Atomic models higher up

There exists a countable structure M of Scott rank ω 1 C K where ω 1 M = ω 1 C K and where the L ω 1 C K , ω -theory of M is not ω -categorical. The Scott rank of a model is the least ordinal β where the model is prime in its L ω β , ω -theory. Most well-known models with unbounded atoms below ω 1 C K also realize a non-principal L ω 1 C K , ω -type; such a model that preserves the Σ 1 -admissibility of ω 1 C K will have Scott rank ω 1 C K + 1 . Makkai [M. Makkai, An example concerning Scott heights, J. Symbolic Logic 46 (1981) 301–318. [4]] produces a hyperarithmetical model of Scott rank ω 1 C K whose L ω 1 C K , ω -theory is ω -categorical. A computable variant of Makkai’s example is produced in [W. Calvert, S.S. Goncharov, J.F. Knight, J. Millar, Categoricity of computable infinitary theories, Arch. Math. Logic (submitted for publication). [1]; J. Knight, J. Millar, Computable structures of rank ω 1 C K J. Math. Logic (2004). [2]].