Blowing up the power of a singular cardinal of uncountable cofinality with collapses

TL;DR

The paper demonstrates how to force a failure of the Singular Cardinal Hypothesis at certain singular cardinals of uncountable cofinality using large cardinal assumptions, while preserving cardinals and cofinalities.

Contribution

It introduces a new forcing method that makes SCH fail at specific singular cardinals of uncountable cofinality under large cardinal assumptions below Woodin cardinals.

Findings

SCH fails at targeted singular cardinals in the constructed model

The forcing preserves cardinals and cofinalities up to a certain point

A very good scale is obtained in the model

Abstract

The {\em Singular Cardinal Hypothesis} (SCH) is one of the most classical combinatorial principles in set theory. It says that if $\kappa$ is singular strong limit, then $2^{\kappa}=\kappa^+$. We prove that given a singular cardinal $\kappa$ of {\em cofinality} $\eta$ in the ground model, which is a limit of suitable large cardinals, and $\eta^+=\aleph_{\gamma}$, then there is a forcing extension which preserves cardinals and cofinalities up to and including $\eta$, such that $\kappa$ becomes $\aleph_{\gamma+\eta}$, and SCH fails at $\kappa$. Furthermore, if $\eta$ is not an $\aleph$-fixed point, then in our model, SCH fails at $\aleph_{\eta}$. Our large cardinal assumption is below the existence of a Woodin cardinal. In our model we also obtain a very good scale.