User blog:Wythagoras/Xkcd forum

This may or may not be material for articles.

Mouffles' 1st number
Mouffles' number is defined as following:

\(f(n) = \underbrace{n\rightarrow n\rightarrow n ... n\rightarrow n\rightarrow n}_{\text{n n's}}\)

\(\text{Mouffles' 1st number} = f^G(G) \approx f_{\omega^2+1}(G)\)

Warriorness' number
\(Q(x) = \underbrace{x\rightarrow x\rightarrow x ... x\rightarrow n\rightarrow x}_{\underbrace{x\rightarrow x\rightarrow x ... x\rightarrow n\rightarrow x}_{\text{x x's}}}\)

\(W(x) = Q^x(x)\)

\(\text{Warriorness' number} = W(G!) \approx f_{\omega^2+1}(G! \cdot 2)\)

Mouffles' 2nd number
\(f1(x) = \underbrace{x\rightarrow x\rightarrow x ... x\rightarrow x\rightarrow x}_{\text{x x's}}\)

\(fy(x) = f^x(y-1)(x)\)

\(\text{Mouffles' 2nd number} = fG(G) \approx f_{\omega^2+\omega}(G)\)

IJmaxwell's number
\(a(0)(n) = \underbrace{n\rightarrow n\rightarrow n ... n\rightarrow n\rightarrow n}_{n^n n's}\)

\(a(m)(n) = a^{a(m-1)(n)}(m-1)(n)\)

\(b(0) = a(xkcd)(xkcd)\)

\(b(n) = a(b(n-1))(b(n-1))\)

\(\text{IJmaxwell's number} = b(xkcd) \approx f_{\omega^2+\omega+1}(xkcd)\)

Actaeus' 1st number & Actaeus' 2nd number
I'm going to include chained arrows here for the easier definition.

X and Y are chains of arrows

\(X rightarrow^a 1 \rightarrow^b Y = X\)

\(X \rightarrow^c (a+1) \rightarrow^c (b+1) = X \rightarrow^c (X \rightarrow^c a \rightarrow^c (b+1)) \rightarrow^c b\)

\(a rightarrow^1 b = a^b\)

\(a rightarrow^{c+1} b = \underbrace{a\rightarrow^c a\rightarrow^c a ... a\rightarrow^c a\rightarrow a}_{\text{b a's}}\)

Longer chains are adapted form above.

\(a \rightarrow_2 b = a \rightarrow^b a\)

\(a \rightarrow_c b = a \rightarrow_{c-1,b} a\) if \(c>2\)

\(a \rightarrow_{d,X,c} b = a \rightarrow^b_{d,X,c-1} a\) if \(len(d,X,c) = d\)

\(a \rightarrow_{d,X,c} b = a \rightarrow_{d,X,c-1,b} a\) if \(len(d,X,c) \leq d\)

\(a \rightarrow_{d,X,1} b = a \rightarrow_{d,X} b\)

\(\text{↻}(a) = a \rightarrow_a a\)

\(\text{Actaeus' 1st number} = \text{↻}(xkcd) \approx f_{\omega^\omega}(xkcd)\)

\(\text{Actaeus' 2nd number} = \text{↻}^{xkcd}(xkcd) \approx f_{\omega^\omega+1}(xkcd)\)

Comments
I really like this one, beacause it has no long array. I also was suprised I could define it formally :P