{
  "id": "1908.09664",
  "version": "v1",
  "published": "2019-08-26T13:18:14.000Z",
  "updated": "2019-08-26T13:18:14.000Z",
  "title": "$C^{(n)}$-Cardinals",
  "authors": [
    "Joan Bagaria"
  ],
  "journal": "Arch. Math. Logic, (2012) 51:213-240",
  "categories": [
    "math.LO"
  ],
  "abstract": "For each natural number $n$, let $C^{(n)}$ be the closed and unbounded proper class of ordinals $\\alpha$ such that $V_\\alpha$ is a $\\Sigma_n$ elementary substructure of $V$. We say that $\\kappa$ is a \\emph{$C^{(n)}$-cardinal} if it is the critical point of an elementary embedding $j:V\\to M$, $M$ transitive, with $j(\\kappa)$ in $C^{(n)}$. By analyzing the notion of $C^{(n)}$-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of superstrong cardinals and up to the level of rank-into-rank embeddings, $C^{(n)}$-cardinals form a much finer hierarchy. The naturalness of the notion of $C^{(n)}$-cardinal is exemplified by showing that the existence of $C^{(n)}$-extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of \\cite{BCMR}, we give new characterizations of Vope\\v{n}ka's Principle in terms of $C^{(n)}$-extendible cardinals.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-26T13:18:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extendible cardinals",
      "large cardinal principles",
      "simple reflection principles",
      "usual hierarchy",
      "unbounded proper class"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}