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  • Hyp cos

    Growth Rate ε(0)−1

    July 13, 2017 by Hyp cos

    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 3-level cost under \(\varepsilon_0\), since \(G(f_3(n))\app…

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  • Hyp cos

    This kind of things have been discussed here.


    Currently, we have these size classes (in ascending order):

    1. Class 0 (< 6)
    2. Class 1 (6 ~ 106)
    3. Numbers with 7 to 21 digits
    4. Numbers with 22 to 100 digits
    5. Numbers with 101 to 309 digits
    6. Numbers with 309 to 4933 digits
    7. Numbers with 4933 to 1000000 digits (#3 ~ #7 are also called "Class 2")
    8. Class 3 (\(10^{10^6}\) ~ \(10^{10^{10^6}}\))
    9. Class 4 (\(10^{10^{10^6}}\) ~ \(10^{10^{10^{10^6}}}\))
    10. Class 5 (\(10^{10^{10^{10^6}}}\) ~ \(10^{10^{10^{10^{10^6}}}}\))
    11. Exponentiation level (\(10^{10^{10^{10^{10^6}}}}\) ~ \(10\uparrow\uparrow10\))
    12. Tetration level
    13. Up-arrow notation level
    14. Chained arrow notation level
    15. 5-6 entry linear array notation level
    16. 7+ entry linear array notation level
    17. Two row array notation level
    18. Planar array notatio…

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  • Hyp cos

    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 second-order 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 second-order arithme…

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  • Hyp cos

    Idea for legion

    August 18, 2016 by Hyp cos

    This blog post is a tip for new googologists who want to create a fast-growing function by recursion. It leads you to \(\psi(\Omega_\omega)\) - the limit of \(\Pi_1^1-CA_0\).


    In BEAF, the symbol "X" can take on some arithmetic, such as exponentiation. Exponentiation is not very fast-growing in googoloical sense. However, when symbol "X" takes on exponentiation, "X^X&" will be a very strong operator, and result very fast-growing functions - much fast-growing than exponentiation. Also, in ExE, when symbol "#" takes on exponentiation, "#^#" will be a very strong separator, and the result functions are much faster-growing than exponentiation. And, in FGH, when symbol "ω" takes on exponentiation, "ω^ω" also results a much faster-growing function…


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  • Hyp cos

    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 fast-growing 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 x-1 &&( y-1 ? y-2 ? x-2 ? G R,r(y)) || R==r(y)&&G L,l(y))  : G x,r(y))  : x-2 && !G y,R)  :1); b(x B E G x,y) ? x-2 ? G d,x) ? b(L B * b(R B  : n && b(L,y,n-1,x) * b(R,y,n-1,x)  : n + G d,x)  :1; t(x,n) E x

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