{
  "id": "2407.20098",
  "version": "v1",
  "published": "2024-07-29T15:26:10.000Z",
  "updated": "2024-07-29T15:26:10.000Z",
  "title": "On the significance of parameters and the projective level in the Choice and Collection axioms",
  "authors": [
    "Vladimir Kanovei",
    "Vassily Lyubetsky"
  ],
  "comment": "124 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "We make use of generalized iterations of a version of the Jensen forcing to define a cardinal-preserving generic model of ZF for any $n\\ge 1$ and each of the following four Choice hypotheses: (1) $\\text{DC}(\\mathbf\\Pi^1_n)\\land\\neg\\text{AC}_\\omega(\\varPi^1_{n+1})\\,;$ (2) $\\text{AC}_\\omega(\\text{OD})\\land\\text{DC}(\\varPi^1_{n+1})\\land \\neg\\text{AC}_\\omega(\\mathbf\\Pi^1_{n+1});$ (3) $\\text{AC}_\\omega\\land\\text{DC}(\\mathbf\\Pi^1_n)\\land\\neg\\text{DC}(\\varPi^1_{n+1});$ (4) $\\text{AC}_\\omega\\land\\text{DC}(\\varPi^1_{n+1})\\land\\neg\\text{DC}(\\mathbf\\Pi^1_{n+1}).$ Thus if ZF is consistent and $n\\ge1$ then each of these four conjunctions (1)--(4) is consistent with ZF. As for the second main result, let PA$^0_2$ be the 2nd-order Peano arithmetic without the Comprehension schema $\\text{CA}$. For any $n\\ge1$, we define a cardinal-preserving generic model of ZF, and a set $M\\subseteq\\mathcal P(\\omega)$ in this model, such that $\\langle\\omega, M\\rangle$ satisfies (5) PA$^0_2$ + $\\text{AC}_\\omega(\\varSigma^1_{\\infty})$ + $\\text{CA}(\\mathbf\\Sigma^1_{n+1})$ + $\\neg\\text{CA}(\\mathbf\\Sigma^1_{n+1})$. Thus $\\text{CA}(\\mathbf\\Sigma^1_{n+1})$ does not imply $\\text{CA}(\\mathbf\\Sigma^1_{n+2})$ in PA$^0_2$ even in the presence of the full parameter-free (countable) Choice $\\text{AC}_\\omega(\\varSigma^1_{\\infty}).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-29T15:26:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E15",
      "03E35"
    ],
    "keywords": [
      "collection axioms",
      "projective level",
      "cardinal-preserving generic model",
      "significance",
      "parameters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 124,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}