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  "body": "\n## Statement\n\nThis shows that the order-structure of Turing degrees is quite complicated, even among enumerable sets.\n\nWLOG, we assume that $C$ is enumerated as $C_0 \\subsetneq C_1 \\subsetneq C_2 \\subsetneq \\cdots$, each $C_n$ being a [[finite set|070R982G4B]].\n\nTo prove the theorem, we define the concept of **permission**. Technically, this is the Yates permitting. There are other kids of permissions.\n\nGiven two sequences of finite sets $A_n \\uparrow A, C_n \\uparrow C$, we say that $\\{C_n\\}_{n \\in \\mathbb{N}}$ *permits* $\\{A_n\\}_{n \\in \\mathbb{N}}$ iff\n\n$$\\forall n, i \\quad C_n \\cap [0:i] = C \\cap [0:i] \\implies i \\not\\in A_{n+1} \\setminus A_n$$\n\nA bit more generally, if $f : \\mathbb{N} \\to \\mathbb{N}$, then we say that $\\{C_n\\}_{n \\in \\mathbb{N}}$ *f*-*permits* $\\{A_n\\}_{n \\in \\mathbb{N}}$ iff\n\n$$\\forall n, i \\quad C_n \\cap [0:f(i)] = C \\cap [0:f(i)] \\implies i \\not\\in A_{n+1} \\setminus A_n$$\n",
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