equivalence class of

(1,2,4, ... ,2", ... )  is a still larger infinite hyperreal number.

EPILOGUE 913 （结束语页码

We can also give examples of nonzero infinitesimals. The equivalence classes

of  1, 1/2, 1/3, ... , 1/n, .. . ), (1, 1/4, 1/9, ... , n-2, .. . ),  and (1, 1/2, 1/4, ... ,2-", ... )  are progressively smaller positive infinitesimals. The mistake of （妙也）Leibniz and his contemporaries was to identify all the infinitesimals with zero. This leads to an immediate contradiction because dyjdx becomes 0/0. In the present treatment the equivalence classes of  (1, 1/2, 1/3, ... , 1/n, .. . ) and (0, 0, 0, ... '0, ... )  are different hyperreal numbers. They are not equal but merely have the same standard part, zero. This avoids the contradiction and once again makes infinitesimals a mathematically sound method（微积分手机版的理论基础就在此！）. For more information about the ideas touched on in this epilogue, see the instructor's supplement, Foundations of Infinitesimal Calculus,（无穷小微积分网站提供此书的下载） which has a self-contained treatment of ultra products 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 fi"om Ancient to Modern Times; Morris Kline, Oxford Univ. Press, New York, 1972. Non-standard Analysis; Abraham Robinson, North-Holland, Amsterdam, London, 1966.

FOR ADVANCED READING ON INFINITESIMAL ANALYSIS SEE NON-STANDARD ANALYSIS BY ABRAHAM ROBINSON AND:

Lectures on Non-standard Analysis; M. Machover and J. Hirschfeld, Springer-Verlag, Berlin, Heidelberg, New York, 1969.

Victoria Symposiwn on Nonstandard Analysis; A. Hurd and P. Loeb, Springer-Verlag, Berlin, Heidelberg, New York, 1973.

Studies in Model Theory; M. Morley, Editor, Mathematical Association of America, 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 of Infinitesimal Stochastic Analysis: K. D. Stroyan and J. M. Bayod, N

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