Bachmann-Howard ordinal is a recursive ordinal. It's not that large compared to those proof-theoretic ordinals of stronger theories, but the definition of BHO is sufficient to illustrate how collapsing functions works. It give us an intuition about how large $\omega^{CK}_1$ is.
I want to understand the structure below larger admissible ordinals.
It seems that, for any admissible ordinal $\alpha$ such that there is no $\beta<\alpha$ such that $L_\beta <_1 L_\alpha$, there is a "BHO alternative" below $\alpha$. Normally it is $\psi_\alpha(\varepsilon_{\alpha+1})$ for some kind of collapsing function $\psi_\alpha$.
Let me take some examples. Because "BHO alternative" is relatively harder to define, so I'll take $\Gamma_0$ alternatives instead.
For the second admissible ordinal $\omega^{CK}_2$, the $\Gamma_0$ alternative below it is just $\Gamma_{\omega^{CK}_1+1}$.
For the least recursively inaccessible ordinal $I$, the $\Gamma_0$ alternative below it is $\Phi(1,0,0)$, where $\Phi$ is similar to Veblen's $\varphi$, the only difference in definition is $\Phi(0,\alpha)=\omega^{CK}_{1+\alpha},\varphi(0,\alpha)=\omega^{\alpha}$.
For the least recursively mahlo ordinal $M$, the $\Gamma_0$ alternative below it is $I(1,0,0)$, where:
If $\beta$ is a successor ordinal or 0, then $I(\alpha,\beta)$ is the $1+\beta$th least admissible ordinal that is also a limit of ordinals of the form $I(\alpha',x)$, for any given $\alpha'<\alpha$. And $I(\alpha,\beta)$ is continuous for $\beta$.
And $I(1,0,0)$ is the least $\alpha$ such that $I(\alpha,0)=\alpha$. Sometimes it is also called recursively hyper-inaccessible ordinal.
My question is, can we define "BHO alternative below some ordinal" in a more uniformly way?
Also some other question. For a theory $T$ s.t. $L\vDash T$ and an ordinal $\kappa$, define $PTO(T,\kappa)=sup\{\alpha<\kappa|\text{for some parameter-free }\Sigma_1\varphi, T\vdash \exists^!\beta(\varphi^L(\beta)),\text{and }L\vDash \varphi(\alpha)\}$.
(We can assume $0^\#$ exists so that this definition can be written in set theory language.)
Then do we have $PTO(KPi,I)$ is the BHO alternative below I, and $PTO(KPM,M)$ is the BHO alternative below M?
Also, even if this is true, it seems that this definition can't be generalized above +1 stable ordinal, because they require larger universes to define the ordinal itself. So maybe PTO(KP+there is a stable ordinal,the least $\Pi^1_1$-reflecting ordinal) is the $\psi(\varepsilon_{\Omega_2+1})$ alternative below the least $\Pi^1_1$-reflecting ordinal. How can we weaken the axiom system so that the previous pattern keep true?
