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  "body": "\n## Definition\n\nLet $M_0, M_1, \\ldots$ be a fixed effective list of all [[oracle machines|Z5CCVBME2V]].\nLet $A \\subseteq \\N$.\nThe *Turing jump* of $A$ is the set\n$$A' = \\set{e \\in \\N : M_e^A \\text{ halts on input } e}.$$\nThe $n$-th jump $A^{(n)}$ is defined by $A^{(0)} = A$ and $A^{(n+1)} = (A^{(n)})'$.\nFor a [[Turing degree|P5N2ND3VCG]] $\\mathbf{a} = \\deg(A)$, the jump is $\\mathbf{a}' = \\deg(A')$.\n\n## Remarks\n\nExample: for $A = \\emptyset$, the machine $M_e^{\\emptyset}$ never gets a \"yes\" answer, so it behaves like an ordinary Turing machine.\nHence $\\emptyset'$ is the set $\\set{e : M_e \\text{ halts on input } e}$, a form of the [[halting problem|Z7TN2B35B0]].\n\nThe definition of $\\mathbf{a}'$ is legitimate: if $A \\le_T B$, then $A' \\le_T B'$, so $A \\equiv_T B$ implies $A' \\equiv_T B'$.\nFor every $A$, one has $A \u003c_T A'$, that is, $A \\le_T A'$ and $A' \\not\\le_T A$; the proof is a diagonal argument as for the halting problem.\nThe jump of $A$ does depend on the chosen list of machines, but only up to a computable permutation, so its degree does not.\nBy [[Post's theorem|NYJX5QXX1T]], $\\emptyset^{(n)}$ is complete for the $\\Sigma^0_n$ sets of the [[arithmetical hierarchy|7CVJCQKY41]] for $n \\ge 1$.\n",
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