$\mathcal{O}$ notation describes an onto function $f:\mathcal{O} \rightarrow \omega_{CK}$. In calculating all values $n \in \mathbb{N}$ such that $f(n)=\alpha$, when $\alpha$ is a limit, all indexes $e$ of ordinary programs are considered such that $\phi_e(i)=n_i$ (with $i \in \mathbb{N}$). The values $n_i$ must satisfy the condition that $f(n_i)=\alpha_i$ with the $\alpha_i$'s forming a (fundamental) sequence for $\alpha$.

I am just interested in the variation where we don't consider indexes of normal programs but instead consider indexes of primitive recursive functions/programs (given a suitable indexing for them).

**(Q1)** How far would such a variation go? That is, what is the smallest ordinal it can't represent.

**(Q2)** Consider the notation system (described by a 1-1 function $g:\beta \rightarrow \mathbb{N}$) assigning a unique number to each ordinal in which on limit values $\alpha<\beta$ we seek the p.r. function with *smallest* index $e$ satisfying $\phi_e(i)=g(\alpha_i)$ where the $\alpha_i$'s must form a (fundamental) sequence for $\alpha$.

At what value $\beta$ would such a system stop?

I don't have proficiency with the underlying theory so if the question seems posed in a strange manner then that might be the reason.

polynomial-time computability(or indeed even less!) - $\omega_1^{CK}$ isunspeakablyrobust. $\endgroup$ – Noah Schweber Mar 2 at 18:26