Maksudov's transfinite arrow notation

View full site to see MathJax equation

This is an extension of arrow notation to transfinitely many arrows by Googology Wiki user Denis Maksudov.

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.