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
satisable i every nite subset is satisable.
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 satisable.
Although we do not here formally dene “consistent”, it does mean
what you think it does. In particular, a set of sentences is
consistent if and only if each nite 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 aeld 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
nite T0 ⊆T such that T0 |=
σ. Recall that T |= σ i T ∪{¬σ} is not
satisable. 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 nite
models, then it has an innite model. Proof. Consider new constant
symbols ci for i ∈
N, the usual natural numbers, and ex
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