Atomic models higher up

Annals of Pure and Applied Logic 155 (3):225-241 (2008)
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Abstract

There exists a countable structure of Scott rank where and where the -theory of is not ω-categorical. The Scott rank of a model is the least ordinal β where the model is prime in its -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 46 301–318. [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 . [1]; J. Knight, J. Millar, Computable structures of rank J. Math. Logic . [2]]

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References found in this work

The α-finite injury method.G. E. Sacks & S. G. Simpson - 1972 - Annals of Mathematical Logic 4 (4):343-367.
Metarecursive sets.G. Kreisel & Gerald E. Sacks - 1965 - Journal of Symbolic Logic 30 (3):318-338.
An example concerning Scott heights.M. Makkai - 1981 - Journal of Symbolic Logic 46 (2):301-318.
Scott sentences and admissible sets.Mark Nadel - 1974 - Annals of Mathematical Logic 7 (2):267.

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