A.E.C. with not too many models

TL;DR

This paper investigates the structure of abstract elementary classes, showing that the number of models of certain sizes behaves predictably under specific conditions, even without categoricity assumptions.

Contribution

It establishes conditions under which the model counting function in A.E.C.s is well-behaved, extending understanding beyond categorical cases.

Findings

For a closed unbounded class of cardinals, the number of models is either large, extendable, or bounded.

Models of large size can be maximal or non-extendable, highlighting differences from elementary classes.

The paper identifies conditions ensuring the 'niceness' of the model counting function in A.E.C.s.

Abstract

Consider an a.e.c. (abstract elementary class), that is, a class K of models with a partial order refining inclusion (submodel) which satisfy the most basic properties of an elementary class. Our test question is trying to show that the function dot I (lambda, K), counting the number of models in K of cardinality lambda up to isomorphism, is "nice", not chaotic, even without assuming it is sometimes 1, i.e. categorical in some lambda's. We prove here that for some closed unbounded class C of cardinals we have (a), (b) or (c) where (a) for every lambda in C of cofinality aleph_0, dot I (lambda, K) greater than or equal to lambda, (b) for every lambda in C of cofinality aleph_0 and M belongs to K_lambda, for every cardinal kappa greater than or equal to lambda there is N_kappa of cardinality kappa extending M (in the sense of our a.e.c.), (c) mathfrak k is bounded; that is, dot I (lambda, K) = 0 for every lambda large enough (equivalently lambda greater than or equal to beth_{delta_*} where delta_* = (2^{LST(mathfrak k)})^+). Recall that an important difference of non-elementary classes from the elementary case is the possibility of having models in K, even of large cardinality, which are maximal, or just failing clause (b).