Takeuti-Feferman-Buchholz ordinal

The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's original set of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals". It is not particularly vastly larger than the Buchholz ordinal, although one could comparatively describe the difference in size as like that between \( \varepsilon_0 \) and the Bachmann-Howard ordinal. It is equal to the proof-theoretic ordinal of \( \Pi^1_1 \mathrm{-CA_0} \) (of which the Buchholz ordinal is the proof-theoretic ordinal) with additional induction schemata. It also is the proof-theoretic ordinal of Peano arithmetic, augmented by iterated inductive definitions of length \( \omega \) (while the Buchholz ordinal has iterated definitions of arbitrary finite lengths).

Connection to Buchholz hydras