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This blog will provide a detailed calculation and references for the growth rate of The S Function that I have developed. The growth rate is comparable to:
\(S(n,T^{T(0)}(0),1) \approx f_{\varphi(1,1,0)}(n)\)
The S function is recursively defined set of two functions \(S()\) and \(T()\) which use string substitution procedures only. S Functions can be either restricted or generalised. Refer to my main blog on The S Function for a full definition of how the function is constructed.
As a simple introduction it will be useful to compare a typical S function with more familiar functions:
\(S(3,2,1) = f_2(3)\) ordinal value
This equivalence is intentional. In fact:
\(S(n,g,h) = f_g^h(n)\) ordinal value
This equivalence will become more obvious if you ref…
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This is my candidate for a large named number. It is called The Last Alpha Number and is calculated using The Alpha Function. It is approximately \(f_{svo}(10000)\) and therefore it is not big (by this website standard).
The Alpha Function calculates an ordinal for a given real number input. For example:
\(\alpha(0.00) = 0\)
\(\alpha(1.00) = 1\)
\(\alpha(2.00) = 8\)
\(\alpha(e) = 9\)
\(\alpha(3.00) = 10\)
\(\alpha(\pi) = 11\)
\(\alpha(4.00) = f_{\omega}(8)\)
The function accepts real numbers up to 10,000 at which point it asymptotically goes to infinity. The Alpha Function is based on my work on the S Function (substitution function). The limit of the Alpha Function is represented as follows:
\(\alpha(10000) = S(2,T^{\omega}(0),1) = \omega\) equals in…
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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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