How to define BHO alternatives below admissible ordinals?

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?

• There are problems with your definition $\mathit{PTO}(T,\kappa)$. For example, $\mathit{PTO}(\mathsf{KP}\omega,\omega_1^{\textit{CK}})=\omega_1^{\textit{CK}}$, since for any $\alpha<\omega_1^{\textit{CK}}$ there is a formula $\psi_\alpha(x)$ such that $L\models \forall x (\psi_\alpha(x)\leftrightarrow x=\alpha)$ and by taking $\varphi_\alpha(x)$ to be $(x=0\land \forall x(\lnot \psi_\alpha(x)))\lor (x\in\mathsf{On}\land \psi_\alpha(x)\land \forall y<x(\lnot \psi_\alpha(x)))$ we will get that $\mathsf{KP}\omega\vdash \exists! \beta(\varphi_\alpha(\beta))$ and $L\models\varphi_\alpha(\alpha)$. Mar 18 at 15:56
• I think that you could get some more reasonable properties if you would restrict $\varphi$'s in the definition of $\mathit{PTO}$ to just $\Sigma_1$-formulas, see www1.maths.leeds.ac.uk/~rathjen/SPECK.pdf . Mar 18 at 16:38
• @FedorPakhomov ok, thank you. Mar 19 at 4:42