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catrincm
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Definition of HYP in $L_{\omega_1^{CK}}[a]$?
@Joel, do you mind explaining your response. I understand this as that it is true in $L_{\omega_1^{CK}}$ that each set is countable.
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Definition of HYP in $L_{\omega_1^{CK}}[a]$?
You claim that $L_{\omega_1^{CK}}$ consists of only HYP sets. I want to make it clear what is meant by this. The reals (i.e. subsets of $\omega$) which are in $L_{\omega_1^{CK}}$ are exactly the hyperarithmetic sets, but $L_{\omega_1^{CK}}$ of course contains other things.