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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Non-isomorphism invariant Borel quantifiers
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by Fredrik Engström and Philipp Schlicht PDF
Proc. Amer. Math. Soc. 139 (2011), 4487-4496 Request permission

Abstract:

Every isomorphism invariant Borel subset of the space of structures on the natural numbers in a countable relational language is definable in ${\mathscr {L}}_{\omega _1\omega }$ by a theorem of Lopez-Escobar. We derive variants of this result for stabilizer subgroups of the symmetric group ${\mathrm {Sym}}(\mathbb {N})$ for families of relations and non-isomorphism invariant generalized quantifiers on the natural numbers such as “for all even numbers”. Moreover we produce a binary quantifier $Q$ for every closed subgroup of ${\mathrm {Sym}}(\mathbb {N})$ such that the Borel sets of structures invariant under the subgroup action are exactly the sets of structures definable in ${\mathscr {L}}_{\omega _1\omega }(Q)$.
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Additional Information
  • Fredrik Engström
  • Affiliation: Department of Philosophy, Linguistics and Theory of Science, University of Gothenburg, Box 200, 405 30 Göteborg, Sweden
  • Email: fredrik.engstrom@gu.se
  • Philipp Schlicht
  • Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
  • Email: schlicht@math.uni-bonn.de
  • Received by editor(s): March 19, 2010
  • Received by editor(s) in revised form: October 19, 2010
  • Published electronically: April 21, 2011
  • Additional Notes: Part of the work in this paper was done while visiting the Institut Mittag-Leffler. The authors would like to thank the Institut Mittag-Leffler for support and also the anonymous referee for several valuable comments and suggestions.
    The first author was partially supported by the EUROCORE LogICCC LINT program and the Swedish Research Council.
    The second author received support from the European Science Foundation for the activity ‘New Frontiers of Infinity: Mathematical, Philosophical and Computational Prospects’
  • Communicated by: Julia Knight
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 4487-4496
  • MSC (2010): Primary 03E15, 03C80
  • DOI: https://doi.org/10.1090/S0002-9939-2011-10849-9
  • MathSciNet review: 2823094