Inner models from extended logics: Part 2

We introduce a new inner model [Formula: see text] arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively PFA, the regular uncountable cardinals of [Formula: see text] are measurable in the inner model [Formula: see text] and [Formula: see text] satisfies CH. Moreover, assuming a proper class of Woodin cardinals, the theory of [Formula: see text] is (set) forcing absolute. We introduce an auxiliary concept that we call Club Determinacy, which simplifies the construction of [Formula: see text] greatly but may have also independent interest. Based on Club Determinacy, we introduce the concept of aa-mouse which we use to prove CH and other properties of the inner model [Formula: see text].