Crossing-out: (How does one prove the existence of Church-Kleene ordinal? Also, why is it labeled as $\omega_1^{CK}$?
And why is it first ordinal not hyperarithmetical, and is the first admissible ordinal after $\omega$?)
Edit: OK, I will cross out the first question and the second question. And I will change my question to this:
How does one prove that $L_{\omega_1^{CK}}$ is the model of Kripke-Platek set theory, and how does one prove that it is first admissible ordinal after $\omega$? (So why can't $\omega+1$ can be a model of KP set theory?) And why is it first ordinal not to be hyperarithmetical?
And somehow extraneous question: Why are admissible sets labeled "admissible"? Curiosity.