Googology Wiki
Googology Wiki

Hyp cos Hyp cos 19 December 2019

Degrees of Recursive Inaccessibility and other OCFs

Degrees of Recursive Inaccessibility is an OCF created by Dmytro Taranovsky, earlier than other ordinal notations. It seems to be the simplest OCF and yield a simplest ordinal notation among this kind.

In this OCF, the a in C(a,b,c) means "admissibility degree"; b means degree and is related to how "rare" this ordinal is; c is related to the absolute size of this ordinal. It is addition-free and yield \(C(0,a,b)=b+\omega^a\) iff \(a\omega\land\beta\) is admissible \(\land\forall\gamma

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Hyp cos Hyp cos 19 April 2019

Googolisms hard to extend?

Here, to extend a googolism (large number) means to make a larger number in a similar way.

Some googolisms from "notations" such as array notations, hydras, FGH/HH/SGH + OCF, are easy to extend. For instance, Bird's \(U^{U(3)}(3)\) can extend to a new part of Bird's array notation, with arrays as subscript of the slash. Extending to large branches of new numbers is not much more difficult than understanding the notation itself.

Some googolisms resulting from "combinatorial functions" are also easy to extend, though naive extension. For instance, from TREE(3) there are simply TREE(4), TREE(5), etc.

But there are still some numbers appearing singly, or even without any "input parameters", thus hard to extend. A cool example is 808017424794512875…

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Hyp cos Hyp cos 25 February 2019

Non-Classic Growth Rate

The post is a continuation of this one.

  • 1 A better definition of "comparable" functions
  • 2 Growth rate (HH) addition and subtraction
  • 3 Growth rate (HH) multiplication and division by ordinal
  • 4 Growth rate (HH) multiplication and division
  • 5 General procedure of growth rate \(\alpha-1\)

There are many definitions of eventual domination. Some are listed below from tighter to looser.

  1. \(f\ge^*g\) (f eventually dominates or is comparable to g) if \(\exists N\forall n>N(f(n)\ge g(n))\)
  2. \(f\ge^*g\) if \(\lim\limits_{n\to\infty}\frac{g(n)}{f(n)}\le1\)
  3. \(f\ge^*g\) if \(\exists a,b,N\forall n>N(f(n+a)+b\ge g(n))\)
  4. \(f\ge^*g\) if \(\exists a,b,N\forall n>N(f(an+b)\ge g(n))\)

And \(f\approx g\) (f is comparable to g) if \(f\ge^*g\land g\ge^*f\). So \(f\approx f\) holds…

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Hyp cos Hyp cos 26 October 2018

Attempt of OCF up to Stability

These OCFs are more complicated than previous OCFs. They are quite different from the OCFs in some literature, which I do not really understand, so it is probable that I do not follow the right way.

Those OCF are defined based on some guesses at reflecting/indescribable ranges. If there are more things, such as

  • \(\Pi_3\)-reflecting ordinals that are also \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings, but are not "\(\Pi_3\)-reflecting ordinals that are also \(\Pi_2\)-reflecting on \(\Pi_2\)-reflectings on \(\Pi_3\)-reflectings"
  • ordinals that are both \(\Pi_3\)-reflecting on \(\Pi_3\)-reflectings, and \(\Pi_2\)-reflecting on \(\Pi_3\)-reflectings on \(\Pi_3\)-reflectings, but are not "\(\Pi_3\)-reflecting on \(\Pi_3\)-reflectings, and \(\Pi_2\)…
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Hyp cos Hyp cos 24 September 2018

TON, stable ordinals, and my array notation

Many people don't understand Taranovsky's ordinal notation (TON). Some of them understand normal ordinal collapsing functions (OCFs), and want to know comparisons between other notations and OCFs. However, OCFs beyond \(\Pi_3\)-reflection suddenly become complicated, such as the ones for \(\Pi_4\)-reflection, 1-stable, \(\beta\)-stable for any constant \(\beta\), and further, yet TON remains simple. Here I show some details of TON with my understanding.

To understand the definition of this kind of notation is not easy, although it might look simpler than normal OCF. We need 5 steps to understand it.

Terms are the things the system work on. In the n-th system of TON, terms are constructed using these following 2 rules.

  1. \(0\) and \(\Omega_n\) a…

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