Bachmann-Howard ordinal

The Bachmann-Howard ordinal is a key milestone in the set of recursive ordinals. It is equal to the limit of the dimensional Veblen function, as well as the Buchholz hydra with only zero and one labels. It was originally discovered as the limit of a basic ordinal collapsing function, namely Bachmann's psi, which was used in ordinal-analysis. In particular, the Bachmann-Howard ordinal is exactly the proof-theoretic ordinal of basic Kripke-Platek set theory, which has the same strength as \( \mathrm{ID}_1 \). This is a system of arithmetic augmented by inductive definitions. The ordinal collapsing function used to give this ordinal analysis had the Bachmann-Howard ordinal as its limit, and it can be represented as the countable collapse of \( \varepsilon_{\Omega+1} \). Buchholz further extended this to his famous set of collapsing functions, whose limit is the much larger Buchholz ordinal.