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  "body": "\n## Definition\n\nLet $A \\subseteq \\N$ be a [[subset|0BJR9QR08E]] of the natural numbers.\nThe characteristic function of $A$ is $\\chi_A \\colon \\N \\to \\set{0, 1}$, with $\\chi_A(n) = 1$ if $n \\in A$ and $\\chi_A(n) = 0$ if $n \\notin A$.\nThe set $A$ is *computable*, or *recursive*, or *decidable*, if $\\chi_A$ is a total [[computable function|KZSB5E471K]].\nThe same definition applies to subsets of $\\N^k$.\n\n## Remarks\n\nExample: the set of prime numbers is computable.\nA machine can test $n \\ge 2$ for divisors $d$ with $2 \\le d \u003c n$, which takes finitely many steps, and so computes $\\chi_A$.\n\nNon-example: the set $K = \\set{e : \\text{program } e \\text{ halts on input } e}$ is not computable, by the undecidability of the [[halting problem|Z7TN2B35B0]].\nIts characteristic function is total but not computable.\n\nA set $A$ is computable if and only if both $A$ and $\\N \\setminus A$ are [[computably enumerable|TN9G24VABN]].\nThe computable sets are closed under union, intersection and complement.\n",
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