This is an extension of arrow notation to transfinitely many arrows by Googology Wiki user Denis Maksudov.[1]

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 \(n\uparrow^\alpha b\ :=\ n\uparrow^{(\alpha}\)\(^{[n]}\)\(^)n\)

Examples

For finite \(\alpha\), this notation exactly corresponds to arrow notation.

Sources

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