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

Extended arrow notation

Extended arrow notation is a notation that was invented by Googology Wikia User Googleaarex: [1].

Basic Notation

Basic Notation is very simple. It generalizes the normal arrow notation.

$$a \uparrow_2 b = a \underbrace{\uparrow\uparrow\dots\uparrow}_b a$$

$$a \uparrow_3 b = a \underbrace{\uparrow_2\uparrow_2\dots\uparrow_2}_b a$$

$$a \uparrow_n b = a \underbrace{\uparrow_{n-1}\uparrow_{n-1}\dots\uparrow_{n-1}}_b a$$

Note that all parts of Extended arrow notation, like Knuth’s up-arrow notation, have expressions that are evaluated from the right.

Limit in FGH: $$f_\omega(n)$$

Nested up-arrow notation

To extend the notation here, we first have to make a change: $$\uparrow_n = \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_{n-1}}$$

Then we turn the problem into Basic notation: $$a \uparrow_{\uparrow_2} b = a \uparrow_{\underbrace{\uparrow\uparrow\dots\uparrow}_b} a = a \uparrow_{b+1} b$$, and $$a \uparrow_{\uparrow\uparrow_2} b = a \underbrace{\uparrow_{\uparrow_2}\uparrow_{\uparrow_2}\dots\uparrow_{\uparrow_2}}_b a$$

Then: $$a \uparrow_{\uparrow_{\uparrow_2}} b = a \uparrow_{\uparrow_{b+1}} a$$ and so on.

Limit: $$\varepsilon_0$$

Array up-arrow notation

$$\Omega$$ typed arrows

Limit: $$\psi(\varepsilon_{\Omega+1})$$

$$\Omega_2$$ typed arrows

Limit: $$\psi(\psi_1(\varepsilon_{\Omega_2+1}))$$

$$\Omega_3$$ typed arrows and beyond

Limit: $$\psi(\psi_I(0))$$

Inaccesible arrows

Limit: $$\psi(\psi_{I(1,0)}(0))$$

1-inaccesible arrows and beyond

Limit: $$\psi(\psi_{I(\omega, 0)}(0))$$

Dimensional array up-arrow notation

Limit: $$\psi(\psi_{\chi(\varepsilon_{M+1})}(0))$$

Hyperarray up-arrow notation

Limit: $$\psi(\psi_{\chi(M(1,0))}(0))$$

Legion array up-arrow notation

Layered arrays

Limit: $$\psi(\psi_{ {\Xi(1)}^\omega}(0))$$

The hyperseparator

Limit: $$\psi(\psi_{M(1,\Xi(1)+1)}(0))$$

Limit:???