Absoluteness for the theory of the inner model constructed from finitely many cofinality quantifiers

TL;DR

This paper establishes that the theory of models constructed with finitely many cofinality quantifiers remains absolute under set forcing, assuming many Woodin cardinals, and is independent of the chosen regular cardinals.

Contribution

It proves set-forcing absoluteness of the theory of models built from finitely many cofinality quantifiers under large cardinal assumptions, extending understanding of inner model theory.

Findings

The theory is set-forcing absolute under class many Woodin cardinals.

The theory is independent of the specific regular cardinals used.

Properties of generic embeddings from the stationary tower are analyzed.

Abstract

We prove that the theory of the models constructible using finitely many cofinality quantifiers - $C_{\lambda_{1},...,\lambda_{n}}^{*}$ and $C_{<\lambda_{1},...,<\lambda_{n}}^{*}$ for $\lambda_{1},...,\lambda_{n}$ regular cardinals - is set-forcing absolute under the assumption of class many Woodin cardinals, and is independent of the regular cardinals used. Towards this goal we prove some properties of the generic embedding induced from the stationary tower restricted to $<\mu$-closed sets.