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This version of The Alpha Function has been rewritten to use Javascript in Google Sheets. The code is available for anybody to use or copy as they like. This code replaces my last version which used VBA in Microsoft Excel.
The function code is still based on The S Function (Version 2), with a growth rate of \(f_{\varphi(1,1,0)}(n)\).
Version 9 has been completely rewritten to use Javascript in Google Sheets. A link to the first draft Google Sheet file is available here:
First Draft Google Sheet File
Version 9 has also been 'recalibrated' to allow an input parameter range from 0 to 100,000 that should be more interesting. The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. The real number is manipulated by Javascrâ€¦
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The SeT function offers a way to generate very large string sequences (representing large numbers). It is the third version of my substitution functions and it has a growth rate \(\approx f_{LVO}(n)\).
The SeT function is based on my other substitution function called The S Function (Version 2). Refer to my other blogs for more information on my work.
The SeT 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. In the SeT function, the \(S()\) function behaves the same as it does in The S Function (Version 2), but the \(T()\) function has been extended. The new substitution function is called S and extâ€¦
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This blog is no longer relevant to my work on the S Function and Alpha Function. Please keep this in mind if you want to review this work.
The S Function grows quickly. Comparing \(T()\) functions, that are used recursively in the S Function definition, directly to ordinals should make it easier to compare the growth rate of the S Function.
Using the Omega Count function of counting the number of \(\omega\) additions that are required to reach larger ordinals is one way of comparing \(T()\) functions to ordinals.
We can start with this comparison:
Let \(n=3\) and diagonalise over \(n\)
\(\Omega(n) = \omega == T(0) = n\)
\(\Omega(n.2) = \omega.2 == S(T(0),1,1) = n.2\)
\(\Omega(n^n) = \omega^{\omega} == S(S(T(0),2,1),0,T(0)) = n.2^n + n = n^n\) whâ€¦
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This blog is no longer relevant to my work on the S Function and Alpha Function. Please keep this in mind if you want to review this work.
I think it is useful to keep track of the number of \(\omega\) additions are required to reach higher ordinals. Here is an example of what I mean:
Let \(n = 2\) and diagonalise over \(n\) then:
\(\omega + \omega = \omega.2 = \omega^2 = \epsilon_0 = \varphi(1,0)\)
Now lets define an Omega Count function that keeps track of this count:
\(\Omega(2) = \omega^2 = \epsilon_0 = \varphi(1,0)\)
\(\Omega(2^2) = \Omega(4) = \epsilon_0^{\epsilon_0} = \epsilon_1 = \varphi(1,1)\)
\(\Omega(4^4) = \epsilon_1^{\epsilon_1} = \epsilon_2 = \varphi(1,2) = \varphi(1,\omega)\)
Here are the Omega Counts for various ordinals and varioâ€¦
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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 refâ€¦
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