{
  "id": "2212.13627",
  "version": "v1",
  "published": "2022-12-27T21:48:38.000Z",
  "updated": "2022-12-27T21:48:38.000Z",
  "title": "Forcing with Urelements",
  "authors": [
    "Bokai Yao"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "ZFCU$_{\\rm R}$ is ZFC (with the Replacement Scheme) modified to allow a class of urelements. I first isolate a hierarchy of axioms based on ZFCU$_{\\rm R}$ and argue that the Collection Principle should be included as an axiom in order to obtain a more robust set theory with urelements. I then turn to forcing over countable transitive models of ZFCU$_{\\rm R}$. A new definition of $\\mathbb{P}$-names is given. The resulting forcing relation is full just in case the Collection Principle holds in the ground model. While forcing preserves ZFCU$_{\\rm R}$ and many axioms in the hierarchy, it can also destroy the DC$_{\\omega_1}$-scheme and recover the Collection Principle. The ground model definability fails when the ground model contains a proper class of urelements.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-27T21:48:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E30",
      "03E40"
    ],
    "keywords": [
      "urelements",
      "ground model definability fails",
      "ground model contains",
      "collection principle holds",
      "robust set theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}