Takeuti–Feferman–Buchholz ordinal

• Let ${\displaystyle \Omega _{\alpha }}$ represent an uncountable ordinal with cardinality ${\displaystyle \aleph _{\alpha }}$.
• Let ${\displaystyle \varepsilon _{\beta }}$ represent the ${\displaystyle \beta }$th epsilon number, equal to the ${\displaystyle 1+\beta }$th fixed point of ${\displaystyle \alpha \mapsto \omega ^{\alpha }}$
• Let ${\displaystyle \psi }$ represent Buchholz's psi function
• The TFBO is equal to ${\displaystyle \psi (\varepsilon _{\Omega _{\omega }+1})}$.

In other words, the TFBO is the smallest ordinal which cannot be expressed from ${\displaystyle 0}$, ${\displaystyle 1}$, ${\displaystyle \omega }$ and ${\displaystyle \Omega }$ using sums, products, exponentials, and the ${\displaystyle \psi }$ function itself, the latter of which only to previously constructed ordinals less than ${\displaystyle \varepsilon _{\Omega _{\omega }+1}}$.