English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

We present Saharon Shelah's Stability Spectrum and Homogeneity Spectrum theorems, as well as the equivalence between the order property and instability in the framework of Finite Diagrams. Finite Diagrams is a context which generalizes the flrst order case. Localized versions of these theorems are presented. Our presentation is based on several papers; the point of view is contemporary and some of the proofs are new. The treatment of local stability in Finite Diagrams is new. Saharon Shelah's Finite Diagrams Stable in Power (Sh3), published in 1970, is one of the seminal articles in model theory. It contains a large number of key ideas which have shaped the development of classiflcation theory. The model-theoretic framework of the paper is more general than the flrst order case. However, while all the particular cases of the results in the flrst-order case can be found in several more recent publications of Saharon Shelah as well as countless expositions, the non flrst-order content of (Sh3) is still not available in a concise form. The primary purpose of this paper is to present, in this more general framework, most of the stability results of (Sh3), together with the order/stability dichotomy from (Sh12), and the homogeneity spectrum appearing in The Lazy Model Theorist's Guide To Stability (Sh54). A secondary purpose is to present the necessary background to (Le1) and (GrLe2). This is done in a contemporary and self-contained manner, and includes improvements and techniques from (Sh b), (Sh300), and (Gr1). Finally, with little additional work, we localize all the theorems and obtain local versions of the Stability Spectrum Theorem and the Homogene- ity Spectrum in Finite Diagrams. This study of local stability in more general frameworks has been started in (GrLe1). The framework introduced by Shelah in (Sh3) is the study of classes of models of a flnite diagram. These classes are described in more detail below. Such classes are examples of nonelementary classes and the results presented in this paper belong to a subfleld of pure model theory named by Shelah: classiflcation theory for nonelementary classes. The word nonelementary refers to the fact that the

Shelah's stability spectrum and homogeneity spectrum in finite diagrams