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I am going to restart Sheet Analysis because it's unclear and unorganized as before.
You might check the brand new 'The Sheet Analysis' out here.
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Here is the proof that \(^{\omega 2}\omega\) = \(\varepsilon_{\varepsilon_0}\) or a weaker ordinal \(\varepsilon_1\).
Ordinal tetration is a type of normal form but we can express ordinals with 0, 1, \(\omega\), addition, multiplication, exponentiation, and tetration.
The proof require to have 1 new property, OPTD (\(\omega\) power tower depth). Now what's OPTD? A natural number have OPTD 0, \(\omega\) to maximum value that less than \(\omega^{\omega}\) have OPTD 1, and \(\omega^{\alpha}\) have OPTD \(\beta+1\), such that \(\alpha\) have OPTD \(\beta\).
Also, I also show that \(1+\omega\) = \(\omega\) and \(\omega^{\varepsilon_02}\) have OPTD \(\omega\).
Imagine 0 is the empty set and \(\alpha+1\) is the union set of all elements of \(\alpha\)â€¦
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Here is reboot version of my array notation, simpler than ever.
Growth rate: 0
The simplest and weakest part in AANR (Aarex's Array Notation Rebooted). There is only 1 rule in this extension: r(a) = a+1
The limit of simple arrays corresponds to FGH level 0 because r(a) evaluates as the same of \(f_0(a)\).
Growth rate: \(\omega\)
Let [A]^B = AAA... with Bs, but [A] evaluates, even inside of any [_]s. r(a,0) = a+1
 r(a,b) = [r(]^{a}a[,[b1])]^{a}
If the syntax concentration is too hard for you, here is rules without syntax concentration:
 r(a,0) = a+1
 r(a,b)[0] = a
 r(a,b)[n] = r(r(a,b)[n1],b1)[n]
 r(a,b) = r(a,b)[a]
To solve r(2,2) (using without syntax concentration):
 r(2,2) =
 r(2,2)[2] =
 r(r(2,2)[1],1) =
 r(r(r(2,2)[0],1),1) =
 r(r(2,1),1) =
 r(r(2,1)[2],1) =
 r(r(r(2,1)â€¦
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You can find the definition of Dropper Ordinal Notation (DON) here.
Dropper Ordinal Notation Other OCFs
\(D[1]\) \(\Omega\)
\(D[2]\) \(\Omega_2\)
\(D[n]\) \(\Omega_n\)
\(D[\omega]\) \(\Omega_{\omega}\)
\(D[D[0]]\) \(\Omega_{\Omega}\)
\(D[D[D[0]]]\) \(\Omega_{\Omega_{\Omega}}\)
\(\psi_{D[D_0]}(0)\) \(\psi_I(0)\)
\(\psi_{D[D_0]}(0)^{\psi_{D[D_0]}(0)}\) \(\psi_I(0)^{\psi_I(0)}\)
\(\psi_{D[\psi_{D[D_0]}(0)+1]}(0)\) \(\psi_{W\_{\psi_I(0)+1}}(0)\)
\(D[\psi_{D[D_0]}(0)+1]\) \(\Omega_{\psi_I(0)+1}\)
\(D[\psi_{D[D_0]}(0)+D[\psi_{D[D_0]}(0)+1]]\) \(\Omega_{\Omega_{\psi_I(0)+1}}\)
\(\psi_{D[D_0]}(1)\) \(\psi_I(1)\)
\(\psi_{D[D_0]}(D[D_0])\) \(\psi_I(I)\)
\(D[D_0]\) \(I\)
\(D[D_0+1]\) \(\Omega_{I+1}\)
\(D[D_0+D[D_0+1]]\) \(\Omega_{\Omega_{I+1}}\)
\(D[D_02]\) \(I_2\)
\(Dâ€¦
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\(D[0]\) is least cardinality and \(D[\alpha+1]\) is greater cardinality than \(D[\alpha]\). Also, \(D[\alpha[D_0]]\) is similar to weakly inaccessible cardinals in OCFs, which:
 \(\psi_{D[\alpha[D_0]]}(0)[1]\) = \(D[\alpha[1]]\)
 \(\psi_{D[\alpha[D_0]]}(0)[n]\) = \(D[\alpha[\psi_{D[\alpha[D_0]]}(0)[n1]]]\)
 \(\psi_{D[\alpha[D_0]]}(\beta+1)[1]\) = \(D[\alpha[\psi_{D[\alpha[D_0]]}(\beta)+1]]\)
 \(\psi_{D[\alpha[D_0]]}(\beta+1)[n]\) = \(D[\alpha[\psi_{D[\alpha[D_0]]}(\beta)+\psi_{D[\alpha[D_0]]}(\beta+1)[n1]]]\)
Then a new definition happen to prove it WAY stronger:
 \(D[\alpha[\beta[D[\alpha[D_{\gamma+1}]]]]]\) can be reduced to
\(D[\alpha[\beta[D_{\gamma}]]]\), where \(\beta\) must be less than
\(D_{\gamma+1}\) and \(\gamma\) > 0.
 Do not follow theâ€¦