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 the limit of Ferferman's theta notation, as well as the limit of Buchholz's psi notation.

It is also the ordinal measuring the strength of Buchholz hydras with \(\omega\) labels, as well as the upper bound of the SCG function.