Chang's conjecture and the non-stationary ideal (Q2732272)

The partial order \(\mathbb P_{\max}\), constructed by Woodin, under large cardinal assumptions produces a forcing extension of \(L(\mathbb R)\) in which ZFC holds and the non-stationary ideal on \(\omega_1\) is \(\omega_2\)-saturated. This result for \(\mathbb P_{\max}\) forcing over \(L(\mathbb R)\) can be carried out assuming only the Axiom of Determinacy (AD). NEWLINENEWLINENEWLINEThe author's main result is that by strengthening the determinacy assumptions the conjecture (Chang) that every finitary algebra on \(\omega_2\) has a subalgebra of order type \(\omega_1\) can be shown to hold in the resulting forcing extension. In particular, he proves that under \(\text{AD}+ V=L(\mathbb R,\mu)\) + \(\mu\) is a normal fine measure on \(\mathcal P_{\omega_1}(\mathbb R)\) Chang's conjecture holds in any \(\mathbb P_{\max}\)-generic extension of \(L(\mathbb R)\). The author notes that by results of Woodin the assumptions in his theorem are equiconsistent with the existence of \(\omega^2\) Woodin cardinals and hence strictly stronger than \(\roman{AD}^{L(\mathbb R)}\).