# cantors-attic

Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

View the Project on GitHub neugierde/cantors-attic

# Knuth's up-arrow notation

Knuth’s up-arrow notation is a notation for expressing large numbers. However, there are some numbers so large that even this up-arrow notation is not enough. You have to use Extended arrow notation.

## Definition

$$a \uparrow b=a^b$$

$$a \uparrow\uparrow b=\underbrace{a^{a^{.^{.^{.^{a}}}}}}_{\text{b a’s}}$$

$$a \uparrow\uparrow\uparrow b=\underbrace{a \uparrow\uparrow a \uparrow\uparrow a\cdots\uparrow\uparrow a}_{\text{b a’s}}$$

$$a \uparrow^{n+1} b=\underbrace{a \uparrow^n a \uparrow^n a\cdots a\uparrow^n a}_{\text{b a’s}}$$

This notation is supposed to be evaluated from the right if there is more than one patch of arrows and there are no brackets.

## Examples

$$3\uparrow\uparrow\uparrow3=3\uparrow\uparrow3\uparrow\uparrow3=3\uparrow\uparrow3\uparrow3\uparrow3=3\uparrow\uparrow3\uparrow27=3\uparrow\uparrow7625597484987=$$a power tower that will reach all the way to the Sun