Inner models from extended logics: Part 2

We introduce a new inner model C(aa) arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively PFA, the regular uncountable cardinals of V are measurable in the inner model C(aa) and C(aa) satisfies CH. Moreover, assuming a proper class of Woodin cardinals, the theory of C(aa) is (set) forcing absolute. We introduce an auxiliary concept that we call Club Determinacy, which simplifies the construction of C(aa) 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 C(aa).