Abstract elementary classes: some answers, more questions

Abstract

We survey some of the recent work in the study of Abstract Elementary Classes focusing on the categoricity spectrum and the introduction of certain conditions (amalgamation, tameness, arbitrarily large models) which allow one to develop a workable theory. We repeat or raise for the first time a number of questions; many now seem to be accessible.Much late 19th and early 20th century work in logic was in a 2nd order framework; infinitary logics in the modern sense were foreshadowed by Schroeder and Pierce before being formalized in modern terms in Poland during the late 20's. First order logic was only singled out as the 'natural' language to formalize mathematics as such authors as Tarski, Robinson, and Malcev developed the fundamental tools and applied model theory in the study of algebra. Serious work extending the model theory of the 50's to various infinitary logics blossomed during the 1960's and 70's with substantial work on logics such as L ω1,ω and L ω1,ω (Q). At the same time Shelah's work on stable theories completed the switch in focus in first order model theory from study of the logic to the study of complete first order theories As Shelah in [She75, She83a] sought to bring this same classification theory standpoint to infinitary logic, he introduced a total switch to a semantic standpoint. Instead of studying theories in a logic, one studies the class of models defined by a theory. He abstracted (pardon the pun) the essential features of the class of models of a first order theory partially ordered by the elementary submodel relation. An abstract elementary class AEC (K, ≺ K ) is a class of models closed under isomorphism and partially ordered under ≺ K , where ≺ K is required to refine the substructure relation, that is closed under unions and satisfies two additional conditions: if each element M i of a chain satisfiesFurther there is a Löwnenheim-Skolem number κ associated with K so that if A ⊆ M ∈ K, there is an M 1 with A ⊂ M 1 ≺ K M and |M 1 | ≤ |A| + κ.

A global approach to AECs

“…1.4. Shelah's Presentation Theorem ( [She], [Bal2]): Is the first occurence of "change of languages" in AECs-theory: Bal1]): There is a cardinal µ(κ) such that for all AEC K with LS(K) κ, if K is categorical in some cardinal greater than µ(κ) then K is categorical in all λ µ(κ).…”

The categoricity spectrum of large abstract elementary classes

“…These questions feature in several prominent lists of open problems, for example in [She00, 6.14], [Gro02, Section 9], [Bal06], [Bal09, Appendix D], or the introduction of [She09a].…”