Cofinality spectrum problems: The axiomatic approach

Our investigations are framed by two overlapping problems: finding the right axiomatic framework for so-called cofinality spectrum problems, and a 1985 question of Dow on the conjecturally nonempty (in ZFC) region of OK but not good ultrafilters. We define the lower-cofinality spectrum for a regular ultrafilter D on λ and show that this spectrum may consist of a strict initial segment of cardinals below λ and also that it may finitely alternate. We define so-called ‘automorphic ultrafilters’ and prove that the ultrafilters which are automorphic for some, equivalently every, unstable theory are precisely the good ultrafilters. We axiomatize a bare-bones framework called “lower cofinality spectrum problems”, consisting essentially of a single tree projecting onto two linear orders. We prove existence of a lower cofinality function in this context and show by example that it holds of certain theories whose model theoretic complexity is bounded.