{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nProving that the axiom of separation, axiom of replacement, and [[axiom of choice|0HTGB13FE2]] hold in $L$ requires (at least as shown above) the use of a reflection principle for $L$. Here we describe such a principle.\n\nBy induction on $n\u003c\\omega$, we can use ZF in $V$ to prove that for any ordinal $\\alpha$, there is an ordinal $\\beta\u003e\\alpha$ such that for any sentence $P(z_1,\\ldots,z_k)$ with $z_1,\\ldots,z_k$ in $L_\\beta$ and containing fewer than $n$ symbols (counting a constant symbol for an element of $L_\\beta$ as one symbol) we get that $P(z_1,\\ldots,z_k)$ holds in $L_\\beta$ if and only if it holds in $L$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "inventory": [
      "NBWMGF4K2E"
    ],
    "kind": "statement",
    "name": "L has a reflection principle",
    "role": "proposition",
    "sources": [
      {
        "locator": "Constructible universe, section L has a reflection principle",
        "mention": "wikipedia://en/343335#4UZO",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:343335@r1372309993"
      }
    ],
    "tag": "ANWT2HEH2Y"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/ANWT2HEH2Y/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/ANWT2HEH2Y.md",
  "msc": null,
  "name": "L has a reflection principle",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ANW/ANWT2HEH2Y.md",
  "status": null,
  "tag": "ANWT2HEH2Y",
  "uri": "m2://ANWT2HEH2Y",
  "url": "https://math2.tamnd.com/wiki/l-has-a-reflection-principle/",
  "writing": {
    "tag": "ANWT2HEH2Y",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}