We answer a question of Usuba by showing that the combinatorial principle $\mathrm {UB}_\lambda $ can fail at a singular cardinal. Furthermore, $\lambda $ can be taken to be $\aleph _\omega .$

Assuming the existence of suitable large cardinals, we show it is consistent that the Provability logic $\mathbf {GL}$ is complete with respect to the filter sequence of normal measures. This result answers a question of Andreas Blass from 1990 and a related question of Beklemishev and Joosten.

We continue our study of the class ${\cal C}\left( D \right)$, where D is a uniform ultrafilter on a cardinal κ and ${\cal C}\left( D \right)$ is the class of all pairs $\left( {{\theta _1},{\theta _2}} \right)$, where $\left( {{\theta _1},{\theta _2}} \right)$ is the cofinality of a cut in ${J^\kappa }/D$ and J is some ${\left( {{\theta _1} + {\theta _2}} \right)^ + }$-saturated dense linear order. We give a combinatorial characterization of the class ${\cal C}\left( D \right)$. We also show that if $\left( {{\theta _1},{\theta _2}} \right) \in {\cal C}\left( D \right)$ and D is ${\aleph _1}$-complete or ${\theta _1} + {\theta _2} > {2^\kappa }$, then ${\theta _1} = {\theta _2}$.

Starting from large cardinals we construct a model of ZFC in which the GCH fails everywhere, but such that GCH holds in its HOD. The result answers a question of Sy Friedman. Also, relative to the existence of large cardinals, we produce a model of ZFC + GCH such that GCH fails everywhere in its HOD.

In this paper, we consider Foreman’s maximality principle, which says that any nontrivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We observe that it is consistent that every c.c.c. forcing adds a real and that for every uncountable regular cardinal κ, every κ-closed forcing of size 2<κ collapses some cardinal.

Shelah-Woodin [10] 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(κ +3)-strong cardinal; by work of Gitik and Mitchell [6] it is known that more than an H(κ ++)-strong cardinal is required.