DECIDABLE MODELS OF ω-STABLETHEORIES

We characterize the ω-stable theories all of whosecountable models admit decidable presentations. In particular, we show that fora countable ω-stable T, everycountable model of T admits a decidable presentation if andonly if all n-types in T are recursive andT has only countably many countable models. We furthercharacterize the decidable models of ω-stabletheories with countably many countable models as those which realize onlyrecursive types.