Countable limit of Extended Buchholz's function

The countable limit \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) of Extended Buchholz's ordinal collapsing function is a large countable ordinal, where \(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}\) denotes the least omega fixed point. It is the proof-theoretic ordinal of \(\Pi_1^1-\text{TR}_0\), a subsystem of second-order arithmetic.

Warning
In this community, it is frequently denoted by \(\psi(\psi_I(0))\), but the expression has a problem. Here \(I\) is the first inaccessible cardinal, and \(\psi\) is an unspecified or undefined ordinal collapsing function, and \(\psi_I(0)\) is supposed to be the least omega fixed point.

Since beginners tend to talk about specific values of ordinal collapsing functions without specifying definitions under the wrong assumptions that the difference of the definition does not cause the difference of specific values or that an ordinal collapsing function just gives roles to symbols to diagonalise other functions, the \(\psi\) function is often confounded with Buchholz's ordinal collapsing function, Extended Buchholz's ordinal collapsing function, and Rathjen's ordinal collapsing functions based on a weakly Mahlo cardinal. However, they do not have either the same expression or the required property: As the last example shows, it is possible (and actually easy) to formulate an ordinal collapsing function such that \(\psi_I(0)\) is a valid expression and its value precisely coincides with the least omega fixed point. Also, Jäger's function and Buchholz's simplification of Jäger's function satisfy the property, although either of them is rarely used in this community. The significant point of this story is that it clearly tells us that arguments on unspecified or undefined ordinal collapsing functions under the assumptions above actually cause serious errors.
 * 1) Buchholz's function does not have the expression \(\psi_I(0)\).
 * 2) Extended Buchholz's function does not satisfy \(\psi_I(0) = \Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}\).
 * 3) Rathjen's standard ordinal collapsing function based on a weakly Mahlo cardinal does not satisfy \(\psi_I(0) = \Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}\).
 * 4) Rathjen's simplified ordinal collapsing function based on a weakly Mahlo cardinal does not have the expression \(\psi_I(0)\).
 * 5) Rathjen's simplfied ordinal notation based on a weakly Mahlo cardinal has a function symbol \(\psi\), but it is not an actual function. Although the notation is associated to a simplified ordinal collapsing function based on a weakly Mahlo cardinal, the ordinal collapsing function is unpublished, and hence cannot be a candidate due to the lack of an explicit source.

Why do people tell others wrong informations about OCFs? It is because there are so many wrong articles written by those who do not know the precise definitions of OCFs. When beginners want to know OCFs, they will check precise articles, and judge such articles useless because the contents are quite difficult for them. Then they will search "introductory" articles written by those who do not know the precide definitions of OCFs, and praise such articles as useful ones. However, such articles are unsourced, and include many wrong articles. After then, the beginners guess that they have already understood OCFs. They try to write useful "introduction" before correcting their understandings by checking sources, and tell others wrong informations.

As a result, beginners should be very careful when other googologists talk about the expression "\(\psi(\psi_I(0))\)" without specifying the definition of \(\psi\), and ask the precise definition of the \(\psi\). The expression "\(\psi(\psi_I(0))\)" is a key which tells us possibility that others are talking about functions whose precise definitions they do not actually know. Of course, professional googologists might use the expression for convenience. In that case, they easily answer the question.