{
  "id": "1104.3077",
  "version": "v2",
  "published": "2011-04-15T14:52:48.000Z",
  "updated": "2018-10-28T15:03:29.000Z",
  "title": "Projective sets, intuitionistically",
  "authors": [
    "Wim Veldman"
  ],
  "comment": "82 pages",
  "doi": "10.13140/RG.2.2.21225.80484",
  "categories": [
    "math.LO"
  ],
  "abstract": "We study `definable' subsets of Baire space $\\mathcal{N}$. The logic of our arguments is intuitionistic and we use L.E.J.~Brouwer's Thesis on bars in $\\mathcal{N}$ and his continuity axioms. We avoid the operation of taking the complement of a subset of $\\mathcal{N}$. A subset of $\\mathcal{N}$ is $\\mathbf{\\Sigma}^1_1$ or: analytic if it is the projection of a closed subset of $\\mathcal{N}$. Important $\\mathbf{\\Sigma}^1_1$ set are the set of the codes of all closed and located subsets of $\\mathcal{N}$ that are positively uncountable and the set of the codes of all located and closed subsets of $\\mathcal{N}$ containing at least one member coding a (positively) infinite subset of $\\mathbb{N}$. A subset of $\\mathcal{N}$ is strictly analytic if it is the projection of a closed and located subset of $\\mathcal{N}$. Brouwer's Thesis on bars in $\\mathcal{N}$ proves separation and boundedness theorems for strictly analytic subsets of $\\mathcal{N}$. A subset of $\\mathcal{N}$ is $\\mathbf{\\Pi}^1_1$ or: co-analytic if it is the co-projection of an open subset of $\\mathcal{N} \\times \\mathcal{N}=\\mathcal{N}$. There is no symmetry between analytic and co-analytic sets like in classical descriptive set theory. An important $\\mathbf{\\Pi}^1_1$ set is the set of the codes of all closed and located subsets of $\\mathcal{N}$ all of whose members code an almost-finite subset of $\\mathbb{N}$. The set of the codes of closed and located subsets of $\\mathcal{N}$ that are almost-countable, or, equivalently, \\{reducible in Cantor's sense, is treated at some length. This set is probably not $\\mathbf{\\Pi}^1_1$. The projective hierarchy collapses: every (positively) projective set is $\\mathbf{\\Sigma}^1_2$: the projection of a co-analytic subset of $\\mathcal{N}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-04-15T14:52:48.000Z",
      "title": "Analytic, co-analytic and projective sets from Brouwer's intuitionistic perspective",
      "abstract": "We study projective subsets of Baire space from Brouwer's intuitionistic point of view, using his Thesis on Bars and his continuity axioms. We first study analytic sets; these are the projections of the closed subsets of Baire space. We consider a number of examples and discover a fine structure in the class of the analytic sets that fail to be positively Borel. A subset of Baire space is strictly analytic if it coincides with the range of a continuous function from Baire space to itself. We prove separation and boundedness theorems for strictly analytic sets. Co-analytic sets are the co-projections of the open subsets of Baire space. We show different ways to prove that some co-analytic sets are not analytic and that some analytic sets are not co-analytic. We consider the set of the codes of the closed and located subsets of Baire space that are almost-countable as an example of a set that is a projection of a co-analytic set. We bring to light the collapse of the projective hierarchy: every (positively) projective set coincides with the projection of a co-analytic set.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2018-10-28T15:03:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03F55",
      "03E15",
      "04A15"
    ],
    "keywords": [
      "baire space",
      "brouwers intuitionistic perspective",
      "projective set",
      "co-analytic set",
      "first study analytic sets"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 82,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1104.3077V"
    }
  }
}