Decomposition Theorem for Abstract Elementary Classes
Pablo Cubides Kovacsics
Abstract:
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: