The strong tree property and the failure of SCH

Abstract

Fontanella (J Symb Logic 79(1):193–207, 2014) showed that if $$\langle \kappa _n:n<\omega \rangle $$ is an increasing sequence of supercompacts and $$\nu =\sup _n\kappa _n$$ , then the strong tree property holds at $$\nu ^+$$ . Building on a proof by Neeman (J Math Log 9:139–157, 2010), we show that the strong tree property at $$\kappa ^+$$ is consistent with $$\lnot SCH_\kappa $$ , where $$\kappa $$ is singular strong limit of countable cofinality.