The Alpha Function

The Alpha Function uses Version 9 code and my S Function (substitution function) to generate every finite integer up to a very large number. The Alpha Function has a growth rate of \(f_{\varphi(1,1,0)}(n)\) for any n.

The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. The real number is manipulated to create a finite sequence of finite integers. The sequences can be translated to represent every unique finite integer.

Refer to the Granularity Examples in my Version 9 blog that illustrate how finely tuned the functions are to access any and every finite integer.

What is the Alpha Function

My motivation to create this function was to develop a finely grained number notation system for really big numbers. \(\alpha(2)\) for example can be used to reference the number 6. Therefore 2 is the Alpha Index for the number 6. Alpha needs to reference big numbers very quickly to be useful, therefore it uses The S Function for this purpose. Alpha should also be monotonically increasing and every input real \(a > b\), results in a larger output number, where \(\alpha(a) >= \alpha(b)\) in all cases. The function is finely grained. It accepts a real number input and depending upon the precision of the real number, it can locate any and every big number.

High Level Description

The Alpha Function is 'calibrated' to accept real number inputs up to 100,000 and generate unique S() function outputs representing any and every big number.

Refer to my Version 9 blog for more information.

Some Calculations

Refer to my Version 9 blog blog for more examples:

\(\alpha(0.00) = S(0,0,0) = 0\)

\(\alpha(1.00) = S(2,1,1) = 4\)

\(\alpha(2.00) = S(S(2,1,1),0,2) = S(4,0,2) = 6\)

\(\alpha(3.00) = S(S(2,1,1),0,S(2,0,1)) = S(4,0,3) = 7\)

\(\alpha(4.00) = S(S(S(2,T(0),1),1,1),0,1) = S(S(8,1,1),0,1) = S(16,0,1) = 17\)

\(\alpha(4.29) = S(S(2,T(0),1),2,1) = f_2(f_{\omega}(2)) = f_2(8) = 2048\)

\(\alpha(4.332) = S(S(S(2,T(0),1),2,1),1,9) = f_1^9(2048) = 2048.2^9 = 1048576 >\) One Million

\(\alpha(4.33681975) = S(S(8,2,1),1,S(S(8,1,5),0,S(S(8,1,3),0,2)))\)

\(= S(2048,1,S(8,1,5) + S(64,0,2)) = 2048.2^{256 + 66} = 17.10^{99} >\) Googol

\(\alpha(4.35) = S(S(2,T(0),1),2,3) = S(8,2,3) = f_2^3(f_{\omega}(2)) = f_2^3(8) > f_2^3(6) >\) Googolplex

\(\alpha(4.39) = S(S(2,T(0),1),3,1) = S(8,3,1)\)

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

\(\alpha(6.2596315) = S(S(2,S(T(0),0,1),1),T(0),S(S(8,1,2),0,S(16,0,S(8,0,7))))\)

\(= S(f_{\omega}(8),T(0),63) = f_{\omega}^{63}(f_{\omega}(8)) = f_{\omega}^{64}(8) > G\) is Graham's number

\(\alpha(13.9) = S(2,S(T(1),0,1),1) = S(2,T(1),2) = f_{\varphi(\omega,0)}^2(2) > f_{\varphi(\omega,0)}(3)\)

\(\alpha(100) = S(S(2,S(S(T(T(0)),S(S(T(1),1,1),0,T(1)),1),0,S(S(T(1),S(T(0),0,1),T(0)),T(0),S(S(T(1),T(0),1),1,1))),1),S(T(T(0)),S(2,S(T(1),1,S(S(T(0),1,1),0,T(0))),1),1),1)\)

\(> S(2,T(T(0)),2) > f_{\varphi(1,0,0)}(3)\)

In general:

\(S(2,b+1,1) = S(2,b,2) = S(S(2,b,1),b,1) > S(n,b,1)\) for any reasonably sized number \(n\)

Here are more comparisons:

\(\alpha(1306.35) = S(2,S(T(T(T(0))),0,1),1) > S(n,T^{3}(0),1) > f_{\varphi(1,1,0)}(3)\)

\(\alpha(9058.76) = S(2,S(T(T(T(T(0)))),0,1),1) > S(n,T^{4}(0),1) > f_{\varphi(1,1,0)}(4)\)

\(\alpha(100000) = S(2,T^{\omega}(0),1) = \omega\) equals infinity by definition

Further References

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

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