9,750 Pages

# Hyp cos

My favorite wikis
• ## General fundamental sequences for OCFs

October 2, 2017 by 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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• ## 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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• ## 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…

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• ## 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…

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