User blog:B1mb0w/Program Code Version 4

Alpha Function
The Alpha Function has been defined using program code shown below.

This VBA visual basic code has been significantly simplified and replaces Version 3 and will be much easier to explain and to understand.

Program Code Version 4
This VBA visual basic code will run as a macro in Microsoft Excel. This function creates a string literal of a \(J_8\) Function equal to the Alpha function with any Real number input. The program does not attempt to evaluate the function and the run time is therefore very fast.

WORK IN PROGRESS

How the Function Works

A description of how the code works will be provided here ... Work in Progress.


 * VBA Constants
 * VBA Data Structures
 * VBA Functions
 * Alpha Function
 * Work In Progress

Test Bed for Version 4
Below is the test bed for version 4.

Various Ordinals

\(\alpha(100) = \varphi(1,0)\)

\(\alpha(110) = (\varphi^{2}(2,2)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2)^{4}.6 + 14}.((\omega\uparrow\uparrow 4)^{4}.2 + (\omega\uparrow\uparrow 2)^{5}.6) + 2\)

\(\alpha(120) = \varphi^{2}(3,(\omega\uparrow\uparrow 4)^{9}.(\omega^{4}.6 + \omega.4 + 4) + 3)^{(\omega\uparrow\uparrow 6)^{3}.(\omega^{3}.5 + \omega^{2}.2 + 4) + (\omega\uparrow\uparrow 3)^{6}.((\omega\uparrow\uparrow 2)^{3}.2 + \omega.2 + 1) + 5}.((\omega\uparrow\uparrow 5).((\omega\uparrow\uparrow 2)^{11}.9 + (\omega\uparrow\uparrow 2)^{9}.(\omega^{3}.5 + 5) + 4) + 8) + \varphi(3,(\varphi^{3}(2,(\omega\uparrow\uparrow 4)^{2} + (\omega\uparrow\uparrow 3)^{11})\uparrow\uparrow 2)^{8}.((\omega\uparrow\uparrow 5)^{6}.((\omega\uparrow\uparrow 2)^{10}.(\omega^{2}.4) + \omega^{6} + \omega^{5}.6 + \omega^{4}.2 + \omega^{2}.4 + \omega)))\)

\(\alpha(130) = (\varphi^{2}(4,21)\uparrow\uparrow 7).5 + (\omega\uparrow\uparrow 16)^{4}.((\omega\uparrow\uparrow 13)^{3}.2 + 7) + 1\)

\(\alpha(140) = (\varphi(5,(\omega\uparrow\uparrow 10)^{2}.((\omega\uparrow\uparrow 4)^{3}.(\omega^{2}.2 + \omega.6 + 2) + \omega^{2}.3 + 7) + (\omega\uparrow\uparrow 6)^{6}.2 + 4)\uparrow\uparrow 4)^{3}.(\varphi^{5}(2,\varphi^{4}(1,0)^{2}.((\varphi^{3}(1,(\omega\uparrow\uparrow 3)^{3} + (\omega\uparrow\uparrow 2)^{5}.(\omega^{5} + 3) + \omega.2 + 6)\uparrow\uparrow 2)^{4}.3 + \varphi^{3}(1,3)^{(\omega\uparrow\uparrow 10)^{4}.((\omega\uparrow\uparrow 5)^{4}.((\omega\uparrow\uparrow 4)^{3}.(\omega^{6}.3 + 1) + 4) + \omega.6 + 4) + (\omega\uparrow\uparrow 9)^{3}.((\omega\uparrow\uparrow 7) + (\omega\uparrow\uparrow 5)^{9}.(\omega^{6}.4 + 3) + (\omega\uparrow\uparrow 5)^{4})})))\)

\(\alpha(150) = (\varphi^{10}(5,(\omega\uparrow\uparrow 6)^{4}.((\omega\uparrow\uparrow 3)^{8}.((\omega\uparrow\uparrow 2)^{3}.(\omega^{5}.4 + \omega.3) + (\omega\uparrow\uparrow 2).(\omega^{5}.4 + \omega.2 + 3) + \omega^{12}.4 + \omega^{6}.5) + (\omega\uparrow\uparrow 3)^{6} + \omega^{5} + \omega^{2}.5 + \omega.2 + 5) + 4)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 6)^{2}.((\omega\uparrow\uparrow 3)^{5}.(\omega^{2}.5 + \omega.10 + 8) + (\omega\uparrow\uparrow 3)^{3}.((\omega\uparrow\uparrow 2)^{4}.5 + (\omega\uparrow\uparrow 2)^{2}.6 + 3) + \omega^{5} + 1) + (\omega\uparrow\uparrow 5)^{12}.10 + (\omega\uparrow\uparrow 5)^{5}.2 + (\omega\uparrow\uparrow 3)^{4}.(\omega)}\)

