无穷小的逻辑相容性

  在传统微积分教课书里面，（实）无穷小是一个导致自相矛盾的概念。

  当前，这种陈旧的观念仍然在国内普通高校课堂里面在灌输给大学生，培养大批“小糊涂”。

  但是，数理逻辑模型理论紧致性定理对此说“不”。

  请见本文附件。

  其他废话就不说了。

袁萌  陈启清  1月11日

附件：

A third application of the compactness theorem is the construction of nonstandard models of the real numbers, that is, consistent extensions of the theory of the real numbers that contain "infinitesimal" numbers. To see this, let Σ be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1/n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε. By the compactness theorem, there is a model *R that satisfies Σ and also contains an infinitesimal element ε. A similar argument, adjoining axioms ω > 0, ω > 1, etc., shows that the existence of infinitely large integers cannot be ruled out by any axiomatization Σ of the reals.