This may or may not be material for articles.

Also, xkcd = A(G,G) in case you forgot.

Mouffles' 1st number

Mouffles' number is defined as following:

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

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

Warriorness' number

\(Q(x) = \underbrace{x\rightarrow x\rightarrow x ... x\rightarrow x\rightarrow x}_{\underbrace{x\rightarrow x\rightarrow x ... x\rightarrow x\rightarrow x}_{x x's}}\)

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

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

Mouffles' 2nd number

\(f1(x) = \underbrace{x\rightarrow x\rightarrow x ... x\rightarrow x\rightarrow x}_{x x's}\)

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

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

IJmaxwell's number

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

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

\(b(0) = a(\text{xkcd})(\text{xkcd})\)

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

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

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

  1. \(X \rightarrow^a 1 \rightarrow^b Y = X\)
  2. \(X \rightarrow^c (a+1) \rightarrow^c (b+1) = X \rightarrow^c (X \rightarrow^c a \rightarrow^c (b+1)) \rightarrow^c b\)
  3. \(a \rightarrow^1 b = a^b\)
  4. \(a \rightarrow^{c+1} b = \underbrace{a\rightarrow^c a\rightarrow^c a ... a\rightarrow^c a\rightarrow a}_{b a's}\)

Longer chains are adapted form above.

  1. \(a \rightarrow_2 b = a \rightarrow^b a\)
  2. \(a \rightarrow_c b = a \rightarrow_{c-1,b} a\) if \(c>2\)
  3. \(a \rightarrow_{d,X,c} b = a \rightarrow^b_{d,X,c-1} a\) if \(len(d,X,c) = d\)
  4. \(a \rightarrow_{d,X,c} b = a \rightarrow_{d,X,c-1,b} a\) if \(len(d,X,c) \leq d\)
  5. \(a \rightarrow_{d,X,1} b = a \rightarrow_{d,X} b\)

\(\text{↻}(a) = a \rightarrow_a a\)

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

\(\text{Actaeus' 2nd number} = \text{↻}^{\text{xkcd}}(\text{xkcd}) \approx f_{\omega^\omega+1}(\text{xkcd})\) I really like this one, beacause it has no long array. I also was suprised I could define it formally :P

Tricky's 1st number and Tricky's 2nd number

It has no complete definition but it looks like Taro's multivariable Ackermann notation.

Therefore \(\text{Tricky's 1st number} \approx f_{\omega^\omega}(20)\)

Tricky's 2nd number is defined as b({xkcd,xkcd}) but the b function isn't defined anywhere as far as I can see.

There are more of such numbers, most of them does not terminate and they are all based on Tricky's undefined multivariable Ackermann notation.

Itaibn's 1st number

The sum of the results of all algorithms that are provably computable in ZFC+I0 with a proof of length less than A(9,9). (Here I0 is a Reinhart cardinal, so we don't know whether it is well-defined or not)

Itaibn's 2nd number

  1. \(f(0,n) = n\)
  2. \(f(\alpha,n) = f(\alpha(n),n+1)\) (he forgot to define \(f(\alpha,n)\) when \(\alpha\) is a successor ordinal)

Define the normal form of an ordinal as following: It is written as \(\alpha_1^{\beta_1}+\alpha_2^{\beta_2}+\alpha_3^{\beta_3}+...\alpha_n^{\beta_n}\) such that every \(\alpha_k\) is as large as possible, \(\alpha_k \in \{\omega, \Omega, p(n)\}\), and \(\forall k (\alpha_k > \alpha_{k+1}) \vee ((\alpha_k = \alpha_{k+1}) \wedge (\beta_k \geq \beta_{k+1}))\)

  1. \(\alpha_1^{\beta_1}+\alpha_2^{\beta_2}+\alpha_3^{\beta_3}+...\alpha_n^{\beta_n+1}(m) = \alpha_1^{\beta_1}+\alpha_2^{\beta_2}+\alpha_3^{\beta_3}+...(\alpha_n(m))^{\alpha_n^{\beta_n}+1})\)
  2. \(\alpha_1^{\beta_1}+\alpha_2^{\beta_2}+\alpha_3^{\beta_3}+...\alpha_n^{\beta_n}(m) = \alpha_1^{\beta_1}+\alpha_2^{\beta_2}+\alpha_3^{\beta_3}+...\alpha_n^{\beta_n(m)}\)
  3. \(p(0)(0) = \omega\)
  4. \(p(b)(0) = p(0)\)
  5. \(p(b+1)(0) = p(b)\)
  6. \(p(b+1)(n+1) = p(b)^{p(b+1)(n)}\)
  7. \(p(b)(n) = p(b(n))\) if \(\text{cof}(b) = \omega\)
  8. \(p(\Omega^{\beta_1}+\Omega^{\beta_2}+\Omega^{\beta_3}+...\Omega^{\beta_n+1})(k+1) = p(\Omega^\beta_1+\Omega^\beta_2+\Omega^\beta_3+...\Omega^{\beta_n}p(\Omega^\beta_1+\Omega^\beta_2+\Omega^\beta_3+...\Omega^{\beta_n})(k))\)
  9. \(g(0) = \Omega\)
  10. \(g(n+1) = \Omega^{g(n)}\)

We also see here an definition up to the BHO in only 9 rules (+3 for FGH, or 2 if you'd use HH). Then \(\text{Itaibn's 2nd number} = f(p(g(67)),68) \approx f_{\vartheta(\varepsilon_{\Omega+1})}(67)\)

Deedlit's 1st number

Uses the fast-growing hierarchy up to \(\psi(\Omega_\omega)\)

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