{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "01BX15HSRR"
    },
    {
      "field": "uses",
      "target": "0GF6AJEBZ0"
    },
    {
      "field": "uses",
      "target": "0KMVFSHFGZ"
    }
  ],
  "body": "\n## Definition\n\nLet $a$ and $b$ be [[natural numbers|0MWKJ44V5A]].\nThe notation $a \\le b$ means that there is a natural number $c$ with $a + c = b$, where $+$ is the [[addition|0W5DSDFQ4G]] of $\\mathbb{N}$.\nThe notation $a \u003c b$ means that $a \\le b$ and $a \\neq b$.\n\n## Remarks\n\nThe relation $\\le$ is a total order on $\\mathbb{N}$: it is reflexive, antisymmetric and transitive, and $a \\le b$ or $b \\le a$ for all $a$ and $b$.\nThe order agrees with the operations: if $a \\le b$, then $a + c \\le b + c$ and $a c \\le b c$ for each natural number $c$.\nEach nonempty set of natural numbers has a least element for this order. This is the [[well-ordering principle|01BX15HSRR]].\n\n## Explanations\n\n### L1 human\n\nThe number $a$ is at most $b$ when you can get from $a$ to $b$ by adding something.\nFor example, $3 \\le 7$ because $3 + 4 = 7$, but $7 \\le 3$ is false because no natural number added to $7$ gives $3$.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0HNATXFEV5",
    "kind": "definition",
    "msc": [
      "03F30"
    ],
    "name": "Order on the natural numbers",
    "sources": [
      {
        "locator": "Order",
        "url": "https://en.wikipedia.org/wiki/Natural_number"
      }
    ],
    "tag": "0Y9H52BT03",
    "uses": [
      "0MWKJ44V5A",
      "0W5DSDFQ4G"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0Y9H52BT03/",
  "links": [
    {
      "field": "defines",
      "target": "0HNATXFEV5"
    },
    {
      "field": "uses",
      "target": "0MWKJ44V5A"
    },
    {
      "field": "uses",
      "target": "0W5DSDFQ4G"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0Y9H52BT03.md",
  "msc": [
    "03F30"
  ],
  "name": "Order on the natural numbers",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0Y9/0Y9H52BT03.md",
  "status": null,
  "tag": "0Y9H52BT03",
  "uri": "m2://0Y9H52BT03",
  "url": "https://math2.tamnd.com/wiki/order-on-the-natural-numbers-definition/",
  "writing": {
    "tag": "0Y9H52BT03",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}