Another $\diamondsuit $-like principle

Michael Hrušák¹

¹ Department of Mathematics and Statistics York University 4700 Keele Street, North York Ontario, Canada, M3J 1P3

Fundamenta Mathematicae, Tome 167 (2001) no. 3, pp. 277-289.

Résumé

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.

Détail

DOI : 10.4064/fm167-3-5

Keywords: diamondsuit like principle diamondsuit mathfrak consistent negation continuum hypothesis introduced studied shown neg diamondsuit mathfrak consistent many models mathfrak omega principle diamondsuit mathfrak holds diamondsuit mathfrak implies there mad family size aleph provides partial answer question roitman who asked whether mathfrak omega implies mathfrak omega proved diamondsuit mathfrak holds model obtained adding single laver real answering question brendle who asked whether mathfrak omega models