Countable limit of Extended Buchholz's function

\(\psi(\psi_I(0))\) is a large countable ordinal. Michael Rathjen's ordinal collapsing function \(\psi\) is used here along with \(I\), the first inaccessible cardinal. \(\psi_I(0)\) is the omega fixed point. It is the proof-theoritic ordinal of \(\Pi_1^1-\text{TR}_0\), a susbystem of second-order arithmetic.

Ψ(ψᵢ(0))