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  "body": "\n## Definition\n\nLet $\\le_T$ be [[Turing reducibility|PA4PYQ1PZ0]] on subsets of $\\N$, and let $A \\equiv_T B$ mean $A \\le_T B$ and $B \\le_T A$.\nThe relation $\\equiv_T$ is an equivalence relation on the power set of $\\N$.\nThe *Turing degree* of a set $A \\subseteq \\N$ is its equivalence class\n$$\\deg(A) = \\set{B \\subseteq \\N : B \\equiv_T A}.$$\nThe degrees are partially ordered by $\\deg(A) \\le \\deg(B)$ if and only if $A \\le_T B$; this does not depend on the choice of representatives.\n\n## Remarks\n\nExample: the set $E$ of even numbers and the set $\\set{0}$ have the same degree.\nBoth are [[computable sets|PPGD5KW8JC]], and every computable set is computable from every set, so $E \\le_T \\set{0}$ and $\\set{0} \\le_T E$.\nThis common degree is written $\\mathbf{0}$; it is the least degree.\n\nNon-example: the [[halting problem|Z7TN2B35B0]] $K$ does not belong to $\\mathbf{0}$, because $K \\not\\le_T \\emptyset$.\nIts degree is written $\\mathbf{0}'$.\n\nEvery degree is a countable set, since there are only countably many oracle machines.\nHence there are uncountably many degrees.\nAny two degrees $\\deg(A)$ and $\\deg(B)$ have a least upper bound, the degree of $A \\oplus B = \\set{2n : n \\in A} \\cup \\set{2n + 1 : n \\in B}$.\n",
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