Maksudov's transfinite arrow notation

This is an extension of arrow notation to transfinitely many arrows by Googology Wiki user Denis Maksudov.D. Maksudov, Traveling to the Infinity
==Definition==
Let \(n\) and \(b\) be natural numbers, and \(\alpha\) be a countable ordinal.
*If \(\alpha=0\), \(n\uparrow^\alpha b\ :=\ n\times b\)
*\(n\uparrow^{\alpha+1}b\ :=\ \begin{cases}n\textrm{ if }b=1 \\ n\uparrow^\alpha (n\uparrow^{\alpha+1}(b-1))\textrm{ if }b>1\end{cases}\)
*If \(\alpha\) is a limit ordinal, then \(n\uparrow^\alpha b\ :=\ n\uparrow^{(\alpha}\)\(^{[b]}\)\(^)n\)
==Examples==
For finite \(\alpha\), this notation exactly corresponds to arrow notation.
*\(3\uparrow^\omega 6=3\uparrow^6 3\)
*\(3\uparrow^{\omega+1} 5=3\uparrow^\omega(3\uparrow^\omega(3\uparrow^\omega(3\uparrow^\omega 3)))\)
*\(3\uparrow^{\Gamma_0}4=3\uparrow^{\varphi(\varphi(\varphi(\varphi(1,0),0),0),0)}3\) using this system of fundamental sequences for Veblen's function.
==Sources==
Category:Notations
Category:Arrow