{
  "id": "2108.03666",
  "version": "v1",
  "published": "2021-08-08T15:29:43.000Z",
  "updated": "2021-08-08T15:29:43.000Z",
  "title": "Are $\\mathfrak a$ and $\\mathfrak d$ your cup of tea? Revisited",
  "authors": [
    "Saharon Shelah"
  ],
  "comment": "revisited version of arXiv:math/0012170",
  "categories": [
    "math.LO"
  ],
  "abstract": "This is a revised version (of late 2020) of [Sh:700], which is arXiv:math/0012170 . First point is noting that the proof of Theorem 4.3 in [Sh:700], which says that the proof giving the consistency $ \\mathfrak{b} = \\mathfrak{d} = \\mathfrak{u} < \\mathfrak{a} $ also gives $ \\mathfrak{s} = \\mathfrak{d} $. The proof uses a measurable cardinal and a c.c.c. forcing so it gives large $ \\mathfrak{d} $ and assumes a large cardinal. Second point is adding to the results of \\S2,\\S3 which say that (in \\S3 with no large cardinals) we can force $ {\\aleph_1} < \\mathfrak{b} = \\mathfrak{d} < \\mathfrak{a}$. We like to have $ {\\aleph_1} < \\mathfrak{s} \\le \\mathfrak{b} = \\mathfrak{d} < \\mathfrak{a} $. For this we allow in \\S2,\\S3 the sets $ K_t $ to be uncountable; this requires non-essential changes. In particular, we replace usually $ {\\aleph_0}, {\\aleph_1} $ by $ \\sigma , \\partial $. Naturally we can deal with $ \\mathfrak{i} $ and similar invariants. Third we proofread the work again. To get $ \\mathfrak{s} $ we could have retained the countability of the member of the $ I_t$-s but the parameters would change with $ A \\in I_t$, well for a cofinal set of them; but the present seems simpler. We intend to continue in [Sh:F2009].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-08T15:29:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large cardinal",
      "first point",
      "second point",
      "non-essential changes",
      "similar invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}