MoL-2009-06: Decomposition Theorem for Abstract Elementary Classes

MoL-2009-06: Kovacsics, Pablo Cubides (2009) Decomposition Theorem for Abstract Elementary Classes. [Report]

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Classical model theory deals essentially with elementary classes, namely, the classes that consist of models of a given complete first-order theory. Yet, many natural mathematical classes are non-elementary; examples include the class of well-ordered sets and the class of Archimedean ordered fields. The concept of abstract elementary classes (AEC) was introduced by Shelah in [12], as a way to lift classical results from elementary classes to classes which, despite being non-elementary, share properties with elementary ones. In [7], Rami Grossberg and Olivier Lessmann proposed a number of axioms in order to lift and generalize the decomposition theorem, first proved by Shelah in [11], to the AEC setting. The decomposition theorem was generated to prove part of the main gap theorem, one of Shelah’s most famous results. Informally, the main gap theorem states that for any first-order theory T, the function I(T, \kappa) –that is, the number of non-isomorphic models of T of cardinality \kappa– takes either its maximum value 2^\kappa or every model of T can be decomposed as a tree of small models; in this case, the number of such trees gives an upper bound to I(T, \kappa) below 2^\kappa. The decomposition theorem deals precisely with assigning such a tree to every model. This thesis has two objectives. The first and key objective is to provide a detailed proof of the abstract version of the decomposition theorem in the spirit of [7]. This detailed proof is provided because, although the results in [7] are correct, some of the proofs contain mistakes and missing details1. In addition, the axiomatic framework outlined here varies slightly from [7], and many proofs differ completely in their approach2. The second objective is to present an application of the abstract version of the decomposition theorem for the class of (D, \aleph_0)-models of a totally transcendental good diagram D. It will be shown that any two models of cardinality \lambda of a totally transcendental good diagram which are L1,\lambda-equivalent, are isomorphic (for a large enough \lambda). This application is an extension of a theorem proved by Shelah for the first-order case (see [12], chapter XII). The text is divided as follows. Section 1 addresses the preliminaries. In subsection 1.1, notation and basic concepts are outlined. Three topics which deserve a special treatment are discussed in subsections 1.2–1.4: trees, infinitary languages and pregeometries. Proofs are presented only for trees given their import to the entire thesis, while for infinitary languages and pregeometries results will be stated with references to proof sources. Section 2 contains the core argumentation and has two parts. First, in subsection 2.1, a brief introduction to abstract elementary classes is presented, bringing in Galois types and the monster model convention. In subsection 2.2, the axiomatic framework for the decomposition theorem is presented together with its revised proof. Finally, in section 3, totally transcendental diagrams are introduced in subsection 3.1 and the above-mentioned application regarding L\infty,\lambda-equivalence as an invariant is proved in subsection 3.2. 3 Keywords:

Item Type: Report
Report Nr: MoL-2009-06
Series Name: Master of Logic Thesis (MoL) Series
Year: 2009
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38

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