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 called Extended Buchholz's ordinal or EBO in Japanese googology, and the term was coined by a Japanese googologist 叢武. In googology, it is sometimes referred to as \(\psi(\psi_I(0))\), but the OCF used in this expression is usually unspecified. (It is not the correct value in Buchholz's function, Extended Buchholz's function, or Rathjen's \(\psi\) function. It is also referred to as \(\psi(OFP)\) (where OFP stands for "omega fixed point"), or sometimes simply called OFP.

Properties
It is the smallest ordinal which is not expressed by using \(0, +, \psi\). It is also the smallest ordinal which is not expressed by the ordinal notation associated to Extended Buchholz's function with respect to the natural correspondence \(o\), but can be expressed as \(\langle \langle, \rangle ,  \rangle\) or \(\psi_{\psi_0(0)}(0)\) with respect to the order type.