Clubs on quasi measurable cardinals

by Kumar and Shelah. [KmSh:1063]
Math Logic Quarterly, 2018
We construct a model satisfying kappa < 2^{aleph_0} + clubsuit_{kappa} + kappa is quasi measurable. Here, we call kappa quasi measurable if there is an aleph_1-saturated kappa-additive ideal {I} over kappa . We also show that, in this model, forcing with {P}(kappa)/ {I} adds one but not kappa Cohen reals. We introduce a weak club principle and use it to show that, consistently, for some aleph_{{1}}-saturated kappa-additive ideal {I} over kappa, forcing with P(kappa) /I adds one but not kappa random reals.

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