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  "body": "\n## Summary\n\nHyperarithmetical theory studies the hyperarithmetical sets of natural numbers, those computable from an iterate of the [[Turing jump|60P2XWY5QX]] along a computable ordinal.\nThe computable ordinals are those below the Church–Kleene ordinal $\\omega_1^{CK}$, the least [[ordinal|0YSCSX5YZC]] that is not the order type of a computable well-ordering of $\\N$.\nBy a theorem of Kleene, the hyperarithmetical sets are exactly the $\\Delta^1_1$ sets of the [[analytical hierarchy|M6JGF4TPM6]].\nThe theory connects [[computability theory|X14Y880PN5]], descriptive set theory and the study of subsystems of second-order arithmetic.\n\n## Explanations\n\n### L1 human\n\nThe [[arithmetical hierarchy|7CVJCQKY41]] uses the jumps $\\emptyset', \\emptyset'', \\ldots$, one for each natural number.\nOne can go further: join all of them into one set $\\emptyset^{(\\omega)}$, take its jump, and continue.\nDoing this along every ordinal that a computer can describe gives the hyperarithmetical sets.\nThe set $\\emptyset^{(\\omega)} = \\set{(n, x) : x \\in \\emptyset^{(n)}}$ is hyperarithmetical but not arithmetical.\nThese are the sets that can be defined by a single existential, and also by a single universal, quantifier over sets of numbers.\n",
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