• Hyp cos

    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, \(\Omega_0=0\) and…

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  • Hyp cos

    Adcanced approximation project

    September 14, 2017 by Hyp cos

    Current approximations can be further improved, and this will take large amount of edits, so I start this discussion.

    BEAF is not well-defined 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 well-defined. But \(X\uparrow\uparrow m\&n\) and \(X\uparrow\uparrow X\&n\) are still considered as well-defined.

    My suggestion about that is:

    1. Approximations of numbers defined in ill-defined BEAF should be removed.
    2. Approximations in ill-defined BEAF of any numbers should be removed.
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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, \(f(n+k)\ge g(n)\)


    \(f\approx g\) 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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