{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nWork in first-order logic with the relation symbol $\\in$.\nLet $\\varphi(x, y, w_1, \\dots, w_n)$ be a formula whose free variables are among $x, y, w_1, \\dots, w_n$, and in which $A$ and $B$ do not occur.\nThe *instance of replacement* for $\\varphi$ is the axiom\n$$\\forall w_1 \\cdots \\forall w_n\\, \\forall A\\, \\Bigl( \\forall x \\in A\\, \\exists! y\\, \\varphi(x, y, w) \\to \\exists B\\, \\forall y\\, \\bigl( y \\in B \\leftrightarrow \\exists x \\in A\\, \\varphi(x, y, w) \\bigr) \\Bigr).$$\nThe *axiom schema of replacement* is the set of these axioms, one for each formula $\\varphi$.\n\nIn words: if $\\varphi$ defines a [[function|0XHDTE5YMZ]] on the [[set|0SJ3RX9TXA]] $A$, then the image of $A$ under this function is a set.\n\n## Remarks\n\nExample: let $\\varphi(x, y)$ say \"$x$ is a natural number and $y$ is the set $\\omega + x$\".\nFor $A = \\omega$ the hypothesis holds, so replacement gives the set $\\set{\\omega + n : n \\in \\omega}$, whose union is the ordinal $\\omega + \\omega$.\nNon-example: the set $V_{\\omega + \\omega}$ of all sets of rank below $\\omega + \\omega$ satisfies all axioms of Zermelo set theory, but not replacement.\nIn it, the instance above fails: each $\\omega + n$ lies in $V_{\\omega + \\omega}$, but the set $\\set{\\omega + n : n \\in \\omega}$ has rank $\\omega + \\omega$ and does not.\nReplacement implies the separation scheme, given the other axioms of [[ZF|0HE5WDQ9FJ]].\nIt is used to justify definitions by transfinite recursion on the ordinals.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "D96K8S8R61",
    "kind": "definition",
    "msc": [
      "03E30"
    ],
    "name": "Axiom schema of replacement",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Axiom_schema_of_replacement"
      }
    ],
    "tag": "9BPQXY11NR",
    "uses": [
      "0HE5WDQ9FJ",
      "0SJ3RX9TXA",
      "0XHDTE5YMZ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/9BPQXY11NR/",
  "links": [
    {
      "field": "defines",
      "target": "D96K8S8R61"
    },
    {
      "field": "uses",
      "target": "0HE5WDQ9FJ"
    },
    {
      "field": "uses",
      "target": "0SJ3RX9TXA"
    },
    {
      "field": "uses",
      "target": "0XHDTE5YMZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/9BPQXY11NR.md",
  "msc": [
    "03E30"
  ],
  "name": "Axiom schema of replacement",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9BP/9BPQXY11NR.md",
  "status": null,
  "tag": "9BPQXY11NR",
  "uri": "m2://9BPQXY11NR",
  "url": "https://math2.tamnd.com/wiki/axiom-schema-of-replacement-definition/",
  "writing": {
    "tag": "9BPQXY11NR",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}