{
  "id": "1902.04146",
  "version": "v1",
  "published": "2019-02-11T21:03:13.000Z",
  "updated": "2019-02-11T21:03:13.000Z",
  "title": "A $\\square(κ)$-like principle consistent with weak compactness",
  "authors": [
    "Brent Cody",
    "Victoria Gitman",
    "Chris Lambie-Hanson"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Sun proved that when $\\kappa$ is weakly compact, the \\emph{$1$-club} subsets of $\\kappa$ provide a filter base for the weakly compact ideal, and hence can also be used to give a characterization of weakly compact sets which resembles the definition of stationarity: a set $S\\subseteq\\kappa$ is weakly compact (or equivalently $\\Pi^1_1$-indescribable) if and only if $S\\cap C\\neq\\emptyset$ for every $1$-club $C\\subseteq\\kappa$. By replacing clubs with $1$-clubs in the definition of $\\Box(\\kappa)$, we obtain a $\\Box(\\kappa)$-like principle we call $\\Box_1(\\kappa)$ that is consistent with the weak compactness of $\\kappa$ but inconsistent with the $\\Pi^1_2$-indescribability of $\\kappa$. By generalizing the standard forcing to add a $\\Box(\\kappa)$-sequence, we show that if $\\kappa$ is $\\kappa^+$-weakly compact and GCH holds then there is a cofinality-preserving forcing extension in which $\\kappa$ remains $\\kappa^+$-weakly compact and $\\Box_1(\\kappa)$ holds. If $\\kappa$ is $\\Pi^1_2$-indescribable and GCH holds then there is a cofinality-preserving forcing extension in which $\\kappa$ is $\\kappa^+$-weakly compact, $\\Box_1(\\kappa)$ holds and every weakly compact subset of $\\kappa$ has a weakly compact proper initial segment. As an application, we prove that, relative to a $\\Pi^1_2$-indescribable cardinal, it is consistent that $\\kappa$ is $\\kappa^+$-weakly compact, every weakly compact subset of $\\kappa$ has a weakly compact proper initial segment, and there exist two weakly compact subsets $S^0$ and $S^1$ of $\\kappa$ such that there is no $\\beta<\\kappa$ for which both $S^0\\cap\\beta$ and $S^1\\cap\\beta$ are weakly compact.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-11T21:03:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E55",
      "03E35"
    ],
    "keywords": [
      "weak compactness",
      "weakly compact proper initial segment",
      "principle consistent",
      "weakly compact subset",
      "cofinality-preserving forcing extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}