Atomic models higher up

There exists a countable structure M of Scott rank where and where the -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 Lωβ,ω-theory. Most well-known models with unbounded atoms below also realize a non-principal -type; such a model that preserves the Σ1-admissibility of will have Scott rank . Makkai [M. Makkai, An example concerning Scott heights, J. Symbolic Logic 46 (1981) 301–318. [4], ] produces a hyperarithmetical model of Scott rank whose -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 J. Math. Logic (2004). [2]].