Toward classification theory of good lambda frames and abstract elementary classes

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.