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