模型论的诞生及其对现代数学的影响

简而言之，在上世纪数学形式化浪潮中，人们的目光对准了用形式语言描述的“理论”，研究这些形式理论的性质。比如，一个形式理论，在什么条件下，具有模型（Model）。由此，产生了所谓的“模型论”。

假若把微积分用形式语言表述出来，作为一种形式理论，那么，实数系就是该“理论”的模型。

在形式理论研究中，人们发现，只要这个形式理论的任意有限子集合（构成所谓“子理论”）有模型，则该理论自身（整体）必有模型。这条定理是模型论的核心定理，叫做“紧致性定理”（Compactness Theorem）。

实际情况是，超实数系*R模型的存在性就是紧致性定理的直接推论！

回顾历史，在上世纪50 – 60年代，美国数学家塔尔斯基及其弟子对数学的形式理论做了深入的研究，发表许多研究论文，最终促成了模型论的诞生。

塔尔斯基的工作对整个现代数学具有极为深远的影响。特别需要提及的是，他的大弟子J.Keisler模型论专家就是无穷小微积分教材的作者。

从历史发展主流来看，无穷小微积分是“根红苗正”，不是流浪狗脏兮兮的，无人理睬。菲氏微积分是十九世纪的数学老黄历，不足道也。

袁萌    11月3日

说明：现将模型论的历史（英文）附后，请读者参考。

Model theory as a subject has existed since approximately the middle of the 20th century.   

Model theory as a subject has existed since approximately the middle of the 20th century.   

However some earlier research, especially in mathematical logic, is often regarded as being of a model-theoretical nature in retrospect. The first significant result in what is now model theory was a special case of the downward Löwenheim–Skolem theorem, published by Leopold Löwenheim in 1915. The compactness theorem was implicit in work by Thoralf Skolem,^[3] but it was first published in 1930, as a lemma in Kurt Gödel's proof of his completeness theorem. The Löwenheim–Skolem theorem and the compactness theorem received their respective general forms in 1936 and 1941 from Anatoly Maltsev.

The development of model theory can be traced to Alfred Tarski, a member of the Lwów–Warsaw school during the interbellum. Tarski's work included logical consequence, deductive systems, the algebra of logic, the theory of definability, and the semantic definition of truth, among other topics. His semantic methods culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s. These modern concepts of model theory influenced Hilbert's program and modern mathematics.（全文完）