无穷小微积分植根于现代模型论沃土之中

当前，国内有人认为，无穷小微积分是邪门怪盗，不是正统数学。这是不正确的。

在国外，无穷小微积分（即鲁宾逊无穷小分析）被认定为，是模型论的一个数学重要分支，鲁宾逊与塔尔斯基、哥德尔齐名，堂堂正正，不是邪门怪盗。

今年9月，大批00后青年学子即将进入大学学习微积分基础课。选择基于极限轮的菲氏微积分，还是选择基于现代模型论的无穷小微积分是一个不可回避的现实问题，不能误人子弟。

关于现代模型论的数学分株，请见本文附件。

袁萌  7月7日

附：Branches of model theory

This article focuses on finitary first order model theory of infinite structures. Finite model theory, which concentrates on finite structures, diverges significantly from the study of infinite structures in both the problems studied and the techniques used. Model theory in higher-order logics or infinitary logics is hampered by the fact that completeness and compactness do not in general hold for these logics. However, a great deal of study has also been done in such logics.

         Informally, model theory can be divided into classical model theory, model theory applied to groups and fields, and geometric model theory. A missing subdivision is computable model theory, but this can arguably be viewed as an independent subfield of logic.

（请读者注意这段话）Examples of early theorems from classical model theory include （哥德尔）Gödel's completeness theorem, the upward and downward Löwenheim–Skolem theorems, Vaught's two-cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem. Examples of early results from model theory applied to fields are （塔尔斯基）Tarski's elimination of quantifiers for real closed fields, Ax's theorem on pseudo-finite fields, and （鲁宾逊）Robinson's development of non-standard analysis. （注意对其评论）An important step in the evolution of classical model theory occurred with the birth of stability theory (through Morley's theorem on uncountably categorical theories and Shelah's classification program), which developed a calculus of independence and rank based on syntactical conditions satisfied by theories.

During the last several decades applied model theory has repeatedly merged with the more pure stability theory. The result of this synthesis is called geometric model theory in this article (which is taken to include o-minimality, for example, as well as classical geometric stability theory). An example of a theorem from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields. The ambition of geometric model theory is to provide a geography of mathematics by embarking on a detailed study of definable sets in various mathematical structures, aided by the substantial tools developed in the study of pure model theory.（全文完）