Killing the GCH everywhere with a single real

Shelah-Woodin investigate the possibility of violating instances of $GCH$ through the addition of a single real. In particular they show that it is possible to obtain a failure of $CH$ by adding a single real to a model of $GCH$, preserving cofinalities. In this article we strengthen their result by showing that it is possible to violate $GCH$ at all infinite cardinals by adding a single real to a model of $GCH.$ Our assumption is the existence of an $H(\kappa^{+3})$-strong cardinal, by work of Gitik and Mitchell it is known that more than an $H(\kappa^{++})$-strong cardinal is required

Singular cofinality conjecture and a question of Gorelic

We give an affirmative answer to a question of Gorelic \cite{Gorelic}, by showing it is consistent, relative to the existence of large cardinals, that there is a proper class of cardinals $\alpha$ with $cf(\alpha)=\omega_1$ and $\alpha^\omega > \alpha.

Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH

Starting from large cardinals we construct a pair $V_1\subseteq V_2$ of models of $ZFC$ with the same cardinals and cofinalities such that $GCH$ holds in $V_1$ and fails everywhere in $V_2$.Comment: arXiv admin note: text overlap with arXiv:1510.0293