We prove the following main theorem: Let be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality μ. Let μ be a cardinal above the the Löwenheim‐Skolem number of the class. If is μ‐Galois‐stable, has no μ‐Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two ‐limits over M, for, are isomorphic over M.

We provide here the first steps toward a Classification Theory ofElementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some λ greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non μ-splitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the o Conjecture for these classes. Further (...) results are in preparation. (shrink)

Theoria, Volume 87, Issue 4, Page 971-985, August 2021.

We study versions of limit models adapted to the context of metric abstract elementary classes. Under categoricity and superstability-like assumptions, we generalize some theorems from 7, 15-17. We prove criteria for existence and uniqueness of limit models in the metric context.

Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained in this fashion (`unfoldable cardinals') lie in the boundary of the propositions consistent with `V = L' and the existence of 0 ♯ . We also provide an `embedding characterisation' of the unfoldable cardinals and study their preservation and destruction by various forcing constructions.

We study notions of independence appropriate for a stability theory of metric abstract elementary classes (for short, MAECs). We build on previous notions used in the discrete case, and adapt definitions to the metric case. In particular, we study notions that behave well under superstability‐like assumptions. Also, under uniqueness of limit models, we study domination, orthogonality and parallelism of Galois types in MAECs.

Theoria, Volume 87, Issue 4, Page 870-873, August 2021.

In this article we present a notion of “logical perfection”. We first describe through examples a notion of logical perfection extracted from the contemporary logical concept of categoricity. Categoricity (in power) has become in the past half century a main driver of ideas in model theory, both mathematically (stability theory may be regarded as a way of approximating categoricity) and philosophically. In the past two decades, categoricity notions have started to overlap with more classical notions of robustness and smoothness. These (...) have been crucial in various parts of mathematics since the nineteenth century. We postulate and present the category of logical perfection. We draw on various notions of perfection from mathematics of the 19th and 20th centuries and then trace the relation to the concept of categoricity in power as a logical notion of what a “mathematically perfect” structure is. (shrink)

In recent years, mathematical logic has developed in many directions, the initial unity of its subject matter giving way to a myriad of seemingly unrelated areas. The articles collected here, which range from historical scholarship to recent research in geometric model theory, squarely address this development. These articles also connect to the diverse work of Väänänen, whose ecumenical approach to logic reflects the unity of the discipline.

We prove a version of a small index property theorem for strong amalgamation classes. Our result builds on an earlier theorem by Lascar and Shelah. We then study versions of the small index property for various non-elementary classes. In particular, we obtain the small index property for quasiminimal pregeometry structures.

A long series of conversations interweaving mathematical, historical and philosophical aspects of categoricity in model theory took place between the author and Saharon Shelah in 2016 and 2017. In this excerpt of that long conversation, we explore the relationship between explicit and implicit aspects of categoricity. We also discuss the connection with definability issues.