\(P\)-points in \(\mathbb Q_{\max}\) models (Q1861538)

Shelah has proven that \(\omega_1\)-density of the nonstationary ideal on \(\omega_1\) implies the failure of the Continuum Hypothesis. The authors show that this density condition on the nonstationary ideal cannot decide the existence of \(P\)-points. The authors assume \(\text{AD}^{L(\mathbb R)}\) and use a method of Woodin for obtaining models from this hypothesis where the nonstationary ideal on \(\omega_1\) is \(\omega_1\)-dense to construct two canonical models, one with no \(P\)-point and one with a unique \(P\)-point.