Strengthening a result of Leshem (J Symb Logic 65(3):1204---1214, 2000), we prove that the consistency strength of $$\textit{GCH}$$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 $$\varPi ^{1}_1$$ź11-reflecting cardinals. Moreover it is proved that if $$\kappa $$ź is a supercompact cardinal and $$\lambda > \kappa $$ź>ź is measurable, then there is a generic extension of the universe in which $$\kappa $$ź is a strong limit singular cardinal of cofinality $$\omega , ~ \lambda =\kappa ^+,$$ź,ź=ź+, and the definable tree property holds at $$\kappa ^+$$ź+. Additionally we can have $$2^\kappa > \kappa ^+,$$2ź>ź+, so that $$\textit{SCH}$$SCH fails at $$\kappa $$ź.
['Ali Sadegh Daghighi', 'Massoud Pourmahdian']
The definable tree property for successors of cardinals