Two questions on E0-like generic equivalence
Abstract
We answer Problems 8.1 and 8.3 from \cite{Tianyuan2026}. We first note that the condition attributed there to ordinary Prikry forcing is not correct as written: the appropriate formulation uses finite symmetric difference of the ranges of the generic sequences, rather than eventual equality at the same coordinates. We record answers to the two problems for both formulations. A length-ω Magidor forcing gives a genuinely Prikry-type example satisfying both conditions. Rigid real-adding forcing gives a broad source of further examples, and a forcing of Jech and Shelah shows that the condition does not carry any large cardinal strength. An E0-invariant Jensen-type forcing of Kanovei and Lyubetsky gives a stronger nontrivial example in which the generic reals in a fixed extension form exactly one full E0-class. Finally, a simple recoding turns the real-forcing previous examples into answers for the version of the problems with corrected condition.
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BibTeX
@misc{chen2026questions,
title = {{Two questions on $E_0$-like generic equivalence}},
author = {Jason Zesheng Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29641},
eprint = {2609.29641},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29641},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Two questions on E_0-like generic equivalence
AU - Jason Zesheng Chen
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.29641
UR - https://arxiv.org/abs/2609.29641
AB - We answer Problems 8.1 and 8.3 from \cite{Tianyuan2026}. We first note that the condition attributed there to ordinary Prikry forcing is not correct as written: the appropriate formulation uses finite symmetric difference of the ranges of the generic sequences, rather than eventual equality at the same coordinates. We record answers to the two problems for both formulations. A length-ω Magidor forcing gives a genuinely Prikry-type example satisfying both conditions. Rigid real-adding forcing gives a broad source of further examples, and a forcing of Jech and Shelah shows that the condition does not carry any large cardinal strength. An E_0-invariant Jensen-type forcing of Kanovei and Lyubetsky gives a stronger nontrivial example in which the generic reals in a fixed extension form exactly one full E_0-class. Finally, a simple recoding turns the real-forcing previous examples into answers for the version of the problems with corrected condition.
ER -