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 \(\kappa \).
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Daghighi, A.S., Pourmahdian, M. The definable tree property for successors of cardinals. Arch. Math. Logic 55, 785–798 (2016). https://doi.org/10.1007/s00153-016-0494-7
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DOI: https://doi.org/10.1007/s00153-016-0494-7
Keywords
- Aronszajn tree
- Definable tree property
- \(\varPi ^{1}_1\)-Reflecting cardinal
- Easton reverse iteration
- Extender based Prikry forcing