{
  "abstract": "The weak set theory $\\mathsf{ReR}$ is obtained from Kripke-Platek Set Theory ($\\mathsf{KP}$) by replacing the bounded collection scheme with the bounded replacement scheme. We show that $\\mathsf{ReR}$ proves $\\mathsf{TCo}$, which asserts that every set is contained in a transitive set. This is used to show that the theories obtained by adding the negation of the axiom of infinity to $\\mathsf{ReR}$ and $\\mathsf{KP}$ have the same consequences. Our proof of $\\mathsf{TCo}$ relies on the availability of a fragment of class foundation in $\\mathsf{ReR}$. To demonstrate the necessity of this reliance, even in the presence of infinity, we build a model of a significant fragment of $\\mathsf{ZF}$ that includes bounded separation and collection, infinity, powerset, regularity and the axiom of choice, in which $\\mathsf{TCo}$ fails.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23398",
  "author_total": 1,
  "authors": [
    {
      "name": "Zachiri McKenzie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9zkxhwkcvmn0a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23398",
  "fulltext_url": null,
  "id": "9zkxhwkcvmn0a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23398",
  "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": "W7204171178",
  "pdf_url": "https://arxiv.org/pdf/2608.23398",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9zkxhwkcvmn0a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2608.23398",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23398"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "On a slight weakening of Kripke-Platek Set Theory",
  "title_text": "On a slight weakening of Kripke-Platek Set Theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/9zkxhwkcvmn0a",
  "venue": "arXiv",
  "versions": []
}