User blog comment:Hyp cos/Attempt of OCF up to Stability/@comment-32697988-20181027113000/@comment-11227630-20181027121254

It is not "first you think of an ordinal, then you check whether it is \(\Pi_3\)-reflecting, then \(\psi_\pi((\alpha),\beta)\) equals it". It works backwards: for ordinals expressed in this OCF, an ordinal is \(\Pi_3\)-reflecting on \(A_\kappa(\xi)\) iff it is in the form \(\psi_\pi((\alpha),\beta)<\pi\) (where \(\alpha>\xi\)) or \(\psi_\pi((\alpha,0,\mathbb X),\beta)<\pi\) (where \(\alpha>\xi\) or \(\psi_\pi(\mathbb X,0)<\pi\) is \(\Pi_3\)-reflecting on \(A_\kappa(\xi)\)).