{
  "id": "1804.02213",
  "version": "v2",
  "published": "2018-04-06T11:54:08.000Z",
  "updated": "2019-12-08T23:47:55.000Z",
  "title": "On $Σ_1^1$-completeness of quasi-orders on $κ^κ$",
  "authors": [
    "Tapani Hyttinen",
    "Vadim Kulikov",
    "Miguel Moreno"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "We prove under $V=L$ that the inclusion modulo the non-stationary ideal is a $\\Sigma_1^1$-complete quasi-order in the generalized Borel-reducibility hierarchy ($\\kappa>\\omega$). This improvement to known results in $L$ has many new consequences concerning the $\\Sigma_1^1$-completeness of quasi-orders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the non-stationary ideal. This serves as a partial or complete answer to several open problems stated in literature. Additionally the theorem is applied to prove a dichotomy in $L$: If the isomorphism of a countable first-order theory (not necessarily complete) is not $\\Delta_1^1$, then it is $\\Sigma_1^1$-complete. We also study the case $V\\ne L$ and prove $\\Sigma_1^1$-completeness results for weakly ineffable and weakly compact $\\kappa$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2019-12-08T23:47:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-stationary ideal",
      "dense linear orders",
      "inclusion modulo",
      "complete quasi-order",
      "completeness results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}