Another $\diamondsuit $-like principle
Volume 167 / 2001
Abstract
A new $\diamondsuit $-like principle $\diamondsuit _{\mathfrak d}$ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that $\neg \diamondsuit _{\mathfrak d}$ is consistent with {\rm CH} and that in many models of ${\mathfrak d}=\omega _1$ the principle $\diamondsuit _{\mathfrak d}$ holds. As $\diamondsuit _{\mathfrak d}$ implies that there is a MAD family of size $\aleph _1$ this provides a partial answer to a question of J. Roitman who asked whether ${\mathfrak d}=\omega _1$ implies ${\mathfrak a}=\omega _1$. It is proved that $\diamondsuit _{\mathfrak d}$ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether ${ \mathfrak a}= \omega _1$ in such models.
