{
  "abstract": "Powell introduced the Axiom of Double Complement ($\\mathsf{DCom}$) to give his double-negation interpretation of $\\mathsf{ZF}$ into $\\mathsf{IZF_{Rep}}$. However, the consistency, strength, and compatibility of $\\mathsf{DCom}$ remain open problems. This article aims to survey the compatibility and consistency strength of $\\mathsf{DCom}$, its consequence and opposites, which will be named $\\mathsf{NDCom}$ and $\\mathsf{ADCom}$. We will also develop Lubarsky's Kripke models over $\\mathsf{CZF}$ to derive these results. We will show that $\\mathsf{DCom}$ proves the Powerset axiom over $\\mathsf{CZF}$ and is independent of $\\mathsf{IZF}$. We will also show that $\\mathsf{ADCom}$ does not add consistency strength over $\\mathsf{CZF}$, by modifying the construction of Lubarsky's model for $\\mathsf{CZF+\\lnot Pow}$. We will also show that $\\mathsf{DCom}$, $\\mathsf{ADCom}$, and $\\mathsf{NDCom}$ are persistent under realizability under modest conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.00861",
  "author_total": 2,
  "authors": [
    {
      "name": "Hanul Jeon"
    },
    {
      "name": "James E. Hanson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6bcwfgx5k2eh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.00861",
  "fulltext_url": null,
  "id": "c6bcwfgx5k2eh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.00861",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7163131283",
  "pdf_url": "https://arxiv.org/pdf/2606.00861",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6bcwfgx5k2eh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2606.00861",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.00861"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "The Axiom of Double Complement and its opposites",
  "title_text": "The Axiom of Double Complement and its opposites",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/c6bcwfgx5k2eh",
  "venue": "arXiv",
  "versions": []
}