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

TL;DR

This paper examines Merimovich's model where the continuum function is fixed at λ^{+3} for all infinite cardinals, revealing the failure of Shelah's strong hypothesis at singular cardinals and the possibility of adding subsets without changing bounded subsets.

Contribution

It demonstrates the failure of Shelah's strong hypothesis at all singular cardinals within Merimovich's model and shows how to add many subsets to a singular cardinal without altering its bounded subsets.

Findings

Shelah's strong hypothesis fails at all singular cardinals in the model.

For each singular λ, an inner model exists with the same bounded subsets but with 2^λ=λ^{+}.

Many new subsets can be added to λ without changing bounded subsets.

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 $\lambda$.