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  "body": "\n## Summary\n\nThe Eilenberg swindle is an argument with infinite direct sums that cancels an object by regrouping an infinite sum.\nIts basic form shows that every [[projective module|BFFA03GE3V]] $P$ becomes free after adding a large enough [[free module|0DZ9DKAKZR]]: $P \\oplus F \\cong F$ for some free $F$.\nIt explains why the Grothendieck group is built only from finitely generated modules: with infinite sums allowed, every class would be zero.\nThe same trick, used by Mazur in topology, shows that no nontrivial knot has an inverse under connected sum.\n\n## Explanations\n\n### L1 human\n\nThe swindle is the old joke $1 = 1 + (-1 + 1) + (-1 + 1) + \\cdots = (1 - 1) + (1 - 1) + \\cdots = 0$, made correct.\nWith numbers the regrouping is not allowed, but with infinite direct sums of modules it is.\nIf $P \\oplus Q$ is free, then adding infinitely many copies of $P \\oplus Q$ to $P$ gives the same as infinitely many copies of $Q \\oplus P$.\nSo $P$ disappears into an infinite free module.\n",
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