Let $cf(\alpha) > \omega$, and $P_\alpha := \langle P_{\beta}, \dot{Q}_{\beta} : \beta < \alpha \rangle$ be a countable support iterated forcing construction (so for each $\beta$, $P_{\beta} = P_{\beta - 1} \ast \dot{Q}_{\beta}$ if $\beta$ is a successor ordinal, $P_{\beta} =$ inverse limit of $\langle P_{\gamma} : \gamma < \beta \rangle$ if $cf(\beta) = \omega$, and $P_{\beta} =$ direct limit of $\langle P_{\gamma} : \gamma < \beta \rangle$ otherwise). Let $\Vdash_{P_{\beta}}$ "$\dot{Q}_{\beta}$ is non-trivial", for all $\beta$. What can we specify for the $\dot{Q}_{\beta}$ (and $\alpha$) so that $\Vdash_{P_\alpha} |\alpha| > \aleph_2$?

Thanks in advance.

6more comments