User:Hyp cos/Catching Function Analysis p2

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From \(C(\Omega^22)\) to \(C(\Omega^3)\)
This block is similar to from \(C(\Omega2)\) to \(C(\Omega^2)\) block.

From \(C(\Omega^3)\) to \(C(\Omega^\omega)\)
Very simple here.

From \(C(\Omega^\omega)\) to \(C(\Omega^{\omega+1})\)
Let [*] donates separator {{0{0,1*}1}*}, but just here.

Here \(\psi_{I(I(1,0,0),1)}(0)\) is the supremum of \(I(\alpha,I(1,0,0)+1)\) for all \(\alpha&lt;I(1,0,0)\). Then \(\psi_{I(I(1,0,0),1)}(1)\) is the supremum of \(I(\alpha,\psi_{I(I(1,0,0),1)}(0)+1)\) for all \(\alpha&lt;I(1,0,0)\). Then \(\psi_{I(I(1,0,0),2)}(0)\) is the supremum of \(I(\alpha,I(I(1,0,0),1)+1)\) for all \(\alpha&lt;I(1,0,0)\).

Next, \(\psi_{I(I(1,0,0),I(1,0,0))}(1)\) is the supremum of \(I(\alpha,\psi_{I(I(1,0,0),I(1,0,0))}(0)+1)\) for all \(\alpha&lt;I(1,0,0)\). And \(\psi_{I(I(1,0,0),I(1,0,0)+1)}(0)\) is the supremum of \(I(\alpha,I(I(1,0,0),I(1,0,0))+1)\) for all \(\alpha&lt;I(1,0,0)\).

But it's a long way to get that. Before it we get \(f_{\psi(\psi_{I(I(1,0,0),I(1,0,0))}(0))}(2n)\approx g_{\psi(\psi_{I(I(1,0,0),I(1,0,0))}(0)2)}(n)\). Here \(\psi(\psi_{I(I(1,0,0),I(1,0,0))}(0)2)=\psi(\alpha\mapsto\psi_{I(1,0,0)}(\psi_{I(I(1,0,0),I(1,0,0))}(0)+\psi_{I(I(1,0,0),\alpha)}(0)))\), so

From \(C(\Omega^\Omega\omega)\) to \(C(\Omega^\Omega(\omega+1))\)
First, some FGH vs. R function analysis. (Oh, maybe it fits better if this is in an analysis part of R function pages) Notice that \(\psi_{\chi(M(1,0))}(1)\) is the limit of sequence {\(\psi_{\chi(M(1,0))}(0)2\), \(M_{\psi_{\chi(M(1,0))}(0)2}\), \(M_{M_{\psi_{\chi(M(1,0))}(0)2}}\), ...}

And please don't mind that I use a \(\bullet\) for \(\psi_{\chi(M(1,0))}(0)\). Note that \(\bullet>M\), and \(\bullet=\alpha\mapsto M_\alpha\).

And next I'll use R-function-look ordinal notation only.

\(f_{C(\Omega^\Omega\omega)}(n)\) is equivalent to SGH ordinal {0{0,1*}0,1}. And \(f_{C(\Omega^\Omega\omega)}(n+1)\) to {0{0,1*}{0{0,1*}{0{0,1*}0,1}}{0{0,1*}0,1}}, and \(f_{C(\Omega^\Omega\omega)}(2n)\) to {0{0,1*}{0{0,1*}0,1}{0{0,1*}0,1}}. Further, we get nR{{0,{0}}{0,1*}0,1} is equivalent to SGH ordinal {{0,{0}}{0,1*}0,1}. For shorthand I use █ for the separator {{{0{0,1*}1}{0,1*}0,1}*} here.

For shorthand I use ▲ for the separator {{0{0,1*}1}{{0{0,1*}1}{0,1*}0,1}*} here.

That means R function string {{0{0,1*}{0}}{0,1*}0,1} is equivalent to \(C(\Omega^\Omega\times(\omega+1))\).

From \(C(\Omega^{\Omega+\omega})\) to \(C(\Omega^{\Omega\omega})\)
Well, what's the next - \(C(\Omega^{\Omega+\omega}\omega+1)\)? Here we meet some difficulties to compare FGH to SGH from "inside". But we can do it from "outside" though it'll be more complex. Here I use R-function-look ordinal notation to represent SGH ordinal, and the original R function to represent FGH (because the rule "nRa+1\(\odot\) = nRa\(\odot\)Ra\(\odot\)...Ra\(\odot\)" is similar to FGH).

And here I use \(n(R\odot)^k\) to represent \((...(nR\odot)R\odot...)R\odot\) with k \(R\odot\)'s, use [k] for fundamental sequence. That means, in step 3 in Case B3 here, apply \(\{\odot_1\odot_3\{\odot_10\{0^{*m+1}\}a+1\odot_2^{*m}\}\odot_4\{0^{*m+1}\}a\odot_2^{*m}\}=\) \(\{\odot_1\odot_3\{\odot_10\{0^{*m+1}\}a+1\odot_2^{*m}\}\odot_4\{0^{*m+1}\}a\odot_2^{*m}\}[n]\), where \(\{\odot_1\odot_3\{\odot_10\{0^{*m+1}\}a+1\odot_2^{*m}\}\odot_4\{0^{*m+1}\}a\odot_2^{*m}\}[0]=0\) and \(\{\odot_1\odot_3\{\odot_10\{0^{*m+1}\}a+1\odot_2^{*m}\}\odot_4\{0^{*m+1}\}a\odot_2^{*m}\}[k+1]=\) \(\{\odot_1\odot_3\{\odot_1\odot_3\{\odot_10\{0^{*m+1}\}a+1\odot_2^{*m}\}\odot_4\{0^{*m+1}\}a\odot_2^{*m}\}[k]\odot_4\{0^{*m+1}\}a\odot_2^{*m}\}\). Also, in Case B2, apply \(\{a+1\odot\}=\{a+1\odot\}[n]\), where \(\{a+1\odot\}[0]=0\) and \(\{a+1\odot\}[k+1]=\{a+1\odot\}[k]0\{a\odot\}\).

