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A robuster Scott rank

Author: Antonio Montalbán
Journal: Proc. Amer. Math. Soc. 143 (2015), 5427-5436
MSC (2010): Primary 03C75; Secondary 03D45
Published electronically: April 14, 2015
MathSciNet review: 3411157
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Abstract: We give a new definition of Scott rank motivated by our main theorem: For every countable structure $ \mathcal {A}$ and ordinal $ \alpha <\omega _1$, we have that: every automorphism orbit is infinitary $ \Sigma _\alpha $-definable without parameters if and only if $ \mathcal {A}$ has an infinitary $ \Pi _{\alpha +1}$ Scott sentence, if and only if $ \mathcal {A}$ is uniformly boldface $ \bf {\Delta }^0_\alpha $-categorical. As a corollary, we show that a structure is computably categorical on a cone if and only if it is the model of a countably categorical infinitary $ \Sigma _3$ sentence.

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Additional Information

Antonio Montalbán
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840

Received by editor(s): March 24, 2014
Received by editor(s) in revised form: November 7, 2014
Published electronically: April 14, 2015
Additional Notes: The author was partially supported by the Packard Fellowship.
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society

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