Measures in Nairian models, I

Abstract

Assuming $\mathrm{AD}^+ + V = L(P(\mathbb{R}))$, we show that the Ultrapower Axiom holds in $\mathrm{HOD}_s$ for every $ω$-sequence of ordinals $s$. We use this result to analyze the structure of Nairian models, showing that they contain many measurable cardinals. As an application, we establish the consistency of the existence of infinitely many consecutive measurable cardinals with ZF, starting with a Woodin limit of Woodin cardinals. This solves a question of Henle, Mathias, and Woodin. From the same large cardinal hypothesis, adapting a forcing technique of Adkisson and Ben-Neria, we produce a model of ZF in which every uncountable cardinal is singular. Gitik's original construction requires a proper class of strongly compact cardinals.