\(\alpha(200) = (\varphi^{3}((\omega\uparrow\uparrow 2)^{4},(\omega\uparrow\uparrow 6).((\omega\uparrow\uparrow 5)^{11}.5 + 5) + 5)\uparrow\uparrow 5)^{5}.(\varphi^{3}(\omega^{6}.3 + \omega^{4}.6 + \omega^{2}.5 + \omega.4,\varphi^{5}(\omega.6 + 2,(\varphi^{4}(\omega.4 + 5,(\omega\uparrow\uparrow 3)^{11}.10 + 4)\uparrow\uparrow 3)^{(\omega\uparrow\uparrow 2).(\omega^{4} + 4) + \omega^{3}.3}.5 + (\varphi^{2}(6,(\varphi^{6}(1,(\omega\uparrow\uparrow 6)^{2}.((\omega\uparrow\uparrow 5)^{4}.4 + 1) + (\omega\uparrow\uparrow 5)^{2}.2 + (\omega\uparrow\uparrow 3)^{4}.(\omega^{4}.6 + \omega^{2}.4 + 3) + (\omega\uparrow\uparrow 2)^{6}.2 + 4)\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 2)^{6}.(\omega^{4}.2 + 5) + \omega.5 + 4}.4 + 5)\uparrow\uparrow 9)^{6}.(\varphi^{3}(1,(\omega\uparrow\uparrow 2)^{6}.4 + \omega)))))\)

\(\alpha(300) = \varphi^{4}((\omega\uparrow\uparrow 9)^{4}.((\omega\uparrow\uparrow 2)^{6}.6) + (\omega\uparrow\uparrow 3)^{6}.(\omega^{2}.2 + \omega + 3) + 4,(\varphi^{2}((\omega\uparrow\uparrow 9)^{3}.3 + (\omega\uparrow\uparrow 5)^{4}.((\omega\uparrow\uparrow 2)^{6}.(\omega^{3}.7 + 7) + (\omega\uparrow\uparrow 2)^{2} + \omega^{5}.6 + \omega^{4}.7 + \omega.6) + (\omega\uparrow\uparrow 2)^{3}.12 + 8,(\omega\uparrow\uparrow 4)^{5}.((\omega\uparrow\uparrow 3).7 + 2) + \omega^{5}.3 + \omega.5 + 2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3)^{6}.2 + (\omega\uparrow\uparrow 3)^{5} + (\omega\uparrow\uparrow 3)^{2}})\)

\(\alpha(400) = (\varphi^{7}(3,0)\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 6).((\omega\uparrow\uparrow 5)^{11}.5 + 5) + 5}.((\omega\uparrow\uparrow 9).((\omega\uparrow\uparrow 2).(\omega^{6}.4 + \omega^{5}.21) + \omega.3 + 3) + 2) + (\varphi^{5}(16,1)\uparrow\uparrow 11)^{6}.(\omega^{4}.4 + \omega.6) + (\omega\uparrow\uparrow 5)^{9}.((\omega\uparrow\uparrow 2)^{13}.(\omega^{11}.4 + \omega.12 + 4) + (\omega\uparrow\uparrow 2)^{3}.4) + 5\)

\(\alpha(500) = (\varphi^{4}(6,(\varphi^{11}(5,4)\uparrow\uparrow 16)^{(\omega\uparrow\uparrow 5).3 + (\omega\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 3)^{12}.4 + (\omega\uparrow\uparrow 3)^{9}.((\omega\uparrow\uparrow 2)^{5}.5 + 2)) + 5}.((\omega\uparrow\uparrow 5)^{8}.12 + (\omega\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 3).(\omega^{2}.6 + \omega.5 + 4) + \omega^{6}.5 + \omega) + (\omega\uparrow\uparrow 2)^{3}.6 + 2) + (\omega\uparrow\uparrow 6)^{2}.4 + (\omega\uparrow\uparrow 3)^{3}.3 + (\omega\uparrow\uparrow 2)^{5}.21 + (\omega\uparrow\uparrow 2))\uparrow\uparrow 5).6 + \varphi^{3}(9,\varphi^{2}(5,\varphi(2,0)))\)

