{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nAssume that [[ZFC|0HE5WDQ9FJ]], written in first-order logic, is consistent.\nIts signature has one binary relation symbol $\\in$, so it is countable.\nBy the [[Löwenheim–Skolem theorem|484M0TZ1S7]], ZFC has a [[countable|01TQB88H4C]] [[model|MVCH6N9SNW]] $M$.\nBy [[Cantor's theorem|0WTEMGYV2C]], ZFC proves that the power set of $\\omega$ is [[uncountable|0W7FBYS1F1]].\nSo $M$ contains an element $p$ that $M$ believes is uncountable: no element of $M$ is a bijection between $\\omega^M$ and $p$.\nBut $M$ is countable, so the set of elements of $p$ in $M$ is countable when seen from outside.\nThere is no contradiction.\nA bijection between $\\omega^M$ and the members of $p$ exists outside $M$, but it is not an element of $M$.\n\n## Remarks\n\nThe example shows that \"uncountable\" is not an absolute notion: it depends on which functions the model contains.\nThe same reasoning shows that no countable first-order theory with an infinite model can pin down the size of its models.\nThoralf Skolem pointed out this situation in the 1920s.\n\n## Explanations\n\n### L1 human\n\nSet theory proves that some sets are uncountable, yet it has a model with only countably many objects.\nThis sounds impossible, but it is not.\nInside the small model, the counting functions that would list an uncountable set are simply missing.\nFrom outside, one can list the set, but the model cannot see this list.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "illustrates": [
      "484M0TZ1S7"
    ],
    "kind": "example",
    "msc": [
      "03C62",
      "03E30",
      "03A05"
    ],
    "name": "Skolem's paradox",
    "same_as": {
      "wikidata": "Q1096353",
      "wikipedia": "Skolem's paradox"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Skolem's_paradox"
      }
    ],
    "tag": "68QZWDXAQ4",
    "uses": [
      "0HE5WDQ9FJ",
      "01TQB88H4C",
      "0W7FBYS1F1",
      "0WTEMGYV2C",
      "MVCH6N9SNW"
    ]
  },
  "kind": "example",
  "link": "https://math2.tamnd.com/node/68QZWDXAQ4/",
  "links": [
    {
      "field": "illustrates",
      "target": "484M0TZ1S7"
    },
    {
      "field": "uses",
      "target": "0HE5WDQ9FJ"
    },
    {
      "field": "uses",
      "target": "01TQB88H4C"
    },
    {
      "field": "uses",
      "target": "0W7FBYS1F1"
    },
    {
      "field": "uses",
      "target": "0WTEMGYV2C"
    },
    {
      "field": "uses",
      "target": "MVCH6N9SNW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/68QZWDXAQ4.md",
  "msc": [
    "03C62",
    "03E30",
    "03A05"
  ],
  "name": "Skolem's paradox",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/68Q/68QZWDXAQ4.md",
  "status": null,
  "tag": "68QZWDXAQ4",
  "uri": "m2://68QZWDXAQ4",
  "url": "https://math2.tamnd.com/wiki/skolems-paradox/",
  "writing": {
    "tag": "68QZWDXAQ4",
    "kind": "example",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}