A version of κ -Miller forcing

Abstract

We consider a version of $\kappa$ -Miller forcing on an uncountable cardinal $\kappa$ . We show that under $2^{<\kappa }=\kappa$ this forcing collapses $2^{\kappa }$ to $\omega$ and adds a $\kappa$ -Cohen real. The same holds under the weaker assumptions that $\mathrm{cf}\left(\kappa \right)>\omega$ , $2^{2^{<\kappa }}=2^{\kappa }$ , and forcing with $({\left[\kappa \right]}^{\kappa },\subseteq )$ collapses $2^{\kappa }$ to $\omega$ .

Keywords: Forcing with higher perfect trees.