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# Hyp cos

My favorite wikis
• ## 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… Read more > • ## Suggestions about Size Classes of Numbers July 1, 2017 by 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…

• ## Taranovsky has updated his ordinal notation page

September 10, 2016 by 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…

• ## 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…

• ## Fundamental Sequences in Taranovsky's Notation

March 28, 2016 by 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