Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Definable maximal independent families


Authors: Jörg Brendle, Vera Fischer and Yurii Khomskii
Journal: Proc. Amer. Math. Soc. 147 (2019), 3547-3557
MSC (2010): Primary 03E15, 03E17, 03E35
DOI: https://doi.org/10.1090/proc/14497
Published electronically: May 9, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 03E15, 03E17, 03E35

Retrieve articles in all journals with MSC (2010): 03E15, 03E17, 03E35


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
Email: vera.fischer@univie.ac.at

Yurii Khomskii
Affiliation: Universität Hamburg, Fachbereich Mathematik, Bundesstraße 55, 20146 Hamburg, Germany
Email: yurii@deds.nl

DOI: https://doi.org/10.1090/proc/14497
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
Article copyright: © Copyright 2019 American Mathematical Society