Buchholz ordinal

The Buchholz ordinal is the limit of Wilfried Buchholz's original set of ordinal collapsing functions, with finite indices, and is equal to the limit of the sequence \( \omega \), \( \varepsilon_0 \), \( \mathrm{BHO} \), \( \psi_0(\Omega_3) \), ... - i.e. it is equal to \( \psi_0(\Omega_\omega) \). It is also equal to the proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \(\Pi^1_1\)-formulae, i.e. \(\Pi^1_1 \mathrm{-CA}_0\), or of Peano arithmetic with finitely iterated inductive definitions, as well as the limit of pair sequence system in Bashicu matrix system. It can be viewed as an extension of the Bachmann-Howard ordinal by allowing higher uncountable cardinals, which iteratively collapse above \( \Omega \).