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

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The upper attic
The middle attic
The lower attic
The parlour
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Cantor's Attic (original site)
Joel David Hamkins blog post about the Attic
Latest working snapshot at the wayback machine

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.


\(a \uparrow b=a^b\)

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

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

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

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


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