Assuming the existence of a supercompact cardinal, we construct a model of ZFC + ). Answering a question of Uri Abraham [A], [A-S], we prove that adding a real to the world always makes P ℵ 1 - V stationary.

We construct a model in which there is a strongly compact cardinal κ such that the set $S(\kappa,\kappa ^{+})=\{a\in P_{\kappa}\kappa ^{-}\colon o,t(a)=(a\cap \kappa)^{+}\})$ is non-stationary.