Two remarks on Merimovich's model of the total failure of GCH

Abstract

Let $M$ denote the Merimovich's model in which for each infinite cardinal $\lambda, 2^\lambda=\lambda^{+3}$. We show that in $M$ the following hold: (1) Shelah's strong hypothesis fails at all singular cardinals, indeed, $\forall \lambda (\lambda$ is a singular cardinal $\Rightarrow pp(\lambda)=\lambda^{+3}).$ (2) For each singular cardinal $\lambda$ there is an inner model $N$ of $M$ such that $M$ and $N$ have the same bounded subsets of $\lambda,$ $\lambda$ is a singular cardinal in $N$, $(\lambda^{+i})^N=(\lambda^{+i})^M$, for $i=1,2,3,$ and $N \models 2^\lambda=\lambda^{+}$. Thus it is possible to add many new fresh subsets to $\lambda$ without adding any new bounded subsets to $\lambda$.