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Alpha Function Code Version 9

This version of The Alpha Function has been re-written 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 Code (Javascript)

Version 9 has been completely re-written 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 're-calibrated' 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 Javascript Code to create a finite sequence of finite integers that represents a unique combination of S and T functions which can be translated into unique finite integers.

The Alpha Function translates unique real numbers into any and every finite integer.


Version 9 Examples

Links to the Google Sheet file will added here shortly. Each combination uniquely belongs to an ascending order of all sequences. Therefore each sequence can be assigned a finite ordinal value.

The following examples (from Version 8) will be updated as soon as the Google Sheet has been linked to this blog.

\(\alpha(0.0) = 0\)

\(\alpha(0.2) = 2\)

\(\alpha(1.0) = S(2,1,1) = 4\)

\(\alpha(2.0) = S(S(2,1,1),0,2) = 6\)

\(\alpha(3.0) = S(S(2,1,1),0,S(2,0,1)) = 7\)

\(\alpha(3.60) = S(2,T(0),1) = 8\)

\(\alpha(3.95) = S(S(2,T(0),1),1,1) = 16\)

\(\alpha(4.09) = S(S(2,T(0),1),1,2) = 24\)

The growth rate can be seen to accelerate when we start introducing more complex T functions:

\(\alpha(4.75) = S(2,S(T(0),0,1),1) = f_{\omega+1}(2) = f_{\omega}(8)\)

\(\alpha(6.42) = S(2,S(T(0),1,1),1) = f_{\omega.2}(2) = f_{\epsilon_0}(2)\)

\(\alpha(12.68) = S(2,T(1),1)\)

\(\alpha(28.72) = S(2,S(T(1),T(0),1),1)\)

\(\alpha(78.75) = S(2,T(T(0)),1)\)

\(\alpha(275.0) = S(2,T(T(1)),1)\)

\(\alpha(1305.25) = S(2,T(T(T(0))),1)\)


Granularity Examples

These examples illustrate the fine detail in real numbers that can be used to access large numbers via the Alpha Function:

\(\alpha(10.60) = S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1)\)

\(\alpha(11.00) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),1,2)\)

\(\alpha(11.30) = S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),5,S(S(2,S(T(0),1,1),1),T(0),S(S(S(8,2,1),1,S(8,0,4)),0,3))),1,1)\)

\(\alpha(11.45) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(S(S(S(2,S(S(T(0),1,1),0,1),1),S(S(2,S(T(0),1,1),1),S(T(0),0,S(16,0,1)),1),1),S(S(S(S(S(2,S(T(0),1,1),1),S(T(0),0,6),S(8,0,1)),T(0),5),1,S(S(8,5,2),0,7)),0,1),1),S(S(2,S(T(0),0,1),1),T(0),2),1),1)\)

\(\alpha(11.48) = S(S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(S(2,S(T(0),0,1),1),0,1),1),0,3),S(S(2,S(S(T(0),1,1),0,T(0)),1),0,3)),1,1),0,S(S(S(2,S(S(T(0),1,1),0,1),1),S(T(0),1,1),S(S(S(2,S(T(0),0,1),1),1,1),0,S(S(8,1,7),0,6))),T(0),1))\)

\(\alpha(11.49) = S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(T(0),0,2),1),1,S(S(2,S(S(T(0),1,1),0,T(0)),1),2,1)),0,7),1),0,5)\)

\(\alpha(11.50) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(S(S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(S(T(0),1,1),0,1),16),4,S(S(S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(8,0,2),S(S(2,S(S(T(0),1,1),0,1),1),1,6)),4,7),1,1),0,1)),3,1),1,2),1)\)

\(\alpha(11.51) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),T(0),1)\)

\(\alpha(11.52) = S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),T(0),1),0,1)\)

\(\alpha(11.55) = S(S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),T(0),1),1,2),0,S(S(2,S(S(T(0),1,1),0,T(0)),1),S(T(0),0,3),S(S(S(S(2,S(T(0),0,1),1),T(0),5),S(S(S(S(S(8,7,1),3,7),2,1),1,1),0,1),2),4,1)))\)

\(\alpha(11.70) = S(S(S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(T(0),0,1),1),1,1),0,1)\)

