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  "body": "\n## Definition\n\nLet $A \\subseteq \\N$ be a [[subset|0BJR9QR08E]] of the natural numbers.\nThe set $A$ is *computably enumerable*, or *recursively enumerable*, if $A$ is the domain of a partial [[computable function|KZSB5E471K]] $f \\colon \\N \\rightharpoonup \\N$.\nThat is, $A = \\set{n \\in \\N : f(n) \\text{ is defined}}$.\n\n## Remarks\n\nThe following conditions on $A \\subseteq \\N$ are equivalent.\n\n1. $A$ is computably enumerable.\n2. $A$ is empty or the range of a total computable function $\\N \\to \\N$.\n3. There is a computable set $R \\subseteq \\N^2$ such that $A = \\set{n : \\exists m \\, (n, m) \\in R}$.\n\nExample: the halting set $K = \\set{e : \\text{program } e \\text{ halts on input } e}$ is computably enumerable.\nIt is the domain of the partial computable function that, on input $e$, runs program $e$ on input $e$ and then outputs $0$.\n\nNon-example: the complement $\\N \\setminus K$ is not computably enumerable.\nOtherwise $K$ and its complement would both be computably enumerable, and then $K$ would be [[computable|PPGD5KW8JC]], which is false.\n",
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