Diamond, scales and GCH down to $$\aleph _{\omega ^2}$$ ℵ ω 2

Abstract

Gitik and Rinot (Trans Am Math Soc 364(4):1771–1795, 2012) proved assuming the existence of a supercompact that it is consistent to have a strong limit cardinal $$\kappa $$ of countable cofinality such that $$2^\kappa =\kappa ^+$$ , there is a very good scale at $$\kappa $$ , and $$\diamond $$ fails along some reflecting stationary subset of $$\kappa ^+\cap \text {cof}(\omega )$$ . In this paper, we force over Gitik and Rinot’s model but with a modification of Gitik–Sharon (Proc Am Math Soc 136(1):311, 2008) diagonal Prikry forcing to get this result for $$\kappa =\aleph _{\omega ^2}$$ .