Clubs on quasi measurable cardinals

摘要

We construct a model satisfying κ < 2~(?0) +?_κ+ "κ is quasi measurable". Here, we call κ quasi measurable if there is an ?_1-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 ?_1-saturated κ-additive ideal I on κ, forcing with ?(κ)/I adds one but not κ random reals.