{
  "id": "1311.0303",
  "version": "v1",
  "published": "2013-11-01T20:56:13.000Z",
  "updated": "2013-11-01T20:56:13.000Z",
  "title": "Easton functions and supercompactness",
  "authors": [
    "Brent Cody",
    "Sy-David Friedman",
    "Radek Honzik"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Suppose $\\kappa$ is $\\lambda$-supercompact witnessed by an elementary embedding $j:V\\rightarrow M$ with critical point $\\kappa$, and further suppose that $F$ is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton's theorem: (1) $\\forall\\alpha$ $\\alpha<\\textrm{cf}(F(\\alpha))$ and (2) $\\alpha<\\beta$ $\\Longrightarrow$ $F(\\alpha)\\leq F(\\beta)$. In this article we address the question: assuming GCH, what additional assumptions are necessary on $j$ and $F$ if one wants to be able to force the continuum function to agree with $F$ globally, while preserving the $\\lambda$-supercompactness of $\\kappa$? We show that, assuming GCH, if $F$ is any function as above, and in addition for some regular cardinal $\\lambda>\\kappa$ there is an elementary embedding $j:V\\rightarrow M$ with critical point $\\kappa$ such that $\\kappa$ is closed under $F$, the model $M$ is closed under $\\lambda$-sequences, $H(F(\\lambda))\\subseteq M$, and for each regular cardinal $\\gamma\\leq \\lambda$ one has $(|j(F)(\\gamma)|=F(\\gamma))^V$, then there is a cardinal-preserving forcing extension in which $2^\\delta=F(\\delta)$ for every regular cardinal $\\delta$ and $\\kappa$ remains $\\lambda$-supercompact. This answers a question of B. Cody, M. Magidor, On supercompactness and the continuum function, Ann. Pure Appl. Logic, (2013).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-01T20:56:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E35",
      "03E55"
    ],
    "keywords": [
      "easton functions",
      "regular cardinal",
      "supercompactness",
      "continuum function",
      "critical point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.0303C"
    }
  }
}