Toward classification theory of good lambda frames and abstract elementary classes

TL;DR

This paper develops a classification theory for lambda-good frames and abstract elementary classes, exploring stability, categoricity, and saturation properties across various cardinalities, aiming to understand the model theory of these classes.

Contribution

It introduces the concept of omega-successful frames and analyzes their model-theoretic properties, extending stability theory to broader classes of models.

Findings

In omega-successful frames, models exist in all higher cardinals.

Understanding lambda^{+omega}-saturated models helps analyze the class K_s.

Moving from lambda to lambda^+ involves complex saturation considerations.

Abstract

lambda-good frame is for us a parallel of the class of models of a superstable theory. Our main line is to start with lambda-good^+ frame s, categorical in lambda, n-successful for n large enough and try to have parallel of stability theory for K_{s(+l)} for l<n not too large. Characteristically from time to time we have to increase n relative to l to get our desirable properties; we do not critically mind the exact n, so one may think of an omega-successful s. A posteriori we are interested in the model theory of such classes K_s per-se, and see as a test for this theory, that in the omega-successful case we can understand also the model in higher cardinals, e.g., prove that K^s_mu is not empty for every mu>=lambda. Recall there are reasonable lambda-frames which are not n-excellent but still we can say alot on models in $K_{s(+l)} for l<n. Moving from lambda to lambda^+ we would have preferred not to restrict ourselves to saturated models but at present we do not know it. However, in the omega-excellent case we can understand the class of lambda^{+omega}--saturated models in $K_s$, i.e., K^{s(+omega)}. This fits well the thesis that it is reasonable to first analyze the quite saturated case.