MoL-2009-06:
Kovacsics, Pablo Cubides
(2009)
*Decomposition Theorem for Abstract Elementary Classes.*
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## 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:

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 |

URI: | https://eprints.illc.uva.nl/id/eprint/813 |

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