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  "body": "\n## Summary\n\nCantor's diagonal argument shows that some infinite sets are [[uncountable|0X55M9XET4]].\nGiven a list $s_0, s_1, s_2, \\dots$ of infinite sequences of $0$ and $1$, define a sequence $t$ by $t(n) = 1 - s_n(n)$.\nThen $t$ differs from each $s_n$ at position $n$, so $t$ is not in the list.\nSo the set of all such sequences is not countable.\n\n## Remarks\n\nGeorg Cantor published the argument in 1891. His first proof that $\\mathbb{R}$ is uncountable, from 1874, used another method.\nThe same idea proves [[Cantor's theorem|0WTEMGYV2C]] for every set.\nDiagonal arguments also appear in the proofs of the halting problem and of Gödel's incompleteness theorems.\n\n## Explanations\n\n### L1 human\n\nWrite the list of sequences as the rows of a table.\nGo down the diagonal and change each entry.\nThe new row differs from row $n$ at column $n$, so it cannot be any row of the table.\n",
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