The effects of forcing on exacting and ultraexacting cardinals
Abstract
Motivated by recent work of Aguilera--Bagaria--Lücke and Aguilera--Bagaria--Goldberg--Lücke (ABGL), we analyze various effects of set-theoretic forcing upon the classes of exacting and ultraexacting cardinals. Using Magidor support products of Prikry forcings, we prove that an I2-embedding yields a transitive, set-sized model of ZFC with a proper class of exacting cardinals. We show that under Woodin's HOD Conjecture, no exacting cardinal can be a limit of exacting cardinals. In contrast, we prove that models of ZF containing large cardinals beyond choice have forcing extensions that are models of ZFC in which a regular cardinal is a stationary limit of ultraexacting cardinals. We also show that, assuming the HOD Conjecture, an analogue of the classical Levy--Solovay theorem holds for exacting and ultraexacting cardinals. Finally, we analyze the large cardinal properties possessed by exacting cardinals in HOD. We prove that if λ is exacting and Vλ satisfies the HOD Hypothesis, then λ has strong large cardinal properties in HOD. Carrying forward forcing constructions from ABGL, we start with a model of ZFC with an I2-embedding and produce a model of ZF in which the successor of an exacting cardinal λ is extendible in all models of the form HODx for x⊆ λ.
Related
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BibTeX
@misc{lucke2026effects,
title = {{The effects of forcing on exacting and ultraexacting cardinals}},
author = {Philipp Lücke and Alejandro Poveda},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32338},
eprint = {2609.32338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32338},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The effects of forcing on exacting and ultraexacting cardinals
AU - Philipp Lücke
AU - Alejandro Poveda
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32338
UR - https://arxiv.org/abs/2609.32338
AB - Motivated by recent work of Aguilera--Bagaria--Lücke and Aguilera--Bagaria--Goldberg--Lücke (ABGL), we analyze various effects of set-theoretic forcing upon the classes of exacting and ultraexacting cardinals. Using Magidor support products of Prikry forcings, we prove that an I2-embedding yields a transitive, set-sized model of ZFC with a proper class of exacting cardinals. We show that under Woodin's HOD Conjecture, no exacting cardinal can be a limit of exacting cardinals. In contrast, we prove that models of ZF containing large cardinals beyond choice have forcing extensions that are models of ZFC in which a regular cardinal is a stationary limit of ultraexacting cardinals. We also show that, assuming the HOD Conjecture, an analogue of the classical Levy--Solovay theorem holds for exacting and ultraexacting cardinals. Finally, we analyze the large cardinal properties possessed by exacting cardinals in HOD. We prove that if λ is exacting and V_λ satisfies the HOD Hypothesis, then λ has strong large cardinal properties in HOD. Carrying forward forcing constructions from ABGL, we start with a model of ZFC with an I2-embedding and produce a model of ZF in which the successor of an exacting cardinal λ is extendible in all models of the form HOD_x for x⊆ λ.
ER -