Omitting types and the real line

Abstract

We investigate some relations between omitting types of a countable theory and some notions defined in terms of the real line, such as for example the ideal of meager subsets ofR. We also try to express connections between the logical structure of a theory and the existence of its countable models omitting certain families of types.It is well known that assuming MA we can omit <nonisolated types. But MA is rather a strong axiom. We prove that in order to be able to omit <nonisolated types it is sufficient to assume that the real line cannot be covered by less thanmeager sets; and this is in fact the weakest possible condition. It is worth pointing out that by means of forcing we can easily obtain the model of ZFC in whichRcannot be covered by <meager sets. It suffices to add to the ground modelCohen generic reals.We also formulate similar results for omitting pairwise contradictory types. It turns out that from some point of view it is much more difficult to find the family of pairwise contradictory types which cannot be omitted by a model ofT, than to find such a family of possibly noncontradictory types. Moreover, for any two countable theoriesT1,T2without prime models, the existence of a family ofκtypes which cannot be omitted by a model ofT1is equivalent to the existence of such a family forT2. This means that from the point of view of omitting types all theories without prime models are identical. Similar results hold for omitting pairwise contradictory types.