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 ${\sf{ZFC}}+{\sf{MM^{++}}}(c)+\neg\square_{\omega_3}+\neg\square(\omega_3)$. Then, fixing $n\in [3, \omega)$, we design a Nairian model and force over it to produce a model of ${\sf{ZFC}}+{\sf{MM^{++}}}(c)+\forall i\in [2, n]\, \neg\square(\omega_i)$. We also build a Nairian model that satisfies ${\sf{ZF}}+"\omega_1$ is a supercompact cardinal." We obtain as corollaries of these constructions (1) the consistent failure of the Iterability Conjecture for the Mitchell-Schindler $\sf{K}^{c}$ construction, (2) the consistent failure of the Iterability Conjecture for the $\sf{K}^{c}$ construction using $2^{2^{\dots 2^{\omega}}}$-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 ${\sf{ZF}}+"\omega_1$ is a supercompact cardinal" is equiconsistent with ${\sf{ZFC}}+"$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.