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  "body": "\n## Statement\n\nLet $\\mathcal{E}$ be an elementary [[topos|6837RS7N82]].\nSuppose that every [[epimorphism|RVXDSY04KG]] in $\\mathcal{E}$ has a section, that is, $\\mathcal{E}$ satisfies the [[axiom of choice|0HTGB13FE2]].\nThen $\\mathcal{E}$ is Boolean: every [[subobject|FS1FKAE89H]] of every object of $\\mathcal{E}$ has a complement.\nEquivalently, the internal logic of $\\mathcal{E}$ satisfies the [[law of excluded middle|0RYYK8PG4S]].\n\n## Remarks\n\nThe same argument works in constructive set theory, where it is often given in the form of Goodman and Myhill.\nLet $P$ be a proposition, and let $U = \\set{x \\in \\set{0,1} : x = 0 \\lor P}$ and $V = \\set{x \\in \\set{0,1} : x = 1 \\lor P}$.\nBoth sets are inhabited, so a choice function on $\\set{U, V}$ gives $u \\in U$ and $v \\in V$.\nEquality on $\\set{0,1}$ is decidable, so either $u \\ne v$ or $u = v$.\nIf $u \\ne v$, then $P$ is false, because $P$ would give $U = V$ and hence $u = v$; if $u = v$, then either $u = 1 \\in U$ or $v = 0 \\in V$, and in both cases $P$ holds.\nSo $P \\lor \\lnot P$ holds.\n\nThe converse fails.\nFor a nontrivial group $G$, the topos of $G$-sets is Boolean.\nBut the epimorphism from $G$, acting on itself, to the one-point $G$-set has no section, since $G$ has no fixed point.\nIn the topos $\\mathbf{Set}$, both choice and excluded middle hold, in line with the theorem.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that the axiom of choice is not a neutral addition to constructive mathematics.\nOnce one can always choose an element from each nonempty set in a family, every statement becomes either true or false.\nSo a constructive setting that wants to avoid excluded middle must also avoid full choice.\n\n### L2 human\n\nIn a [[topos|6837RS7N82]] the logic is intuitionistic in general: the [[subobject classifier|NKP5TPQ79X]] is an internal [[Heyting algebra|NG943XNQWH]], not a Boolean algebra.\nDiaconescu's theorem shows that choice forces this algebra to be Boolean.\nThis is why constructive set theories and type theories use weaker forms such as countable choice or dependent choice.\n",
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