喜讯传来，无穷小微积分要“试点”

7月17日，在我八十岁生日这一天，上级传来喜讯，无穷小微积分要“试点”。

“试点”意味着什么？无穷小微积分来源于哥德尔、塔尔斯基与鲁滨逊创立的模型论紧致性定理及其推论即将在中国生根。。

2010年，美国数学家pete L. Clark在其夏季模型论讲义中给出了紧致性定理的简要证明，提及了这段历史。

袁萌  7月17日

附：

The Compactness Theorem.

Poof：

Theorem 6. (Godel’s Compactness Theorem) A theory T（形式语言的句子集合） is satisable i every nite subset of T is satisable. Proof. （证明）

The implication =⇒ is certainly clear. Conversely, assume that every nite subset of T is satisable but T itself is not satisable. As in the proof of Corollary 3 above, let φ be any sentence; then T (φ∧¬φ). But by the nite character of proof, this implies that there exists a nite subsetF of T that proves (φ∧¬φ). Thus F is syntactically inconsistent, so it certainly is not satisable, contradiction. Remark: Dene a theory T to be nitely satisable if every nite subset F ⊂T is satisable. Then the compactness theorem can be stated more...compactly?...as follows: a theory T is satisable i it is nitely satisable. One might think that the compactness theorem allows us to dispense with the term “nitely satisable” but this turns out not quite to be the case – the term is used later on in these now。