Title: Dimension theory and homogeneity for elementary extensions of a model
Abstract: We take a fixed countable model M 0 , and we look at the structure of and number of its countable elementary extensions (up to isomorphism over M 0 ). Assuming that S ( M 0 ) 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 M 0 ), then N is homogeneous over M 0 . Moreover the condition that ∣ S ( M 0 )∣ = ℵ 0 cannot be removed. Under the hypothesis that M 0 contains no infinite set of tuples ordered by a formula, we prove that M 0 has infinitely many countable elementary extensions up to isomorphism over M 0 . A preliminary result is that all types over M 0 are definable, and moreover is definable over M 0 if and only if is definable over M 0 (forking symmetry). We also introduce a notion of relative homogeneity, and show that a large class of elementary extensions of M 0 are relatively homogeneous over M 0 (under the assumptions that M 0 has no order and S ( M 0 ) 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 M 0 . First, our result that if M 0 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.
Publication Year: 1982
Publication Date: 1982-03-01
Language: en
Type: article
Indexed In: ['crossref']
Access and Citation
Cited By Count: 27
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot