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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 \(aE_{n+1}=aE_n...E_n\) with \(aE_n\,E_n\)s.
 * 5) Define Church-Kleene-Bachmann-Howard Oodles as \(10^{100}E_{\vartheta^\text{CK}(\varepsilon_{\Omega+1})}\), that is, with the "Church-Kleene-Bachmann-Howard 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 \(b(n,m)=10^{10^{10^{\cdots^{10^m3+1}\cdots}3+1}3-30}155\) with \(n+3\) 10s (default: \(m=31\) instead of the "default of defaults" when the value is set to 1 or 0 by default) Define \(c(n)=10^{10^{\ddots^{10^{31}3+1}\cdots}3+1}+9\) with n+3 10s. Methynillion refers to the group methyne group and equals \(10^{a_{b(10^{150}+9)}}\), methanoicillion refers to formic acid equaling \(10^{a_{b(10^{180}+9)}}\).

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

\(\{\omega,\omega,1,2\}=\vartheta(\Omega^3)=\vartheta(\Omega\Omega\Omega)\).

Extra-Fast Growing Hierarchy

 * 1) \(h_0(n)=n+1\).
 * 2) \(h_{\alpha+1}(n)=h_\alpha^{h_\alpha^{\cdots^{h_\alpha(n)}\cdots}(n)}(n)\) with (\(h_\alpha^{h_\alpha^{\cdots^{h_\alpha(n)}\ddots}(n)}(n)\) with (... with \(h_\alpha^{h_\alpha^{\cdots^{h_\alpha(n)}\ddots}(n)}(n)\) \(h\)s) \(h\)s)...)\(h\)s; with \(h_\alpha(n)\) lines (1D spaces), then similarly defining planes (2D spaces), 3D, ..., defining superdimensions as dimension-defining lines, super-superdimensions as superdimension-defining, ..., trimensions as \(h_\alpha^{h_\alpha(n)}(n)\)-ex-superdimensions, similarly defining supertrimensions, super-supertrimensions,..., tetramensions, pentamensions, hexamensions,..., continue with \(h_\alpha^{h_\alpha(n)}(n)\)th element of dimension, trimension, tetramension,... sequence.
 * 3) \(h_\alpha(n)=h_{\alpha[h_{\alpha[\cdots_{h_{\alpha[n]}(n)}\cdots]}]}(n)\) with (\(h_{\alpha[h_{\alpha[\cdots_{\alpha[n]}\cdots]}(n)}(n)\) with ... with \(h_{\alpha[h_{\alpha[\cdots_{\alpha[n]}\cdots]}(n)}(n)\) \(h\)s) ...) \(h\)s) with (\(h_{\alpha[...]}(n)\) with (... (with \(h_{\alpha[...]}\) \(h\)s) lines (1D spaces))...) then similarly defining 2D, 3D, ..., define superdimensions, super-superdimensions,..., trimensions, tetramensions,..., as in the successor rule, but place after each alpha a bracket containing the rest of values and remove superscripts. Continue with \(h_{\alpha[h_{\alpha[n]}(n)]}(n)\)th element of dimension, trimension,... sequence iff \(\alpha\) is limit.

Define surprimitive recursive functions to be \(<^*h_\omega\), and surrecursive ones to be \(<^*h_{\omega_1^\text{CK}}\). Then,

Tasks

 * 1) where in the FGH, HH, MGH and SGH is \(h_\omega\), the first non-surprimitive-recursive function?
 * 2) where in the FGH is \(h_{\omega_1^\text{CK}}\), the first non-surrecursive function?
 * 3) Does the FGH catch up* to the EFGH, and if yes, at what ordinal?
 * 4) If the FGH catches up* to the EFGH, does the HH catch up to the EFGH, and if yes, at what ordinal?
 * 5) If the HH catches up* to the EFGH, does the MGH catch up to the EFGH, and if yes, at what ordinal?
 * 6) If the MGH catches up* to the EFGH, does the SGH catch up to the EFGH, and if yes, at what ordinal?

&#42; using the most natural definition as in: \(\exists\alpha<\omega_1:h_\alpha\approx^*f_\alpha\).

Googolisms
Default for self-defined googolisms is 1 in function argument. Not because I don't want too large growth, but it's still large enough.

Ordinals beyond Church-Kleene fixed point

 * φCK(1,0), CHURCH-KLEENE FIXED POINT also ε0CK
 * φCK(1,1), also ε1CK
 * φCK(2,0), also ζ0CK
 * φCK(3,0), also η0CK
 * φCK(ω,0)
 * φCK(ε0,0)
 * φCK(φ(ω,0),0)
 * φCK(Γ0,0)
 * φCK(Γ1,0)
 * φCK(ϑ(Ω^3),0)
 * φCK(ϑ(Ω^ω),0)
 * φCK(ϑ(Ω^Ω),0)
 * φCK(ϑ(ε(Ω+1)),0)
 * φCK(ω1Ch,0)
 * φCK(ω1CK,0)
 * φCK(ε0CK,0)
 * φCK(ζ0CK,0)
 * φCK(η0CK,0)
 * φCK(φCK(ω,0),0)
 * φCK(φCK(ε0,0),0)
 * Γ0CK, also ϑCK(Ω^2), also φCK(1,0,0)
 * Γ1CK, also φCK(1,0,1)
 * φCK(1,1,0)
 * φCK(1,0,0,0), also ϑCK(Ω^3)
 * φCK(ω&α), also ϑCK(Ω^ω)
 * φCK(ω&α&α), ...it's possible to extend Veblen arrays just as most arrays
 * φCK(ω&α&α&...), also ϑCK(Ω^Ω)
 * ϑCK(ε(Ω+1)), What do we get when combining BHO with Church-Kleene ordinal? This ordinal.

SbS' errors

 * 1) He uses unary (as in, he writes 1+1+1+1+1 instead of 5) in FGH.
 * 2) He doesn't un-parenthesize the omega powers with either 1 element or only exponentiation.
 * 3) His notation uses carets instead of superscripts.
 * 4) He uses +, *, \(x^y\) as function instead of as operator.