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PreprintCC BY 4.0arXiv

Preserving the Ultrapower Axiom by Forcing over Nonoverlapping Extender Models

  • Eyal Kaplan
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Abstract

We identify a class of simple yet nontrivial forcing notions that preserve the Ultrapower Axiom (UA) over canonical inner models, reaching the level of a strong cardinal. The forcings we consider are discrete product forcings whose index sets are chosen and spaced according to the structure of the underlying nonoverlapping extender sequence. As an application, we show that UA is consistent with the existence of a strong cardinal κ such that V≠ HODX for every X∈ Vκ, answering a question of Goldberg.

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Cite

BibTeXRIS
BibTeX
@misc{kaplan2026preserving,
  title = {{Preserving the Ultrapower Axiom by Forcing over Nonoverlapping Extender Models}},
  author = {Eyal Kaplan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06803},
  eprint = {2609.06803},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06803},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Preserving the Ultrapower Axiom by Forcing over Nonoverlapping Extender Models
AU  - Eyal Kaplan
PY  - 2026
DA  - 2026/09/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.06803
UR  - https://arxiv.org/abs/2609.06803
AB  - We identify a class of simple yet nontrivial forcing notions that preserve the Ultrapower Axiom (UA) over canonical inner models, reaching the level of a strong cardinal. The forcings we consider are discrete product forcings whose index sets are chosen and spaced according to the structure of the underlying nonoverlapping extender sequence. As an application, we show that UA is consistent with the existence of a strong cardinal κ such that V≠ HOD_X for every X∈ V_κ, answering a question of Goldberg.
ER  -