{
  "id": "1701.04358",
  "version": "v1",
  "published": "2017-01-16T17:20:43.000Z",
  "updated": "2017-01-16T17:20:43.000Z",
  "title": "Adding a non-reflecting weakly compact set",
  "authors": [
    "Brent Cody"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "For $n<\\omega$, we say that the $\\Pi^1_n$-reflection principle holds at $\\kappa$ and write $Refl_n(\\kappa)$ if and only if every $\\Pi^1_n$-indescribable subset of $\\kappa$ has a $\\Pi^1_n$-indescribable proper initial segment. The $\\Pi^1_n$-reflection principle $Refl_n(\\kappa)$ generalizes a certain stationary reflection principle and implies that $\\kappa$ is $\\Pi^1_n$-indescribable of order $\\omega$. We prove that the converse of this implication is consistently false in the case $n=1$. Moreover, we prove that if $\\kappa$ is $(\\alpha+1)$-weakly compact then there is a forcing extension in which there is a weakly compact set $W\\subseteq\\kappa$ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and $\\kappa$ remains $(\\alpha+1)$-weakly compact.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-16T17:20:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E35",
      "03E55"
    ],
    "keywords": [
      "non-reflecting weakly compact set",
      "weakly compact proper initial segment",
      "reflection principle holds",
      "indescribable proper initial segment",
      "stationary reflection principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}