{
  "abstract": "We investigate the iterative conception of set in its most general form, allowing urelements without assuming that they form a set. We formulate this conception in two ways, as stage theory and as level theory, and develop a general theory of levels with urelements. Unlike their pure-set counterparts, the resulting stage and level theories are not set-theoretically equivalent; moreover, second-order level theory with urelements is not weakly quasi-categorical. We then consider further principles governing stages and levels, motivated by directedness, unboundedness, and reflection. Some of these principles restore set-theoretic equivalence between the corresponding theories, while their level-theoretic versions yield forms of quasi-categoricity. These principles form strict implication hierarchies, thereby revealing distinct stronger conceptions of set beyond the basic iterative conception.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30422",
  "author_total": 1,
  "authors": [
    {
      "name": "Bokai Yao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/433mt9esxbhe2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30422",
  "fulltext_url": null,
  "id": "433mt9esxbhe2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30422",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7204942055",
  "pdf_url": "https://arxiv.org/pdf/2608.30422",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/433mt9esxbhe2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30422",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30422"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "The Iterative Conception Reconsidered",
  "title_text": "The Iterative Conception Reconsidered",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/433mt9esxbhe2",
  "venue": "arXiv",
  "versions": []
}