Archive for Mathematical Logic

, Volume 55, Issue 5–6, pp 785–798 | Cite as

The definable tree property for successors of cardinals

Article

Abstract

Strengthening a result of Leshem (J Symb Logic 65(3):1204–1214, 2000), we prove that the consistency strength of \(\textit{GCH}\) together with the definable tree property for all successors of regular cardinals is precisely equal to the consistency strength of existence of proper class many \(\varPi ^{1}_1\)-reflecting cardinals. Moreover it is proved that if \(\kappa \) is a supercompact cardinal and \(\lambda > \kappa \) is measurable, then there is a generic extension of the universe in which \(\kappa \) is a strong limit singular cardinal of cofinality \(\omega , ~ \lambda =\kappa ^+,\) and the definable tree property holds at \(\kappa ^+\). Additionally we can have \(2^\kappa > \kappa ^+,\) so that \(\textit{SCH}\) fails at \(\kappa \).

Keywords

Aronszajn tree Definable tree property \(\varPi ^{1}_1\)-Reflecting cardinal Easton reverse iteration Extender based Prikry forcing 

Mathematics Subject Classification

03E35 03E55 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceAmirkabir University of TechnologyTehranIran

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