A Version of $\kappa$-Miller Forcing

Abstract

Let $\kappa$ be an uncountable cardinal such that $2^{<\kappa }=\kappa$ or just $\mathrm{cf}(\kappa )>\omega$, $2^{2^{<\kappa }}=2^{\kappa }$, and $([\kappa {]}^{\kappa },\supseteq )$ collapses $2^{\kappa }$ to $\omega$. We show under these assumptions the $\kappa$-Miller forcing with club many splitting nodes collapses $2^{\kappa }$ to $\omega$ and adds a $\kappa$-Cohen real.