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 \(\omega\) labels, as well as the upper bound of the SCG function. It is speculated to be the upper bound of BEAF, but this has not been confirmed.