{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nAssume the [[axiom of choice|0HTGB13FE2]].\nLet $A$ be an [[infinite set|01MKW4R283]].\nThen there is an [[injective function|0PVG1CRAJN]] $f \\colon \\mathbb{N} \\to A$, so $A$ has a [[countably infinite|0SEZ3JGZP7]] subset $f(\\mathbb{N})$.\n\n## Remarks\n\nThe proof chooses $a_0 \\in A$, then $a_1 \\in A \\setminus \\{a_0\\}$, and so on. Each set $A \\setminus \\{a_0, \\dots, a_n\\}$ is nonempty because $A$ is infinite.\nThe axiom of countable choice is enough, but some choice is needed: in ZF alone, the statement cannot be proved.\nIt follows that $\\aleph_0$ is the least infinite cardinal.\n\n## Explanations\n\n### L1 human\n\nIn an infinite set you can always pick one more element.\nPicking forever gives a sequence of different elements, which is a copy of $\\mathbb{N}$ inside the set.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "03E10"
    ],
    "name": "Every infinite set has a countably infinite subset",
    "role": "theorem",
    "sources": [
      {
        "locator": "Properties",
        "url": "https://en.wikipedia.org/wiki/Infinite_set"
      }
    ],
    "tag": "0RWQ5VZ6VQ",
    "uses": [
      "01MKW4R283",
      "0SEZ3JGZP7",
      "0PVG1CRAJN",
      "0HTGB13FE2"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0RWQ5VZ6VQ/",
  "links": [
    {
      "field": "uses",
      "target": "01MKW4R283"
    },
    {
      "field": "uses",
      "target": "0SEZ3JGZP7"
    },
    {
      "field": "uses",
      "target": "0PVG1CRAJN"
    },
    {
      "field": "uses",
      "target": "0HTGB13FE2"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0RWQ5VZ6VQ.md",
  "msc": [
    "03E10"
  ],
  "name": "Every infinite set has a countably infinite subset",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0RW/0RWQ5VZ6VQ.md",
  "status": null,
  "tag": "0RWQ5VZ6VQ",
  "uri": "m2://0RWQ5VZ6VQ",
  "url": "https://math2.tamnd.com/wiki/every-infinite-set-has-a-countably-infinite-subset/",
  "writing": {
    "tag": "0RWQ5VZ6VQ",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}