{
  "id": "1306.0449",
  "version": "v3",
  "published": "2013-06-03T15:07:53.000Z",
  "updated": "2013-09-11T13:49:04.000Z",
  "title": "On supercompactness and the continuum function",
  "authors": [
    "Brent Cody",
    "Menachem Magidor"
  ],
  "comment": "12 pages",
  "categories": [
    "math.LO"
  ],
  "abstract": "Given a cardinal $\\kappa$ that is $\\lambda$-supercompact for some regular cardinal $\\lambda\\geq\\kappa$ and assuming $\\GCH$, we show that one can force the continuum function to agree with any function $F:[\\kappa,\\lambda]\\cap\\REG\\to\\CARD$ satisfying $\\forall\\alpha,\\beta\\in\\dom(F)$ $\\alpha<\\cf(F(\\alpha))$ and $\\alpha<\\beta$ $\\implies$ $F(\\alpha)\\leq F(\\beta)$, while preserving the $\\lambda$-supercompactness of $\\kappa$ from a hypothesis that is of the weakest possible consistency strength, namely, from the hypothesis that there is an elementary embedding $j:V\\to M$ with critical point $\\kappa$ such that $M^\\lambda\\subseteq M$ and $j(\\kappa)>F(\\lambda)$. Our argument extends Woodin's technique of surgically modifying a generic filter to a new case: Woodin's key lemma applies when modifications are done on the range of $j$, whereas our argument uses a new key lemma to handle modifications done off of the range of $j$ on the ghost coordinates. This work answers a question of Friedman and Honzik [FH2012]. We also discuss several related open questions.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-09-11T13:49:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E35",
      "03E55"
    ],
    "keywords": [
      "continuum function",
      "supercompactness",
      "argument extends woodins technique",
      "regular cardinal",
      "work answers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1306.0449C"
    }
  }
}