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Some new computable structures of high rank


Authors: Matthew Harrison-Trainor, Gregory Igusa and Julia F. Knight
Journal: Proc. Amer. Math. Soc. 146 (2018), 3097-3109
MSC (2010): Primary 03D45, 03C57
DOI: https://doi.org/10.1090/proc/13967
Published electronically: March 19, 2018
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Abstract: We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank $ \omega _1^{CK}$, the computable infinitary theory is $ \aleph _0$-categorical. Millar and Sacks asked whether this was always the case. We answer this question by constructing an example whose computable infinitary theory has non-isomorphic countable models. The standard known computable structures of Scott rank $ \omega _1^{CK}+1$ have infinite indiscernible sequences. We give two constructions with no indiscernible ordered triple.


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Additional Information

Matthew Harrison-Trainor
Affiliation: Group in Logic and the Methodology of Science, University of California, Berkeley, California 94703
Address at time of publication: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: maharris@uwaterloo.ca

Gregory Igusa
Affiliation: Department of Mathematics, University of Notre Dame, South Bend, Indiana 46556 – and – Department of Mathematics, Victoria University of Wellington, Wellington 6012, New Zealand
Email: gigusa@nd.edu

Julia F. Knight
Affiliation: Department of Mathematics, University of Notre Dame, South Bend, Indiana 46556
Email: j1knight@nd.edu

DOI: https://doi.org/10.1090/proc/13967
Received by editor(s): June 2, 2016
Received by editor(s) in revised form: September 29, 2017
Published electronically: March 19, 2018
Additional Notes: The first author was supported by NSERC PGSD3-454386-2014.
The second author was supported by EMSW21-RTG-0838506.
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2018 American Mathematical Society

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