菲氏微积分为什么说落后于时代的发展？

大家知道，菲氏微积分学教程完成于上世纪前半叶，而且是在前苏联的特定环境下完成的，拒绝接受西方的先进数学思想，有一定的历史局限性。

与此相反，J.Keisler精心撰写的无穷小微积分是基于创建于上世纪后半叶的现代数学基础（公理化数学与模型论）之上的微积分学教程，与菲氏微积分相比，显然是两个不同时代的基础数学教材，培养下一代接班人，不得不考虑这个前提。

口说无凭。下面是创建模型论早期的论文（部分），请注意论文发表时间，其中有不少塔尔斯基的学生。论文排序，按照作者氏字母顺序排列。

    请见本文附件。

袁萌  7月5日

附：

References（部分参考文献）

[Ax68] J. Ax, The elementary theory of finite fields. Ann. of Math. (2) 88 (1968), 239–271.

[AK65] J. Ax and S. Kochen, Diophantine problems over local fields. I. Amer. J. Math. 87 (1965) 605–630.

[BS] J.L. Bell and A.B. Slomson, Models and ultraproducts: An introduction. North-Holland Publishing Co., Amsterdam-London 1969

[BCR] J.Bochnak, M. Coste and M.-F. Roy, Real algebraic geometry. Translated from the 1987 French original. Revised by the authors. Ergebnisse der Mathematik und ihrer Grenzgebiete (3)   

[Results in Mathematics and Related Areas (3)], 36. Springer-Verlag, Berlin, 1998.

[BR66] A.R. Bernstein and A. Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific J. Math. 16 (1966) 421–431.

[Bor69] A. Borel, Injective endomorphisms of algebraic varieties. Arch. Math. (Basel) 20 (1969), 531–537.

[CK90] C.C. Chang and H.J. Keisler, Model theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73. North-Holland Publishing Co., Amsterdam, 1990.

[FT] P.L. Clark, Field Theory, lecture notes available at http://math.uga.edu/∼pete/FieldTheory.pdf [GT] P.L. Clark, General Topology, lectures notes available at http://math.uga.edu/∼pete/expositions.html [Hod] W. Hodges, A shorter model theory. Cambridge University Press, Cambridge, 1997.

[Lam] T.Y. Lam, Introduction to quadratic forms over fields. Graduate Studies in Mathematics, 67. American Mathematical Society, Providence, RI, 2005. [Lan52] S. Lang, On quasi algebraic closure. Ann. of Math. (2) 55, (1952). 373–390.

[Mar] D. Marker, Model theory. An introduction. Graduate Texts in Mathematics, 217. Springer-Verlag, New York, 2002.

[Mat] Ju. V. Matijasevic, The Diophantineness of enumerable sets. (Russian) Dokl. Akad. Nauk SSSR 191 1970 279–282.

[Poi] B. Poizat, A course in model theory. (English summary) An introduction to contemporary mathematical logic. Translated from the French by Moses Klein and revised by the author. Universitext. Springer-Verlag, New York, 2000.

[ARob55] A. Robinson, On ordered fields and definite functions. Math. Ann. 130 (1955), 275–271.

[JRob49] J. Robinson, Definability and decision problems in arithmetic. J. Symbolic Logic 14 (1949), 98–114.

[Rud] W. Rudin, Injective polynomial maps are automorphisms. Amer. Math. Monthly 102 (1995), no. 6, 540–543.

[Ser09] J.-P. Serre, How to use finite fields for problems concerning infinite fields. Arithmetic, geometry, cryptography and coding theory, 183–193, Contemp. Math., 487, Amer. Math. Soc., Provide……（省略）