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Intrinsic bounds on complexity and definability at limit levels

Published online by Cambridge University Press:  12 March 2014

John Chisholm ,
Ekaterina B. Fokina ,
Sergey S. Goncharov ,
Valentina S. Harizanov ,
Julia F. Knight  and
Sara Quinn
John Chisholm
Affiliation:
Department of Mathematics, Western Illinois University, Macomb, Il 61455, USA, E-mail: JA-Chisholm@wiu.edu
Ekaterina B. Fokina
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Strasse 25, A-1090 Wien, Austria, E-mail: efokina@logic.univie.ac.at
Sergey S. Goncharov
Affiliation:
Sobolev Institute of Mathematics, Siberian Branch of Ras, 4 Acad, Koptyug Ave. 630090 Novosibirsk, Russia, E-mail: gonchar@math.nsc.ru
Valentina S. Harizanov
Affiliation:
Department of Mathematics, George Washington University, Government Hall, Room 220, Washington, Dc 20052, USA, E-mail: harizanv@gwu.edu
Julia F. Knight
Affiliation:
Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, In 46556, USA, E-mail: knight.1@nd.edu
Sara Quinn
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Il 60208-2730, USA, E-mail: squinn@Math.Northwestern.edu
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Abstract

We show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 3 , September 2009 , pp. 1047 - 1060
Copyright
Copyright © Association for Symbolic Logic 2009

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References

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