Takeuti-Feferman-Buchholz ordinal

\(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (called "Takeuti-Feferman-Buchholz ordinal") is a large countable ordinal. It is the proof-theoretic ordinal measuring the strength of, a subsystem of second-order arithmetic.

It is also the ordinal measuring the strength of Buchholz hydras with ordinal labels.