I asked this question on MSE some days ago but I have not received any answer so I have decided to post it here.

I would like to consider a generalization of the notation $\Pi$ and $\Sigma$ used for the arithmetical hierarchy $(\Pi^0_n$, $\Sigma^0_n)$ and the analytical hierarchy $(\Pi^1_n$, $\Sigma^1_n)$ to set of formulas $\Pi^\alpha_\beta$, $\Sigma^\alpha_\beta$, where $\alpha$ and $\beta$ range over ordinal numbers.

I suppose that the set of formulas of $n$-th order arithmetic is the set $\Pi^{n-1}_\omega \vee \Sigma^{n-1}_\omega$.

The superscript ranging over the ordinals corresponds to the transfinite recursion of the power set operator, similarly to the definition of the Von Neumann universe, but starting with the set of the natural numbers instead of the empty set.

About the subscript (which is the number of quantifiers) I am not sure. I know that it can be defined at least to $\omega_1^{CK}$ as you can see in this table where additionally it is written that $\Pi^0_{\omega_1^{CK}} =\Sigma^0_{\omega_1^{CK}}$ corresponds to a set of formulas $(\Delta^1_1)$ of second order arithmetic. This fact is not clear to me, it would be nice if somebody can elaborate about it in the answer.

Can you use ordinals beyond $\omega_1^{CK}$ as subscript?

If the superscript or the subscript is beyond $\omega_1^{CK}$ will you have non recursive theories with non recursive proof theoretic ordinals? Given any ordinal $\gamma$ is there an ordinal $\alpha$ such that the proof theoretic ordinal of $\Pi^\alpha_1-CA_0$ is higher than $\gamma$. (What about the subscript, what if you replace $\Pi^\alpha_1$ with $\Pi^1_\alpha$)

The same question but restricted to recursive ordinals: given any ordinal $\gamma<\omega_1^{CK}$ is there an ordinal $\alpha<\omega_1^{CK}$ such that the proof theoretic ordinal of $\Pi^\alpha_1-CA_0$ is higher than $\gamma$.