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  "body": "\n## Statement\n\nLet $\\emptyset^{(0)} = \\emptyset$, and let $\\emptyset^{(n+1)}$ be the [[Turing jump|60P2XWY5QX]] of $\\emptyset^{(n)}$.\nLet $A \\subseteq \\N$ and $n \\in \\N$.\n\n1. $A$ is $\\Sigma^0_{n+1}$ in the [[arithmetical hierarchy|7CVJCQKY41]] if and only if $A$ is [[computably enumerable|TN9G24VABN]] relative to the oracle $\\emptyset^{(n)}$.\n2. If $n \\ge 1$, the set $\\emptyset^{(n)}$ is $\\Sigma^0_n$-complete: it is $\\Sigma^0_n$, and every $\\Sigma^0_n$ set is [[many-one reducible|JTAH37JTHD]] to it.\n3. $A$ is $\\Delta^0_{n+1}$ if and only if $A$ is [[Turing reducible|PA4PYQ1PZ0]] to $\\emptyset^{(n)}$.\n\n## Remarks\n\nThe proof is by induction on $n$.\nFor $n = 0$, part 1 says that $\\Sigma^0_1$ sets are the computably enumerable sets, which follows from Kleene's normal form.\nFor the inductive step, an existential quantifier over a set computable from $\\emptyset^{(n)}$ is a search with that oracle.\nA $\\Pi^0_n$ condition can be answered by one question to $\\emptyset^{(n)}$, by the completeness in part 2.\n\nExample: the set of programs that compute total functions is $\\Pi^0_2$, so it is computable from $\\emptyset''$.\n\n## Explanations\n\n### L1 human\n\nThe Turing jump of a set is the halting problem for machines that may consult that set.\nPost's theorem says that each extra quantifier alternation in a definition corresponds to one more jump.\nSo the logical complexity of a definition matches the computational complexity of deciding the set.\n\n### L2 human\n\nThe theorem is the bridge between definability in arithmetic and relative computability.\nIt shows that the arithmetical hierarchy is proper, because $\\emptyset^{(n+1)}$ is not computable from $\\emptyset^{(n)}$.\nIn computability arguments, one often bounds the quantifier complexity of a set to know which oracle can decide it.\nIterating the jump through computable ordinals beyond $\\omega$ leads to the [[hyperarithmetical sets|NS1HWGTD8E]].\n",
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