PreprintCC BY 4.0arXiv
Measures in Nairian models, I
Abstract
Assuming AD+ + V = L(P(ℝ)), we show that the Ultrapower Axiom holds in HODs 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.
Related
BibTeX
@misc{goldberg2026measures,
title = {{Measures in Nairian models, I}},
author = {Gabriel Goldberg and Grigor Sargsyan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19088},
eprint = {2610.19088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19088},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Measures in Nairian models, I
AU - Gabriel Goldberg
AU - Grigor Sargsyan
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19088
UR - https://arxiv.org/abs/2610.19088
AB - Assuming AD^+ + V = L(P(ℝ)), we show that the Ultrapower Axiom holds in 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.
ER -