
3
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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This blog post is a tip for new googologists who want to create a fastgrowing function by recursion. It leads you to \(\psi(\Omega_\omega)\)  the limit of \(\Pi_1^1CA_0\).
In BEAF, the symbol "X" can take on some arithmetic, such as exponentiation. Exponentiation is not very fastgrowing in googoloical sense. However, when symbol "X" takes on exponentiation, "X^X&" will be a very strong operator, and result very fastgrowing functions  much fastgrowing than exponentiation. Also, in ExE, when symbol "#" takes on exponentiation, "#^#" will be a very strong separator, and the result functions are much fastergrowing than exponentiation. And, in FGH, when symbol "Ï‰" takes on exponentiation, "Ï‰^Ï‰" also results a much fastergrowing functionâ€¦
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Here come two sets of fundamental sequences (FS) on Taranovsky's ordinal notation. One is for ideal use, and the other is for reality use.
Here's a code to calculate fastgrowing function based on Taranovsky's notation. This code is used for Bignum Bakeoff.
#define E return #define R r(x) #define L l(x) #define B ,y,n,d) #define G g( L E x%2 ? 0 : 1+l(x/2) ; R E x >> 1+L ; G x,y) E x1 &&( y1 ? y2 ? x2 ? G R,r(y))  R==r(y)&&G L,l(y)) : G x,r(y)) : x2 && !G y,R) :1); b(x B E G x,y) ? x2 ? G d,x) ? b(L B * b(R B : n && b(L,y,n1,x) * b(R,y,n1,x) : n + G d,x) :1; t(x,n) E x
â€¦
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