User blog comment:P進大好きbot/Please Help me on study of Pair Sequence System (2-rowed Bashicu Matrix System)/@comment-35870936-20180813215517/@comment-35470197-20180815080207

Thank you for the description. I am a little confused about what \(\psi(\Omega_{\omega})\) appearing in the left hand side in the equality \(\psi(\Omega_{\omega}) = \psi(\psi_1(\psi_2(\cdots)))\) means. Is it Deedlit's \(\psi\)? Or is it Buchholz's \(\psi\)? (Of course, they would coincide with each other. But in the argument about how to check the equality, the ambiguity is actually confusing for me.)

Could you denote \(\psi\) in KOCF by another symbol such as \(K\) in order to distinguish Buchholz's \(\psi\) here?

To summarise, you are explaining the argument in the following way, right?
 * 1) It is obvious \(\psi_0(\varepsilon_{\Omega+1}) = U(\varepsilon_{\Omega+1}) = U(U_1(\Omega_2)) = U(U_1(U_2(0)))\) and \(\psi_0(\varepsilon_{\Omega+1}) = K(K_1(0))\).
 * 2) In general, it is easy to show \(K(K_1(\cdots K_n(0) \cdots )) = U(U_1(\cdots U_{n+1}(0) \cdots))\).
 * 3) Moreover, the limit can be written in the following way:
 * 4) \(K(\Omega_{\omega}) = K(K_1(\cdots))\)
 * 5) \(U(\Omega_{\omega}) = U(U_1(\cdots))\)
 * 6) Therefore we obtain \(K(\Omega_{\omega}) = U(\Omega_{\omega})\).
 * 7) Also, it is easy to see \(K(\Omega_{\omega}) = \psi_0(\Omega_{\omega})\). So we obtain \(\psi_0(\Omega_{\omega}) = U(\Omega_{\omega})\), which was what we wanted.

I appreciate if you give me which definition of KOCF you are refering to, because Deedlit has constructed several OCFs.

By the way, I noticed that UNOCF works in a similar way to Bashicu's OCF, which can be found here, in the analysis of pair sequence system. It is interesting.