HIGHER MILLER FORCING MAY COLLAPSE CARDINALS

Abstract

AbstractWe show that it is independent whether club $\kappa $ -Miller forcing preserves $\kappa ^{++}$ . We show that under $\kappa ^{<\kappa }> \kappa $ , club $\kappa $ -Miller forcing collapses $\kappa ^{<\kappa }$ to $\kappa $ . Answering a question by Brendle, Brooke-Taylor, Friedman and Montoya, we show that the iteration of ultrafilter $\kappa $ -Miller forcing does not have the Laver property.