Abstract

We construct a model satisfying “κ is quasi measurable”. Here, we call κ quasi measurable if there is an ℵ1‐saturated κ‐additive ideal on κ. We also show that, in this model, forcing with 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 on κ, forcing with adds one but not κ random reals.