Dimension theory and homogeneity for elementary extensions of a model
Dimension theory and homogeneity for elementary extensions of a model
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模型基本扩展的维数论和齐次性
DOI：
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发表时间：
1982
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通讯作者：
A. Pillay
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作者：
A. Pillay
我们采取一个固定的可数模型M0，我们看看它的可数基本扩展的结构和数量（直到M0上的同构）。设S（M0）是可数的，我们证明了：如果N是M（M0）的弱极小扩张，并且如果N在M（M0）上有初等嵌入，则N在M0上是齐次的.此外，不能去除条件S（M0）= S（M0）。在M0不包含按公式排序的元组的无限集合的假设下，我们证明了M0有无穷多个可数的初等扩张，直到M0上的同构。一个初步的结果是，M0上的所有类型都是可定义的，而且在M0上是可定义的当且仅当在M0上是可定义的（分叉对称）。我们还引入了相对齐性的概念，并证明了M0的一大类初等扩张在M0上是相对齐性的（假设M0没有顺序和S（M0）是可数的）。我现在将讨论本文结果背后的背景和动机，以及本文相对于其他文献和调查的位置。为了简化符号，让T表示M0的完整图。首先，我们的结果，如果M0没有秩序，那么T有无穷多个可数模型是有关以下猜想：任何理论与有限数量（一个以上）的可数模型是不稳定的。
We take a fixed countable model M0, and we look at the structure of and number of its countable elementary extensions (up to isomorphism over M0). Assuming that S(M0) is countable, we prove that if N is a weakly minimal extension of , and if then there is an elementary embedding of N into M over M0), then N is homogeneous over M0. Moreover the condition that ∣S(M0)∣ = ℵ0 cannot be removed. Under the hypothesis that M0 contains no infinite set of tuples ordered by a formula, we prove that M0 has infinitely many countable elementary extensions up to isomorphism over M0. A preliminary result is that all types over M0 are definable, and moreover is definable over M0 if and only if is definable over M0 (forking symmetry). We also introduce a notion of relative homogeneity, and show that a large class of elementary extensions of M0 are relatively homogeneous over M0 (under the assumptions that M0 has no order and S(M0) is countable). I will now discuss the background to and motivation behind the results in this paper, and also the place of this paper relative to other conjectures and investigations. To simplify notation let T denote the complete diagram of M0. First, our result that if M0 has no order then T has infinitely many countable models is related to the following conjecture: any theory with a finite number (more than one) of countable models is unstable.