Inner Models from Extended Logics: Part 2

TL;DR

This paper constructs a new inner model $C(aa)$ from stationary logic, showing under certain large cardinal assumptions that it satisfies CH and has measurable cardinals, with a novel concept called club determinacy aiding the construction.

Contribution

It introduces the inner model $C(aa)$ from stationary logic, along with the concept of club determinacy and aa-mouse, advancing inner model theory under large cardinal assumptions.

Findings

$C(aa)$ satisfies CH under large cardinal assumptions.

Regular uncountable cardinals in $V$ become measurable in $C(aa)$.

The theory of $C(aa)$ is forcing absolute.

Abstract

We introduce a new inner model $C(aa)$ arising from stationary logic. We show that assuming a proper class of Woodin cardinals, or alternatively $MM^{++}$, the regular uncountable cardinals of $V$ are measurable in the inner model $C(aa)$, the theory of $C(aa)$ is (set) forcing absolute, and $C(aa)$ satisfies CH. 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 $C(aa)$.