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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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