\(\alpha(12.00) = S(S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1),S(T(0),0,S(S(S(2,S(T(0),1,1),1),1,1),0,1)),S(S(S(S(2,S(S(T(0),1,1),0,T(0)),1),S(2,S(S(T(0),1,1),0,1),1),7),S(S(2,S(T(0),0,1),1),0,1),4),S(2,S(T(0),0,1),1),S(S(S(2,S(S(T(0),1,1),0,1),1),S(T(0),1,1),2),T(0),8)))\)

\(\alpha(12.7) = S(S(2,T(1),1),0,1)\)


Comparing Alpha Function Values

From the above examples, it is interesting to compare:

\(\alpha(8.9) = S(2,S(S(T(0),1,1),0,T(0)),1)\)

\(\alpha(10.6) = S(2,S(S(T(0),1,1),0,S(T(0),0,1)),1)\)

\(\alpha(11.00) = S(\alpha(10.6),1,2)\)

\(\alpha(11.30) = S(S(\alpha(10.6),5,S(S(2,S(T(0),1,1),1),T(0),S(S(S(8,2,1),1,S(8,0,4)),0,3))),1,1)\)

\(\alpha(11.45) = S(\alpha(10.6),S(S(S(S(2,S(S(T(0),1,1),0,1),1),S(S(2,S(T(0),1,1),1),S(T(0),0,S(16,0,1)),1),1),S(S(S(S(S(2,S(T(0),1,1),1),S(T(0),0,6),S(8,0,1)),T(0),5),1,S(S(8,5,2),0,7)),0,1),1),S(S(2,S(T(0),0,1),1),T(0),2),1),1)\)

\(\alpha(11.48) = S(S(S(\alpha(10.6),S(S(\alpha(8.9),S(S(2,S(T(0),0,1),1),0,1),1),0,3),S(\alpha(8.9),0,3)),1,1),0,S(S(S(2,S(S(T(0),1,1),0,1),1),S(T(0),1,1),S(S(S(2,S(T(0),0,1),1),1,1),0,S(S(8,1,7),0,6))),T(0),1))\)

\(\alpha(11.49) = S(S(\alpha(10.6),S(S(S(\alpha(8.9),S(T(0),0,2),1),1,S(\alpha(8.9),2,1)),0,7),1),0,5)\)

\(\alpha(11.50) = S(\alpha(10.6),S(S(S(S(\alpha(8.9),S(S(T(0),1,1),0,1),16),4,S(S(S(S(\alpha(8.9),S(8,0,2),S(S(2,S(S(T(0),1,1),0,1),1),1,6)),4,7),1,1),0,1)),3,1),1,2),1)\)

\(\alpha(11.51) = S(\alpha(10.6),T(0),1) = f_{\omega}(\alpha(10.6))\)

\(\alpha(11.52) = S(\alpha(11.51),0,1)\)

\(\alpha(11.55) = S(S(\alpha(11.51),1,2),0,S(\alpha(8.9),S(T(0),0,3),S(S(S(S(2,S(T(0),0,1),1),T(0),5),S(S(S(S(S(8,7,1),3,7),2,1),1,1),0,1),2),4,1)))\)

\(\alpha(11.7) = S(S(S(\alpha(10.6),S(T(0),0,1),1),1,1),0,1)\)

\(\alpha(12) = S(\alpha(10.6),S(T(0),0,S(S(S(2,S(T(0),1,1),1),1,1),0,1)),S(S(S(\alpha(8.9),S(2,S(S(T(0),1,1),0,1),1),7),S(S(2,S(T(0),0,1),1),0,1),4),S(2,S(T(0),0,1),1),S(S(S(2,S(S(T(0),1,1),0,1),1),S(T(0),1,1),2),T(0),8)))\)

\(\alpha(12.68) = S(2,T(1),1)\)


Growth Rate of the Alpha Function

The Alpha Function is now 're-calibrated' to accept real number inputs up to 100,000 at which point the Alpha Function will generate an S Function approaching:

\(\alpha(100,000) = S(2,T^{\omega}(0),1) = \omega\)

In other words, the Alpha Function has been hard-coded to asymptotically reach infinity when 100,000.


Further References

Further references to relevant blogs can be found here: User:B1mb0w

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