\(\alpha(900) = \varphi(\varphi((\omega\uparrow\uparrow 3)^{4}.4 + (\omega\uparrow\uparrow 3)^{3}.(\omega.3 + 5) + \omega^{2}.3 + 5,(\varphi^{7}(6,(\varphi^{2}(5,(\varphi^{7}(2,2)\uparrow\uparrow 6)^{6}.((\omega\uparrow\uparrow 2)^{6}.(\omega^{6}.2 + \omega^{5}.3 + \omega.4) + 3) + 5)\uparrow\uparrow 4).((\omega\uparrow\uparrow 4)^{6}.4 + (\omega\uparrow\uparrow 3)^{6}.4 + \omega + 4) + (\omega\uparrow\uparrow 4)^{11}.(\omega^{6}.6 + 1) + (\omega\uparrow\uparrow 4)^{5}.((\omega\uparrow\uparrow 2)^{5}.(\omega^{2}.9 + 1) + 3) + (\omega\uparrow\uparrow 2)^{5}.(\omega^{4}.3 + 1) + 3)\uparrow\uparrow 3)^{4}.((\omega\uparrow\uparrow 6)^{2}.(\omega^{5} + \omega^{2}.5 + \omega.5 + 2) + (\omega\uparrow\uparrow 2)^{4}.(\omega))),0)\)

or

\(\alpha(900) = \varphi(\varphi(\beta,\gamma),0)\) where

\(\beta = (\omega\uparrow\uparrow 3)^{4}.4 + (\omega\uparrow\uparrow 3)^{3}.(\omega.3 + 5) + \omega^{2}.3 + 5\)

\(\gamma = (\varphi^{7}(6,(\varphi^{2}(5,(\varphi^{7}(2,2)\uparrow\uparrow 6)^{6}.((\omega\uparrow\uparrow 2)^{6}.(\omega^{6}.2 + \omega^{5}.3 + \omega.4) + 3) + 5)\uparrow\uparrow 4).((\omega\uparrow\uparrow 4)^{6}.4 + (\omega\uparrow\uparrow 3)^{6}.4 + \omega + 4) + (\omega\uparrow\uparrow 4)^{11}.(\omega^{6}.6 + 1) + (\omega\uparrow\uparrow 4)^{5}.((\omega\uparrow\uparrow 2)^{5}.(\omega^{2}.9 + 1) + 3) + (\omega\uparrow\uparrow 2)^{5}.(\omega^{4}.3 + 1) + 3)\uparrow\uparrow 3)^{4}.((\omega\uparrow\uparrow 6)^{2}.(\omega^{5} + \omega^{2}.5 + \omega.5 + 2) + (\omega\uparrow\uparrow 2)^{4}.(\omega))\)

Various Root Ordinals

\(\alpha(100) = \varphi(1,0)\)

\(\alpha(110) = \varphi^{2}(2,2_*)\)

\(\alpha(120) = \varphi^{2}(3,(\omega\uparrow\uparrow 4)^{9}.(\omega^{4}.6 + \omega.4 + 4) + 3)\)

or

\(\alpha(120) = \varphi^{2}(3,\beta_*)\) where \(\beta = (\omega\uparrow\uparrow 4)^{9}.(\omega^{4}.6 + \omega.4 + 4) + 3\)

\(\alpha(130) = \varphi^{2}(4,21_*)\)

\(\alpha(140) = (\varphi(5,(\omega\uparrow\uparrow 10)^{2}.((\omega\uparrow\uparrow 4)^{3}.(\omega^{2}.2 + \omega.6 + 2) + \omega^{2}.3 + 7) + (\omega\uparrow\uparrow 6)^{6}.2 + 4)\uparrow\uparrow 4)^{3}.(\varphi^{5}(2,\varphi^{4}(1,0)^{2}.((\varphi^{3}(1,(\omega\uparrow\uparrow 3)^{3} + (\omega\uparrow\uparrow 2)^{5}.(\omega^{5} + 3) + \omega.2 + 6)\uparrow\uparrow 2)^{4}.3 + \varphi^{3}(1,3)^{(\omega\uparrow\uparrow 10)^{4}.((\omega\uparrow\uparrow 5)^{4}.((\omega\uparrow\uparrow 4)^{3}.(\omega^{6}.3 + 1) + 4) + \omega.6 + 4) + (\omega\uparrow\uparrow 9)^{3}.((\omega\uparrow\uparrow 7) + (\omega\uparrow\uparrow 5)^{9}.(\omega^{6}.4 + 3) + (\omega\uparrow\uparrow 5)^{4})})))\)

