Clubs on quasi measurable cardinals

Abstract

We construct a model satisfying κ<2ℵ0+♣κ+“κ“κ is quasi measurable”. Here, we call κ quasi measurable if there is an ℵ1-saturated κ-additive ideal I on κ. We also show that, in this model, forcing with ℘(κ)\I adds one but not κ Cohen reals. We introduce a weak club principle and use it to show that, consistently, for some ℵ1-saturated κ-additive ideal I on κ, forcing with ℘(κ)\I adds one but not κ random reals.