Extended Buchholz ordinal

The Extended Buchholz ordinal, sometimes known as OFP (short for omega-fixed-point), is the limit of an extension of Buchholz's original set of ordinal collapsing functions, defined by Denis Maksudov, which allows to collapse ordinals such as \( \Omega_{\omega + 1} \) (which corresponds to the Takeuti-Feferman-Buchholz ordinal), \( \Omega_{\omega^2} \) (which is believed to correspond to the BMS matrix (0,0,0)(1,1,1)(2,1,1)), or \( \Omega_{\Omega} \) (which corresponds to the Bird ordinal). It has not been widely studied in the literature, but is common in amateur apeirological discussions, and is known to correspond to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-TR}_0 \), a second-order strengthening of arithmetical transfinite recursion (the proof theoretic ordinal of which is the Feferman-Schütte ordinal). Thus, one could claim that it is to the Buchholz ordinal as the Feferman-Schütte ordinal is to \( \varepsilon_0 \), although this may be an understatement.