\(\alpha(150) = \varphi^{10}(5,(\omega\uparrow\uparrow 6)^{4}.((\omega\uparrow\uparrow 3)^{8}.((\omega\uparrow\uparrow 2)^{3}.(\omega^{5}.4 + \omega.3) + (\omega\uparrow\uparrow 2).(\omega^{5}.4 + \omega.2 + 3) + \omega^{12}.4 + \omega^{6}.5) + (\omega\uparrow\uparrow 3)^{6} + \omega^{5} + \omega^{2}.5 + \omega.2 + 5) + 4)\)

or

\(\alpha(150) = \varphi^{10}(5,(\omega\uparrow\uparrow 6)^{4}.(\beta.\gamma + \delta) + 4)_*)\) where

\(\beta = (\omega\uparrow\uparrow 3)^{8}\)

\(\gamma = (\omega\uparrow\uparrow 2)^{3}.(\omega^{5}.4 + \omega.3) + (\omega\uparrow\uparrow 2).(\omega^{5}.4 + \omega.2 + 3) + \omega^{12}.4 + \omega^{6}.5\)

\(\delta = (\omega\uparrow\uparrow 3)^{6} + \omega^{5} + \omega^{2}.5 + \omega.2 + 5\)

\(\alpha(200) = \varphi^{3}((\omega\uparrow\uparrow 2)^{4},(\omega\uparrow\uparrow 6).((\omega\uparrow\uparrow 5)^{11}.5 + 5) + 5)\)

or

\(\alpha(200) = \varphi^{3}(\beta,\gamma_*)\) where

\(\beta = (\omega\uparrow\uparrow 2)^{4}\)

\(\gamma = (\omega\uparrow\uparrow 6).((\omega\uparrow\uparrow 5)^{11}.5 + 5) + 5\)

\(\alpha(300) = \varphi^{4}((\omega\uparrow\uparrow 9)^{4}.((\omega\uparrow\uparrow 2)^{6}.6) + (\omega\uparrow\uparrow 3)^{6}.(\omega^{2}.2 + \omega + 3) + 4,(\varphi^{2}((\omega\uparrow\uparrow 9)^{3}.3 + (\omega\uparrow\uparrow 5)^{4}.((\omega\uparrow\uparrow 2)^{6}.(\omega^{3}.7 + 7) + (\omega\uparrow\uparrow 2)^{2} + \omega^{5}.6 + \omega^{4}.7 + \omega.6) + (\omega\uparrow\uparrow 2)^{3}.12 + 8,(\omega\uparrow\uparrow 4)^{5}.((\omega\uparrow\uparrow 3).7 + 2) + \omega^{5}.3 + \omega.5 + 2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 3)})\)

\(\alpha(400) = (\varphi^{7}(3,0)\uparrow\uparrow 4)^{(\omega\uparrow\uparrow 6).((\omega\uparrow\uparrow 5)^{11}.5 + 5) + 5}.((\omega\uparrow\uparrow 9).((\omega\uparrow\uparrow 2).(\omega^{6}.4 + \omega^{5}.21) + \omega.3 + 3) + 2) + (\varphi^{5}(16,1)\uparrow\uparrow 11)^{6}.(\omega^{4}.4 + \omega.6) + (\omega\uparrow\uparrow 5)^{9}\)

\(\alpha(500) = (\varphi^{4}(6,(\varphi^{11}(5,4)\uparrow\uparrow 16)^{(\omega\uparrow\uparrow 5).3 + (\omega\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 3)^{12}.4 + (\omega\uparrow\uparrow 3)^{9}.((\omega\uparrow\uparrow 2)^{5}.5 + 2)) + 5}.((\omega\uparrow\uparrow 5)^{8}.12 + (\omega\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 3).(\omega^{2}.6 + \omega.5 + 4) + \omega^{6}.5 + \omega) + (\omega\uparrow\uparrow 2)^{3}.6 + 2) + (\omega\uparrow\uparrow 6)^{2}.4 + (\omega\uparrow\uparrow 3)^{3}.3 + (\omega\uparrow\uparrow 2)^{5}.21 + (\omega\uparrow\uparrow 2))\uparrow\uparrow 5)\)