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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 andâ€¦
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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})â€¦
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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.
I finally got around to correcting some errors I made in the YouTube video. The following are displayed as captions:
29:06 f2(2048) is 'only' equal to 10^619 and is much smaller than 10^10^620
29:13 f2(f2(14)) = f2(229376) is bigger but only equal to 10^10^4
29:18 f2(f2(16)) is only equal to 10^10^5
29:21 f2(f2(24)) is only equal to 10^10^8
29:26 f2(f2(64)) is only equal to 10^10^20 bâ€¦
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The S function offers a way to generate very large string sequences (representing large numbers). The function grows at the rate \(f_{\varphi(1,1,0)}(n)\) .
Refer to my The Alpha Function blogs for more information on my work.
This blog is a significant update to the original Version 1 of this function. Please keep this in mind if you refer to the earlier blog.
The S function is recursively defined set of two functions \(S()\) and \(T()\) which use string substitution procedures only. They do not explicitly use any mathematical or transfinite ordinal theory. The String substitution procedures have been converted into a program. Refer to my blogs on The Alpha Function for more information and presentation of the results.
The S function is definedâ€¦
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I have written a Version 2 of this function that generalises the function and significantly increases its growth rate. Please refer to that version instead.
The S function offers a way to generate very large string sequences (representing large numbers) which grows at a faster rate than \(f_{SVO}(n)\), and may be possible to extend its range to grow at a comparable rate to Ordinal Collapsing Functions.
Refer to my The Alpha Function blogs for more information on my work.
The S function is a string substitution function, and does not explicitly use any mathematical or transfinite ordinal theory. It is effectively a computer algorithm that can be converted to a program. My next version of my Alpha Function will do this and present the results.
Theâ€¦
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