{
  "id": "0812.0656",
  "version": "v3",
  "published": "2008-12-03T05:42:22.000Z",
  "updated": "2023-05-01T18:42:06.000Z",
  "title": "Black Boxes",
  "authors": [
    "Saharon Shelah"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We shall deal comprehensively with Black Boxes, the intention being that provably in ZFC we have a sequence of guesses of extra structure on small subsets, where the guesses are pairwise almost disjoint; by this we mean they have quite little interaction, and are far apart but together are dense. We first deal with the simplest case, where the existence comes from winning a game by just writing down the opponent's moves. We show how it helps when instead of orders we have trees with boundedly many levels, having freedom in the last. After this we quite systematically look at existence of black boxes, and make connection to non-saturation of natural ideals and diamonds on them.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-06-21T03:02:26.000Z",
      "abstract": "We introduce Black Boxes, the intention being provably in ZFC, we have guesses of extra structure. The guesses are pairwise quite far but together are \"dense\". We first deal with the simplest case, were the existence comes from winning a game by just writing down the opponent's moves. We show how it helps when instead orders we have tress with boundedly many levels, having freedom in the last. After this we quite systematically look at existence of black boxes, and make connection to non-saturation of natural ideals and diamonds on them. Chapter IV of upcoming book \"Nonstructure Theory\".",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2023-05-01T18:42:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E05",
      "03C55",
      "03C45"
    ],
    "keywords": [
      "black boxes",
      "extra structure",
      "pairwise quite far",
      "first deal",
      "nonstructure theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.0656S"
    }
  }
}