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Computable trees of Scott rank ω1CK, and computable approximation

Published online by Cambridge University Press:  12 March 2014

Wesley Calvert ,
Julia F. Knight  and
Jessica Millar
Wesley Calvert
Affiliation:
Murray State University, Department of Mathematics and Statistics, Murray, Kentucky 42071, USA. E-mail: [email protected]
Julia F. Knight
Affiliation:
University of Notre Dame, Department of Mathematics, Notre Dame, Indiana 46556, USA. E-mail: [email protected]
Jessica Millar
Affiliation:
University of British Columbia, Department of Mathematics, Vancouver, B.C., V6T 1Z2, Canada. E-mail: [email protected]
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Abstract

Makkai [10] produced an arithmetical structure of Scott rank ω1CK. In [9], Makkai's example is made computable. Here we show that there are computable trees of Scott rank ω1CK. We introduce a notion of “rank homogeneity”. In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated “group trees” of [10] and [9], Using the same kind of trees, we obtain one of rank ω1CK that is “strongly computably approximable”. We also develop some technology that may yield further results of this kind.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 1 , March 2006 , pp. 283 - 298
Copyright
Copyright © Association for Symbolic Logic 2006

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