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  "body": "\n## Example\n\nA hotel has rooms numbered $1, 2, 3, \\dots$, one for each positive integer, and each room has a guest.\nA new guest arrives. The manager moves the guest of room $n$ to room $n + 1$ for each $n$. Then room $1$ is free for the new guest.\nNext, infinitely many new guests $g_1, g_2, \\dots$ arrive. The manager moves the guest of room $n$ to room $2n$, and gives room $2k - 1$ to $g_k$.\nIn both cases, each guest has a room and no two guests share a room.\n\n## Remarks\n\nThe maps $n \\mapsto n + 1$ and $n \\mapsto 2n$ are [[injections|0PVG1CRAJN]] of $\\mathbb{N}_{\u003e0}$ into a proper subset of itself. So the set of rooms is equinumerous with a proper subset.\nThis cannot happen for a [[finite set|0BBTH3YSZA]].\nDavid Hilbert introduced the hotel in a lecture of 1924 to 1925, and George Gamow made it popular in a book of 1947.\n\n## Explanations\n\n### L1 human\n\nA full finite hotel cannot take a new guest, but a full infinite hotel can.\nThe example shows that \"as many as\" for infinite sets does not behave like counting.\n",
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