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The Alpha Function can generate every finite integer up to a very large number. The Alpha Function has a growth rate faster than \(f_{LVO}(n)\) for any n.
The Alpha Function is 'calibrated' to accept real number inputs up to 10,000 and generate unique S() function outputs representing any and every big number up to the size of \(f_{LVO}(v)\) for any n.
I have started work on a set of Ruler Functions based on the Alpha function that will be useful to measure the size of very large numbers. Different 'rulers' can be used as required. A rough sketch of how this will look is:
\(\alpha_0\) Ruler Function
\(\alpha_0(100) = \alpha(31.6) >> f_{svo}(3)\) approximately
\(\alpha_1\) Ruler Function
\(\alpha_1(100) = \alpha(5.79955) = f_{\omega + 1}(64) >> g_{64} =…
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The Alpha Function can generate every finite integer up to a very large number. The Alpha Function has a growth rate faster than \(f_{LVO}(n)\) for any n.
The Alpha Function is 'calibrated' to accept real number inputs up to 10,000 and generate unique S() function outputs representing any and every big number up to the size of \(f_{LVO}(v)\) for any n.
I have started work on a set of Ruler Functions based on the Alpha function that will be useful to measure the size of very large numbers. Different 'rulers' can be used as required. A rough sketch of how this will look is:
\(\alpha_0\) Ruler Function
\(\alpha_0(100) = \alpha(31.6) >> f_{svo}(3)\) approximately
\(\alpha_1\) Ruler Function
\(\alpha_1(100) = \alpha(5.79955) = f_{\omega + 1}(64) >> g_{64} =…
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This version of The Alpha Function has been changed to fix some errors from the last version and to 'recalibrate' the function and make the range more interesting.
The function code is still based on The S Function (Version 2), with a growth rate of \(f_{svo}(n)\).
Version 8 has been changed from Version 7 to fix some errors and to 'recalibrate' the function and make the range more interesting.
The errors in the last version were due to the simple code used to generate the T() functions. The S() functions were successfully generating to their maximum range. The T() functions were not. Code changes to \(t_x\) and \(u_x\) have been made to correct this.
The next change was to reduce the range of the input real number parameter to between 0 and…
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The following table provides a comparison of ordinal collapsing function. It is intended to be a quick reference.
Comparison Ordinal Collapsing Functions
The table is based on the following comparative formulas presented in the comments section of that blog:
\(\vartheta(\Omega.\alpha + \beta) = \theta(\alpha,\beta)\)
\(\vartheta(1 + \Omega.\alpha + \beta) = \psi(\Omega.\alpha(1 + \beta))\)
Comparison Table
Ordinal \(\theta()\) \(\vartheta()\) \(\psi()\)
\(\epsilon_0\) \(\theta(0,1)\) \(\vartheta(1)\) \(\psi(0)\)
\(\zeta_0\) \(\theta(1,0)\) \(\vartheta(\Omega)\) \(\psi(\Omega)\)
\(\Gamma_0\) \(\theta(\Omega,0)\) \(\vartheta(\Omega^2)\) \(\psi(\Omega^{\Omega})\)
Ackermann Ordinal \(\theta(\Omega^2,0)\) \(\vartheta(\Omega^3)\) \(\psi(\Omega^{\Omega^2})…
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Hi. I am putting together a YouTube video to celebrate the field of Googology and to give the public an introduction to the amazing range of large finite numbers as far as our present day knowledge allows.
The video is meant to be accessible to the average person, but the nature of the subject is mindboggling, so it is hard to guess the level of interest out there.
There is a new version of this YouTube video with the music by Steve Reich deleted. Unfortunately, the video is blocked worldwide by the copyright holder, and I can't find a way around this. The new video has no audio which is unfortunate but at least it can still be watched.
I finally got around to correcting some errors I made in the YouTube video. The following are displayed …
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