Therefore, \(C(\Omega^{\Omega+\omega}\omega+1)\) is {0{0,1*}0{{0{0,1*}0{0,1*}1}*}{0,{0}}}, not {{0,{0}}{0,1*}0{0,1*}1} or {0{{0{0,1*}0{0,1*}1}*}{0,{0}}}.

And then,

Also, in normal notation \(C(\Omega^{\Omega\omega})=\psi(\Psi_{\Xi(\omega,0)}(0,0))\).

From \(C(\Omega^{\Omega\omega})\) to \(C(\Omega^{\Omega^2})\)
In normal notation, the \(C(\Omega^{\Omega^2})=\psi(\Psi_{\Xi(K_\omega)}(0,0))\).

From \(C(\Omega^{\Omega^2})\) to \(C(\Omega^{\Omega^\omega})\)
Image this, level-1 is inaccessible ordinal, level-2 is mahlo, and level-3 is compact... The \(C(\Omega^{\Omega^\omega})\) is at the start of level-\(\omega\). Err, I don't know how to translate this ordinal into normal notation, it's too big. Maybe Taranovsky's ordinal notation can hold it, but how to represent it in that notation?

From \(C(\Omega^{\Omega^\omega})\) to \(C(\Omega^{\Omega^\Omega})\)
For shorthand I use ■ for the array {0{0,0,1*}1} here.

From \(C(\Omega^{\Omega^\Omega})\) to \(C(\Omega^{\Omega^\Omega}(\omega+1))\)
For shorthand I use ■ for the array {{0{0,0,1*}1}{0,0,1*}0,1} here.

From \(C(\Omega^{\Omega^{\Omega^\Omega}})\) to \(C(\varepsilon_{\Omega+1})\)
So \(C(\varepsilon_{\Omega+1})\) is equal to the ordinal limit of pseudo nested array notation.

From \(C(\varepsilon_{\Omega+1})\) to \(C(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}\omega)\)
The ordinal {0{0{0{0**}1*}2*}1} is the limit of sequence {0{0{0{0**}1*}1*}1}, {0{0{0{0{0**}1*}1*}1{0{0**}1*}1*}1}, {0{0{0{0{0{0**}1*}1*}1{0{0**}1*}1*}1{0{0**}1*}1*}1}, ... And if we use this ordinal in SGH, its growth rate will be equivalent to (2n)R{0{0{0{0**}1*}1*}1}.

For shorthand I use █ for the separator {0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*} here.

Now we get \(C(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}\omega)\) here, but the R function seems erratic. The {0{0{0{0**}1*}1{0{0{0{0**}1*}0,1*}1}*}1} has higher level than {0{0{0{0{0**}1*}0,1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}, and the encloser { ____ {0{0**}1*}{0{0{0{0**}1*}0,1*}1}*} (where the ____ can contain separators up to (but excluding) {0{0**}1*}) also has higher level than {0{0{0{0**}1*}0,1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}. However, if it reduces, the {0 Sn 1}, where S0 = 0 and Sk+1 = {0 Sk 1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}, all have lower level than {0{0{0{0{0**}1*}0,1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}. In R function, the "level" is not completely equivalent to growth rate. Next, I'll find out what's {0{0{0{0{0**}1*}0,1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} in the form {0{0{0{0**}1*} something *}1}. Start from {0{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}. So \(C(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}\omega)\) is also {0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}.
 * {0{0{0{0{0**}1*}{0█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0█1}*}2{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}2{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0█1}*}0,1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}0,1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{1{0{0**}1*}{0█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0{0**}1*}1*}1{0{0**}1*}{0█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0{0{0**}1*}1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0{0**}1*}{0█1}*}1{0{0**}1*}{0█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}1{0█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}1{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0█1}{0█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0█1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{1█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{1█1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0,1█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0,1█1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0█2}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0█2}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0█0,1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0█0,1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0█0█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0█0█1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0{1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0{1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0{0{0{0{0**}1*}1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0{0{0{0{0**}1*}1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0{0{0{0{0**}1*}1{0█1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0{0{0{0**}1*}1{0{0{0{0**}1*}0,1*}1}*}1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}0,1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}

From \(C(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}}\omega)\) to \(C(\varepsilon_{\Omega+2})\)
Here I change {0{0{0{0{0**}1*}0,1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1} into the form {0{0{0{0**}1*} something *}1}. So \(C(\varepsilon_{\Omega+1}^{\varepsilon_{\Omega+1}^2}\omega)\) = {0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}.
 * {0{0{0{0{0**}1*}{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}1{0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}1{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}{0{0{0{0**}1*}1{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0{0{0{0**}1*}1{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}
 * {0{0{0{0{0**}1*}0,1*}1{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1} = {0{0{0{0**}1*}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}{0{0{0{0**}1*}0,1*}1}*}1}