User blog comment:MilkyWay90/A new ordinal notation/@comment-32697988-20180718002157/@comment-35392788-20180718082509

Here's the best definition he could use :

\(\omega \uparrow\uparrow 1 = \omega\)

\(\omega \uparrow\uparrow \alpha+\beta = (\omega \uparrow\uparrow \alpha)\uparrow\uparrow(1+\beta)\) for limit \(\alpha\) and finite \(\beta\)

\(\omega \uparrow\uparrow \alpha+1 = \omega^{\omega \uparrow\uparrow \alpha}\)

\(\omega \uparrow\uparrow \alpha = \lim\limits_{n \rightarrow \infty}\omega \uparrow\uparrow \alpha[n]\) for limit \(\alpha\)