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Index sets and Scott sentences

Index sets and Scott sentences For a computable structure $${\mathcal{A}}$$ A , there may not be a computable infinitary Scott sentence. When there is a computable infinitary Scott sentence $${\varphi}$$ φ , then the complexity of the index set $${I(\mathcal{A})}$$ I ( A ) is bounded by that of $${\varphi}$$ φ . There are results (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012; Knight and Saraph in Scott sentences for certain groups, pre-print) giving “optimal” Scott sentences for structures of various familiar kinds. These results have been driven by the thesis that the complexity of the index set should match that of an optimal Scott sentence (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012). In this note, it is shown that the thesis does not always hold. For a certain subgroup of $${\mathbb{Q}}$$ Q , there is no computable d- $${\Sigma_2}$$ Σ 2 Scott sentence, even though (as shown in Ash and Knight in Scott sentences for certain groups, pre-print) the index set is d- $${\Sigma^0_2}$$ Σ 2 0 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Index sets and Scott sentences

Archive for Mathematical Logic , Volume 53 (6) – Mar 4, 2014

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References (8)

Publisher
Springer Journals
Copyright
Copyright © 2014 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/s00153-014-0377-8
Publisher site
See Article on Publisher Site

Abstract

For a computable structure $${\mathcal{A}}$$ A , there may not be a computable infinitary Scott sentence. When there is a computable infinitary Scott sentence $${\varphi}$$ φ , then the complexity of the index set $${I(\mathcal{A})}$$ I ( A ) is bounded by that of $${\varphi}$$ φ . There are results (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012; Knight and Saraph in Scott sentences for certain groups, pre-print) giving “optimal” Scott sentences for structures of various familiar kinds. These results have been driven by the thesis that the complexity of the index set should match that of an optimal Scott sentence (Ash and Knight in Computable structures and the hyperarithmetical hierarchy. Elsevier, Amsterdam, 2000; Calvert et al. in Algeb Log 45:306–315, 2006; Carson et al. in Trans Am Math Soc 364:5715–5728, 2012; McCoy and Wallbaum in Trans Am Math Soc 364:5729–5734, 2012). In this note, it is shown that the thesis does not always hold. For a certain subgroup of $${\mathbb{Q}}$$ Q , there is no computable d- $${\Sigma_2}$$ Σ 2 Scott sentence, even though (as shown in Ash and Knight in Scott sentences for certain groups, pre-print) the index set is d- $${\Sigma^0_2}$$ Σ 2 0 .

Journal

Archive for Mathematical LogicSpringer Journals

Published: Mar 4, 2014

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