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On the non-existence of mad families

  • Haim Horowitz
  • Saharon Shelah
Article
  • 6 Downloads

Abstract

We show that the non-existence of mad families is equiconsistent with \(\textit{ZFC}\), answering an old question of Mathias. We also consider the above result in the general context of maximal independent sets in Borel graphs, and we construct a Borel graph G such that \(\textit{ZF}+\textit{DC}+\) “there is no maximal independent set in G” is equiconsistent with \(\textit{ZFC}+\) “there exists an inaccessible cardinal”.

Keywords

Forcing Mad families Amalgamation Borel graphs Inaccessible cardinals 

Mathematics Subject Classification

03E35 03E15 03E25 03E55 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Einstein Institute of Mathematics, Edmond J. Safra CampusThe Hebrew University of JerusalemGivat Ram, JerusalemIsrael
  2. 2.Department of Mathematics, Hill Center - Busch CampusRutgers, The State University of New JerseyPiscatawayUSA

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