The Cofinal Strong Chang Conjecture from Models of Determinacy

In chapter 9 of his book “The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal”, Woodin shows how to force the Strong Chang Conjecture over models of determinacy using ${\mathbb{P}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$. We show here how a modification of the proof implies that such extensions actually verify the stronger cofinal version of the conjecture. This stronger version has important consequences on the semi-properness of small forcing, allowing us to prove the consistency of the theory “ZFC + Namba forcing is semiproper + ${\mathrm{\Theta }}^{UB}={\omega }_3$”. We then use the constructions of this proof to also show that Woodin $(\ast {)}_{UB}$ axiom implies the conjecture.