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In definitions of some ordinal collapsing functions (OCFs), there is always a series of \(C_n(\text{some arguments})\) sets. \(C_0(\text{something})\) may contains some argument in the "something", and may contains a "large" ordinal for collapsing; \(C_{n+1}(\text{something})\) is usually obtained from \(C_n(\text{something})\) by applying some operations with some limitations (avoiding "loop" definition); and \(C(\text{something})\) is the union of all \(C_n(\text{something})\). By this structure, we can define general fundamental sequences for OCFs.
Here this "general" definition works on the notation using a weakly compact cardinal. The definition of the ordinal notation is:
 Let \(K\) denote the weakly compact cardinal, \(\Omega_0=0\) andâ€¦
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Current approximations can be further improved, and this will take large amount of edits, so I start this discussion.
BEAF is not welldefined beyond tetrational arrays. More accurately, \(X\uparrow\uparrow A\&n\) where \(A\) is not a single number (e.g. \(X+1\), \(2X\), \(X^2\), \(X^X\), etc.) and beyond are not welldefined. But \(X\uparrow\uparrow m\&n\) and \(X\uparrow\uparrow X\&n\) are still considered as welldefined.
My suggestion about that is:
 Approximations of numbers defined in illdefined BEAF should be removed.
 Approximations in illdefined BEAF of any numbers should be removed.

What's the growth rate of Goodstein function? \(\varepsilon_0\)?
No. Unlike Hydra function and Worm function, which are comparable to \(f_{\varepsilon_0}(n)\), Goodstein function has some "cost"  \(G(2\uparrow\uparrow n)\approx f_{\varepsilon_0}(n)\). By the definition of "functional approximation":
 \(f\ge^*g\) if there exists k such that for all n > 0, \(f(n+k)\ge g(n)\)
and
 \(f\approx g\) if \(f\ge^*g\) and \(g\ge^*f\)
Goodstein function is not comparable to \(f_{\varepsilon_0}(n)\). Currently we don't have a compact notation to express the growth rate of it. However, in this blog post I won't consider this thing. Instead, I'll consider what it "costs" and extend it. Goodstein has a 3level cost under \(\varepsilon_0\), since \(G(f_3(n))\appâ€¦
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This kind of things have been discussed here.
Currently, we have these size classes (in ascending order):
 Class 0 (< 6)
 Class 1 (6 ~ 10^{6})
 Numbers with 7 to 21 digits
 Numbers with 22 to 100 digits
 Numbers with 101 to 309 digits
 Numbers with 309 to 4933 digits
 Numbers with 4933 to 1000000 digits (#3 ~ #7 are also called "Class 2")
 Class 3 (\(10^{10^6}\) ~ \(10^{10^{10^6}}\))
 Class 4 (\(10^{10^{10^6}}\) ~ \(10^{10^{10^{10^6}}}\))
 Class 5 (\(10^{10^{10^{10^6}}}\) ~ \(10^{10^{10^{10^{10^6}}}}\))
 Exponentiation level (\(10^{10^{10^{10^{10^6}}}}\) ~ \(10\uparrow\uparrow10\))
 Tetration level
 Uparrow notation level
 Chained arrow notation level
 56 entry linear array notation level
 7+ entry linear array notation level
 Two row array notation level
 Planar array notatioâ€¦
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Recently, Taranovsky has updated his ordinal notation page. In the last section he defined another ordinal notation system. In that system he used a ternary function C, and \(C_i(a,b)\) corresponds to \(C(\Omega_2\times i+a,b)\) in the main system.
Taranovsky conjectured that using \(i\le n\) (where n is a positive integer), the last system reaches \(\Pi_n^1\text{TR}_0\). As a result, in the main system, \(C(C(\Omega_2\omega,0),0)\) reaches secondorder arithmetic.
I've done some comparisons between array notation and Taranovsky's ordinal notation (but only a part of them published on my site). If the conjecture and the comparisons work, s(n,n{1{1(2{2^{+}}2)+2}2(2{2^{+}}2)+2}2) will surpass all functions provably recursive in secondorder arithmeâ€¦
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