{
  "id": "1707.08506",
  "version": "v1",
  "published": "2017-07-26T15:43:12.000Z",
  "updated": "2017-07-26T15:43:12.000Z",
  "title": "How much weak compactness does the weakly compact reflection principle imply?",
  "authors": [
    "Brent Cody",
    "Hiroshi Sakai"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "The weakly compact reflection principle $\\text{Refl}_{\\text{wc}}(\\kappa)$ states that $\\kappa$ is a weakly compact cardinal and every weakly compact subset of $\\kappa$ has a weakly compact proper initial segment. The weakly compact reflection principle at $\\kappa$ implies that $\\kappa$ is an $\\omega$-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that $\\kappa$ is $(\\omega+1)$-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at $\\kappa$ then there is a forcing extension preserving this in which $\\kappa$ is the least $\\omega$-weakly compact cardinal. Along the way we generalize the well-known result which states that if $\\kappa$ is a regular cardinal then in any forcing extension by $\\kappa$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $\\kappa$ is a weakly compact cardinal then after forcing with a `typical' Easton-support iteration of length $\\kappa$ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-26T15:43:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03E35",
      "03E55"
    ],
    "keywords": [
      "weakly compact reflection principle",
      "weakly compact cardinal",
      "compact reflection principle holds",
      "weak compactness",
      "compact proper initial segment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}