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  "body": "\n## Statement\n\nThere are [[computably enumerable sets|TN9G24VABN]] $A, B \\subseteq \\N$ such that $A \\not\\le_T B$ and $B \\not\\le_T A$, where $\\le_T$ is [[Turing reducibility|PA4PYQ1PZ0]].\n\n## Remarks\n\nThe sets are built in stages by the [[priority method|3BZFSC5QZ2]].\nThe requirements are $R_{2e}$: $A \\ne \\Phi_e^B$ and $R_{2e+1}$: $B \\ne \\Phi_e^A$, where $\\Phi_e$ is the $e$-th oracle machine.\nTo meet $R_{2e}$, one picks a witness $x$ and puts $x$ into $A$ once $\\Phi_e^B(x) = 0$ appears, then tries to keep $B$ unchanged below the use of that computation.\nActions for requirements of lower priority may be undone, but each requirement is injured only finitely often, so all of them are met.\n\nAs a consequence, the degrees $\\deg(A)$ and $\\deg(B)$ are incomparable, so neither is $\\mathbf{0}$ or $\\mathbf{0}'$.\nSo both lie strictly between $\\mathbf{0}$ and $\\mathbf{0}'$, which solves [[Post's problem|1RVF7DWG8C]].\n\n## Explanations\n\n### L1 human\n\nSome sets of numbers can be listed by a computer but not decided by one.\nThe theorem says that two such sets can be built so that neither helps to compute the other.\nSo the hardness of listable sets is not a single ladder: there are incomparable levels.\n\n### L2 human\n\nThis is the result that solved Post's problem, and its proof introduced the finite-injury [[priority method|3BZFSC5QZ2]].\nThe same method, refined, shows that the computably enumerable [[Turing degrees|P5N2ND3VCG]] are dense (Sacks) and that every countable partial order embeds in them.\nThe construction is also a model for many other constructions in computability theory, where infinitely many conflicting requirements must be met at once.\n",
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