Definable maximal independent families
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- by Jörg Brendle, Vera Fischer and Yurii Khomskii PDF
- Proc. Amer. Math. Soc. 147 (2019), 3547-3557 Request permission
Abstract:
We study maximal independent families (m.i.f.) in the projective hierarchy. We show that (a) the existence of a $\boldsymbol {\Sigma }^1_2$ m.i.f. is equivalent to the existence of a $\boldsymbol {\Pi }^1_1$ m.i.f., (b) in the Cohen model, there are no projective maximal independent families, and (c) in the Sacks model, there is a $\boldsymbol {\Pi }^1_1$ m.i.f. We also consider a new cardinal invariant related to the question of destroying or preserving maximal independent families.References
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Additional Information
- Jörg Brendle
- Affiliation: Graduate School of System Informatics, Kobe University, Rokkodai 1-1, Nada, Kobe 657-8501, Japan
- Email: brendle@kobe-u.ac.jp
- Vera Fischer
- Affiliation: Kurt Gödel Research Center, Universität Wien, Währinger Straße 25, 1090 Vienna, Austria
- MR Author ID: 854652
- Email: vera.fischer@univie.ac.at
- Yurii Khomskii
- Affiliation: Universität Hamburg, Fachbereich Mathematik, Bundesstraße 55, 20146 Hamburg, Germany
- MR Author ID: 891379
- Email: yurii@deds.nl
- Received by editor(s): September 8, 2018
- Received by editor(s) in revised form: November 10, 2018
- Published electronically: May 9, 2019
- Additional Notes: The first author was partially supported by Grants-in-Aid for Scientific Research (C) 15K04977 and 18K03398, Japan Society for the Promotion of Science
The second author was partially supported by the Austrian Science Foundation (FWF) by the START Grant number Y1012-N35
The third author was partially supported by the European Commission under a Marie Curie Individual Fellowship (H2020-MSCA-IF-2015) through the project number 706219, acronym REGPROP
The first and third authors were partially supported by the Isaac Newton Institute for Mathematical Sciences in the programme Mathematical, Foundational and Computational Aspects of the Higher Infinite (HIF) funded by EPSRC grant EP/K032208/1 - Communicated by: Heike Mildenberger
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3547-3557
- MSC (2010): Primary 03E15, 03E17, 03E35
- DOI: https://doi.org/10.1090/proc/14497
- MathSciNet review: 3981132