Universality: new criterion for non-existence

Abstract

We find new “reasons” for a class of models for not having a universal model in a cardinal \(\lambda \). This work, though has consequences in model theory, is really in combinatorial (set theory). We concentrate on a prototypical class which is a simply defined class of models, of combinatorial character—models of \(T_{\mathrm{ceq}}\) (essentially another representation of \(T_{\mathrm{feq}}\) which was already considered but the proof with \(T_{\mathrm{ceq}}\) is more transparent). Models of \(T_{\mathrm{ceq}}\) consist essentially of an equivalence relation on one set and a family of choice functions for it. This class is not simple (in the model theoretic sense) but seems to be very low among the non-simple (first order complete countable) ones. We give sufficient conditions for the non-existence of a universal model for it in \(\lambda \). This work may be continued in Shelah et al. (Tba, In preparation. Preliminary number: Sh:F2150).