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It is well-known that a real $x$ is contained in $L_{\omega_{n+1}^{CK}}$ iff it is hyperarithmetic in the $n$-th hyperjump. I suppose that the relativization of this also holds and $x$ is contained in $L_{\omega_{n+1}^{CK,Y}}[Y]$ iff $x$ is recursive in the $n$-th hyperjump of $Y$. However, I am unsure whether there are subtleties to the proof that do not relativize. Can someone confirm/refute this?

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    $\begingroup$ I don't think the well-known fact is true... With $n=0$, we get: the reals in $L_{\omega_1^{\mathrm{CK}}}$ are exactly the recursive reals, which is false. $\endgroup$ Oct 19, 2012 at 13:07
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    $\begingroup$ I think you want $x$ hyperarithmetic in the $n$-th hyperjump. $\endgroup$ Oct 19, 2012 at 13:44

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I think the relativisation is completely straightforward (but Dave Marker's comment is correct). One can piece together the arguments (as in Steve Simpson's book) and just add the parameter $Y$ at all stages.

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