{
  "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 $V_λ$ 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 $\\mathrm{HOD}_x$ for $x\\subseteq λ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32338",
  "author_total": 2,
  "authors": [
    {
      "name": "Philipp Lücke"
    },
    {
      "name": "Alejandro Poveda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9x25my8mhtd1y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32338",
  "fulltext_url": null,
  "id": "9x25my8mhtd1y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32338",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7214732760",
  "pdf_url": "https://arxiv.org/pdf/2609.32338",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9x25my8mhtd1y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32338",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32338"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "The effects of forcing on exacting and ultraexacting cardinals",
  "title_text": "The effects of forcing on exacting and ultraexacting cardinals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/9x25my8mhtd1y",
  "venue": "arXiv",
  "versions": []
}