Nairian Models

Abstract

We introduce a hierarchy of models of the Axiom of Determinacy called \emph{Nairian models}. Forcing over the simplest Nairian model, we obtain a model of $\mathsf{ZFC}+\mathsf{M}{\mathsf{M}}^{++}(c)+\neg {\square }_{\ddot{I}{‰}_3}+\neg \square (\ddot{I}{‰}_3)$. Then, fixing $n\in [3,\ddot{I}‰)$, we design a Nairian model and force over it to produce a model of $\mathsf{ZFC}+\mathsf{M}{\mathsf{M}}^{++}(c)+\forall i\in [2,n]\neg \square (\ddot{I}{‰}_i)$. We also build a Nairian model that satisfies $\mathsf{ZF}+"\ddot{I}{‰}_1$ is a supercompact cardinal." We obtain as corollaries of these constructions (1) the consistent failure of the Iterability Conjecture for the Mitchell-Schindler ${\mathsf{K}}^{\mathsf{c}}$ construction, (2) the consistent failure of the Iterability Conjecture for the ${\mathsf{K}}^{\mathsf{c}}$ construction using $2^{2^{\dots 2^{\ddot{I}}‰}}$-complete (for any finite stack of exponents) background extenders, answering a strong version of a question asked by Steel, and (3) a negative answer to Trang's question whether $\mathsf{ZF}+"\ddot{I}{‰}_1$ is a supercompact cardinal" is equiconsistent with $\mathsf{ZFC}+"$there is a proper class of Woodin cardinals that are limits of Woodin cardinals." These corollaries identify obstructions to extending the methods of (descriptive) inner model theory past a Woodin cardinal which is a limit of Woodin cardinals.