{
  "id": "1609.00093",
  "version": "v1",
  "published": "2016-09-01T02:44:50.000Z",
  "updated": "2016-09-01T02:44:50.000Z",
  "title": "From geometry to geology: An invitation to mathematical pluralism through the phenomenon of independence",
  "authors": [
    "Jonas Reitz"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "This paper explores how a pluralist view can arise in a natural way out of the day-to-day practice of modern set theory. By contrast, the widely accepted orthodox view is that there is an ultimate universe of sets $V$, and it is in this universe that mathematics takes place. From this view, the purpose of set theory is \"learning the truth about $V$.\" It has become apparent, however, that the phenomenon of independence - those questions left unresolved by the axioms - holds a central place in the investigation. This paper introduces the notion of independence, explores the primary tool (\"soundness\") for establishing independence results, and shows how a plurality of models arises through the investigation of this phenomenon. Building on a familiar example from Euclidean geometry, a template for independence proofs is established. Applying this template in the domain of set theory leads to a consideration of forcing, the tool par excellence for constructing universes of sets. Fifty years of forcing has resulted in a profusion of universes exhibiting a wide variety of characteristics - a multiverse of set theories. Direct study of this multiverse presents technical challenges due to its second-order nature. Nonetheless, there are certain nice \"local neighborhoods\" of the multiverse that are amenable to first-order analysis, and \\emph{set-theoretic geology} studies just such a neighborhood, the collection of grounds of a given universe $V$ of set theory. I will explore some of the properties of this collection, touching on major concepts, open questions, and recent developments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-01T02:44:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "00A30",
      "03E35"
    ],
    "keywords": [
      "independence",
      "mathematical pluralism",
      "phenomenon",
      "invitation",
      "multiverse"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}