In this [note][1] written by Philip Welch, I'm trying to solve exercise 3.5(i), but I don't know how to solve it.

The problem is: assume that $L_{\gamma_0}<_{Σ₂}L_{\gamma_1}$, and $\gamma_1$ is admissible, and $\gamma_0$ is the least such ordinal. Prove that in some (or any) $KP$ model $M$ which end-extends $L_{\gamma_1}$, and $\gamma_1\notin M$, there is an infinite sequence (in $V$) of $M$-ordinals $\alpha_1<\alpha_2<\alpha_3<...<\beta_3<\beta_2<\beta_1$, such that for any $i$, $\alpha_i<\gamma_1$, $\beta_i$ is not well founded, and $M\vDash L_{\alpha_i}<_{\Sigma_2}L_{\beta_i}$.

There is a hint in the note about there are unboundly many $\gamma$ below $\gamma_1$ such that $L_{\gamma_0}<_{\Sigma_2}L_\gamma$. This is very easy because admissible is equivalent with $\Pi_2$-reflecting, and $L_\alpha<_{\Sigma_2}L_\beta$ is a $\Pi_2$ property of $L_\beta$.

But then, I don't know how to go further. Yes $\gamma_0$ can play as $\alpha_1$, but I can't find more $\Sigma_2$ links.

How to solve this problem?


  [1]: https://people.maths.bris.ac.uk/~mapdw/TheoryMachine-Final2020.pdf