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Non-isomorphism invariant Borel quantifiers

Authors: Fredrik Engström and Philipp Schlicht
Journal: Proc. Amer. Math. Soc. 139 (2011), 4487-4496
MSC (2010): Primary 03E15, 03C80
Published electronically: April 21, 2011
MathSciNet review: 2823094
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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 $ {\mathop\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 $ {\mathop\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

Philipp Schlicht
Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.