A Version of $κ$-Miller Forcing

Heike Mildenberger, Saharon Shelah

Published 2018-02-22Version 1

Let $\kappa$ be a regular uncountable cardinal such that $2^{<\kappa} = \kappa$ or just $2^{(\kappa^{<\kappa})} = 2^\kappa$ and there is a $\kappa$-mad family of size $2^\kappa$. We show under these assumptions the forcing order $\kappa$-Miller with club many splitting nodes collapses $2^\kappa$ to $\omega$ and adds a $\kappa$-Cohen real.