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Diagonalizing over oodle theory
For sake of this, \(E\) is shorthand for \(E_0\). \(E\geq^*f_{\vartheta^\text{CK}(\Omega^\Omega)}\), with \(\vartheta^\text{CK}(\Omega^\alpha)\) enumerating the first ordinal that isn't reachable within \(\alpha\)-ary Church-Kleene fixed point function, if \(\alpha<\Omega\), and predicatively many arguments if \(\alpha=\Omega\), and the Church-Kleene fixed points are \(\varphi^\text{CK}(\alpha,\beta,\cdots)\)-enumerated.
 * 1) \(E\) is an operator hierarchy, defined as:\(aEb\) is the largest finite ordinal expressible in \(a\) symbols in \(b\)th order oodle theory.
 * 2) And \(aE\)is\((...((aEa)E(aEa))E...E((aEa)E(aEa))...)E(...((aEa)E(aEa))E...E((aEa)E(aEa))...)\) with \(2^{aEa}\) parens and \(2^{aEa+1}\) operands in total.
 * 3) Define \(aE_\alpha\) as \(aE_{\alpha[\cdots[aE_{\alpha[a]}]}\) with \(aE_{\alpha[aE_{\alpha[a]}]}\) brackets, iff \(\alpha\) is limit.
 * 4) Define Church-Kleene-Veblen Oodles as \(10^{100}E_{\vartheta^\text{CK}(\Omega^\Omega)}\), that is, the "large Church-Kleene-Veblen ordinal".

Large \(n\)umbers
\(a_1=(10^{(10^{421290}+1)421290}+1)421290, a_{n+1}=(10^{a_n}+1)421290\) (we can extend backwards, \(a_0=(10^{421290}+1)421290\)). This goes beyond tetration level, and even tritri, but not up-arrow notation level. Also note that I use the xenna- prefix as \(10^{27}\), and xentillion for \(10^{10^{27}n+6-n}\), where \(n=6\) in long scale and \(n=3\) in short scale. Define \(b(n,m)=10^{10^{10^{\cdots^{10^m3+1}\cdots}3+1}3-30}149\) with \(n+3\) 10s (default: \(m=31\) instead of the "default of defaults" when the value is set to 1 or 0 by default) There're other names, like Kaliumillion for kalium. Thus, it's a "bit" beyond all other naming systems.

Omega pentations and beyond
\(\omega\uparrow^3\omega=\zeta_0\), \(\omega\uparrow^3(\omega+1)=^\omega\zeta_0=\varepsilon_{\zeta_0+1}\), \(\omega\uparrow^{\omega+1}\omega=\Gamma_0\).

Extra-Fast Growing Hierarchy
\(h_0(n)=n+1\).

\(h_{\alpha+1}(n)=h_\alpha^{h_\alpha^{\ddots^{h_\alpha(n)}\ddots}(n)}(n)\).

\(h_\alpha(n)=h_{\alpha[h_{\alpha[\ddots_{h_{\alpha[n]}(n)}\ddots]}]}(n)\) with \(h_{\alpha[n]}(n)\) \(h\)s iff \(\alpha\) is a limit ordinal.