User blog:B1mb0w/Alpha 1 Ruler Function

\(\alpha_1\) Ruler Function
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.

Ruler Functions
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

\(\alpha_1(100) = \alpha(5.79955) = f_{\omega + 1}(64) >> g_{64} = G\) is Graham's number

\(\alpha_2\) Ruler Function

\(\alpha_2(100) = \alpha(\) TBA \ = f_3(4) >>\) Googolplex

\(\alpha_3\) Ruler Function

\(\alpha_3(100) = \alpha(4.9140375) = f_1^{322}(f_{\omega + 1}(2)) >>\) Googol

Examples of \(\alpha_1\) Ruler Function
The function will be 'calibrated' to provide an 'interesting' range of large numbers up to Graham's number:

\(\alpha_0(0) = 0\)

\(\alpha_0(1) = 1\)

\(\alpha_0(2) = 2\)

\(\alpha_0(3) = S(2,T(0),1) = 8\)

\(\alpha_0(4) = S(2,T(0),1) + 2 = 8 + 2\)

\(\alpha_0(5) = S(2,T(0),1) + S(2,1,1) = 8 + 4\)

\(\alpha_0(6) = S(S(2,T(0),1),1,1) = 16\)

\(\alpha_0(7) = S(S(2,T(0),1),1,1) + 2 = 16 + 2\)

\(\alpha_0(8) = S(S(2,T(0),1),1,2) = 32\)

\(\alpha_0(9) = S(S(2,T(0),1),1,S(2,1,1)) = S(8,1,4)\)

\(\alpha_0(10) = S(2,T(0) + 1,1) = 2048\)

\(\alpha_0(11) = S(2,T(0) + 1,1) + 3 = 2048 + 3\)

\(\alpha_0(12) = S(S(2,T(0) + 1,1),1,1) + 1 = S(2048,1,1) + 1\)

\(\alpha_0(13) = S(S(2,T(0) + 1,1),1,S(2,T(0),1) + 1) + S(S(2,T(0),1),1,1) + 1\)

\(= S(2048,1,8 + 1) + 16 + 1\)

\(\alpha_0(14) = S(S(2,T(0) + 1,1),T(0),1) + S(S(2,T(0),1),1,1) = S(2048,T(0),1) + 16\)

\(\alpha_0(15) = S(S(2,T(0) + 1,1),T(0),2) = S(2048,T(0),2)\)

\(\alpha_0(16) = S(S(S(2,T(0) + 1,1),T(0),S(2,T(0),1) + 1),1,S(2,1,1)) + S(2,1,1) + 2\)

\(= S(S(2048,T(0),8 + 1),1,4) + 4 + 2\)

\(\alpha_0(17) = S(2,S(T(0),1,1),1) + 1\)

\(\alpha_0(18) = S(S(2,S(T(0),1,1),1),1,S(2,1,1)) + 1 = S(S(2,S(T(0),1,1),1),1,4) + 1\)

\(\alpha_0(20) = S(2,S(T(0),1,1) + 1,1)\)

\(\alpha_0(21) = S(S(2,S(T(0),1,1) + 1,1),1,1) + 2\)

\(\alpha_0(23) = S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),S(2,1,1) + 1) + 2\)

\(= S(S(2,S(T(0),1,1) + 1,1),S(T(0),1,1),4 + 1) + 2\)

\(\alpha_0(25) = S(S(2,S(T(0),1,1) + T(0),1),T(0),1) + 2\)

\(\alpha_0(30) = S(2,T(1),1)\)

\(\alpha_0(35) = S(S(2,T(1) + 1,1),T(1),1) + S(2,1,1) = S(S(2,T(1) + 1,1),T(1),1) + 4\)

\(\alpha_0(55) = S(S(2,S(T(1),1,T(0)) + 1,1),T(0),1)\)

\(\alpha_0(65) = S(S(2,S(T(1),T(0),1) + T(1) + 1,1),1,3)\)

\(\alpha_0(70) = S(2,S(T(1),T(0),S(T(0),1,1)),1) + 1\)

\(\alpha_0(90) = S(S(2,S(S(S(T(1),S(T(0),1,1) + 1,S(T(0),1,1) + T(0)),T(0) + 1,S(T(0),1,1)),T(0),1) + 1,1),1,1) + 1\)

\(\alpha_0(99.5) = S(2,T(T(0)),1) + 2\)

\(\alpha_0(99.6) = S(S(2,T(T(0)),1),1,1) + 2\)

\(\alpha_0(99.92) = S(S(2,T(T(0)),1),S(T(1),T(0),1) + S(T(1),1,S(T(0),1,1)) + S(T(1),1,T(0) + 1) + T(1),2) + 3\)

\(\alpha_0(99.94) = S(S(2,T(T(0)),1),S(S(T(1),T(0) + 1,1),T(0),T(0)),1) + 1\)

\(\alpha_0(99.97) = S(S(S(2,T(T(0)),1),S(S(T(1),S(T(0),1,1) + T(0),1),1,1) + 1,1),1,1) + 2\)

\(\alpha_0(99.98) = S(S(2,T(T(0)),1),S(S(S(T(1),S(T(0),1,1) + T(0) + 1,1),T(0),1),1,T(0)) + S(T(0),1,1),2)\)

\(\alpha_0(100) = S(2,T(T(0)) + 1,1)\)

Comments and Questions
Look forward to comments and questions.

Cheers B1mb0w.