模型论处于现代数学的中心位置

模型论处于现代数学的中心位置，为什么？

进入二十世纪，数学理论是不是含有内部矛盾（相容性）成为一个首要问题，数学不是文艺小说。

问题是，一个无限；理论系统的无矛盾性是很难判定的。

哥德尔紧致性定理解决了这个问题。

哥德尔的核心思想是把无限转化为有限来解决问题。这就是紧致性定理的中心意思。

模型论处于现代数学的中心位置，为什么？因为模型论就是围绕紧致性定理展开的。

模型论装入手机，现代数学就在你的手中！

请见本文附件。

袁萌陈启清 1月10日

附件：

模型论第一定理原文

Theorem 1. The Compactness Theorem (Malcev) A set of sentences is satisﬁable iﬀ every ﬁnite subset is satisﬁable.

Proof. There are several proofs. We only point out here that it is an easy consequence of the following theorem which appears in all elementary logic texts:

Proposition. The Completeness Theorem (G¨odel, Malcev) A set of sentences is consistent if and only if it is satisﬁable.

Although we do not here formally deﬁne “consistent”, it does mean what you think it does. In particular, a set of sentences is consistent if and only if each ﬁnite subset is consistent.

Remark. The Compactness Theorem is the only one for which we do not give a complete proof. For the reader who has not previously seen the Completeness Theorem, there are other proofs of the Compactness Theorem which may be more easily absorbed: set theoretic (using ultraproducts), topological (using compact spaces, hence the name) or Boolean algebraic. However these topics are too far aﬁeld to enter into the proofs here. We will use the Compactness Theorem as a starting point — in fact, all that follows can be seen as its corollaries. Exercise 6. Suppose T is a theory for the language L and σ is a sentence of L such that T |= σ. Prove that there is some ﬁnite T0 ⊆T such that T0 |= σ. Recall that T |= σ iﬀ T ∪{¬σ} is not satisﬁable. Definition 15. If L, and L0 are two languages such that L⊆L0 we say that L0 is an expansion of L and L is a reduction of L0. Of course when we say that L⊆L0 we also mean that the constant, function and relation symbols of L remain (respectively) constant, function and relation symbols of the same type in L0. Definition 16. Given a model A for the language L, we can expand it to amodel A0 ofL0, whereL0 is an expansion ofL, by giving appropriate interpretations to the symbols in L0\L. We say that A0 is an expansion of A to L0 and that A is a reduct of A0 to L. We also use the notation A0|L for the reduct of A0 to L. Theorem 2. If a theory T has arbitrarily large ﬁnite models, then it has an inﬁnite model. Proof. Consider new constant symbols ci for i ∈ N, the usual natural numbers, and ex

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