User blog:B1mb0w/The Alpha Function

The Alpha Function
The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. It is derived from the The J Function and in particular the Sandpit \(J_8\) function.

This blog replaces two other previous attempts at this function. Links to the original blogs are available in the References section at the end of this blog. However, they will provide links back to here, and this blog will not use any of the material from those older attempts.

What is the Alpha Function
Edited: Late Feb 2016 to align to changes to my blog on Fundamental Sequences.

My motivation to create this function was to develop a finely grained number notation system for really big numbers. \(\alpha(1)\) for example can be used to reference the number 0. Therefore 1 is the Alpha Index for the number 0. Alpha needs to reference big numbers very quickly to be useful, therefore it uses the J Function for this purpose. Alpha should also be strictly hierarchical and every number \(a > b\), must reference larger numbers, so that \(\alpha(a) >> \alpha(b)\) in all cases. The function is finely grained. It accepts a real number input and offers some finesse to locate really big numbers.

The Alpha Function has a growth rate of up to the Small Veblen Ordinal (SVO).

Some Calculations
Refer to Sandpit \(J_8\) blog for all definitions and explanations:

\(\alpha(1.00) = J_8(<0,0>,0,0) = f_0^0(0) = 0\)

\(\alpha(1.05) = J_8(<0,0>,0,1) = f_0^0(1) = 1\)

\(\alpha(1.20) = J_8(<0,0>,2,2) = f_0^2(2) = 4\)

\(\alpha(1.46) = J_8(<0,1>,3,3) = f_1^3(3) = 24\)

\(\alpha(1.94) = J_8(<0,2>,2,3) = f_2^2(3) >>\) 400 Million

\(\alpha(1.97) = J_8(<0,2>,2,6) = f_2^2(6) >>\) Googol

\(\alpha(2.13) = J_8(<0,2>,4,4) = f_2^4(4) = f_3(4) >>\) Googolplex

\(\alpha(e) = \alpha(2.71828182845905) = J_8(<0,3>,2,5) = f_{3}^{2}(5)\)

\(\alpha(2.82) = J_8(<0,3>,3,4) = f_3^3(4) >> g_1\) where \(g_{64} = G\) is Graham's number

\(\alpha(\pi) = \alpha(3.14159265358979) = J_8(<0,3>,5,7) = f_{3}^{5}(7)\)

\(\alpha(5.00) = J_8(<0,4>,9,14) = f_{4}^{9}(14)\)

\(\alpha(10.0) = J_8(<1,1,<0,1>,<0,1>,<0,0>>,2,2) = f_{\omega}^2(2) = f_{\omega}(8)\)

\(\alpha(10.0096) = J_8(<1,1,<0,1>,<0,1>,<0,1>>,2,2) = f_{\omega+1}^2(2) = f_{\epsilon_0}(2)\)

\(>> g_{64} = G\) is Graham's number

\(\alpha(16.434006) = J_8(<1,2,<1,1,<0,1>,<0,2>,<0,0>>,<0,1>,<0,0>>,2,3)\)

\(= f_{(\omega\uparrow\uparrow 2)^{\omega.2}}^{2}(3) = f_{\omega^{\omega.\omega.2}}^{2}(3) = f_{\omega^{\omega^2.2}}^2(3)\)

\(\alpha(100.78626719) = J_8(<2,<0,0,<0,1>,<0,0>,0>,0,<0,0>,<0,0>,<0,0>>,2,3)\)

\(= f_{\varphi(2,0)}^2(3) = f_{\zeta_0}^2(3)\) More examples of how to calculate Alpha numbers are available here.

\(\alpha(1000)\)

\(= J_8(<3,<0,0,<0,0>,<0,0>,<0,0>,0>,0,<0,0>,<0,0>,<0,0>>,2,4)\)

\(= f_{\varphi(1,0,0)}^2(4) = f_{\Gamma_0}^2(4) >> tree(3)\) i.e. the weak tree function

\(\alpha(Googol) = f_{\varphi(1,0_{[99]})}^2(100)\)

tree and TREE Functions
The Alpha function grows fast enough to reach the weak tree function but not the TREE function which grows faster than SVO.

\(\alpha(1000) = f_{\varphi(1,0,0)}^2(4) = f_{\Gamma_0}^2(4) >> tree(3)\)

The weak tree function grows at the rate of SVO but a lower bound for tree(3) is stated as \(2^{18}-4\) which is much much smaller. Therefore

\(\alpha(1000) >> tree(3) >> 2^{18}-4 >> 2^{17} >> 2^4.2^{13} >> 13.2^{13} >> f_2(13)\)

and

\(tree(3) >> f_2(13) = f_1^{13}(13) = J_8(<0,1>,13,13) = \alpha(1.905)\)

Appreciate some help here with better lower and upper bounds for tree(3).

WORK IN PROGRESS

Granularity Examples of this Function
The Alpha Function uses the full depth of the Real Numbers to enable almost every ordinal and big number to be described. In these examples, the highest ordinal rises from \(\omega^{\omega.2}\) to \(\omega^{\omega.5}\)

\(\alpha(15) = f_{(\omega\uparrow\uparrow 2)^{2}.(\omega^{2}.6 + \omega.2 + 1) + (\omega\uparrow\uparrow 2).6 + 2}^{3}(10)\)

\(\alpha(16) = f_{(\omega\uparrow\uparrow 2)^{4}}^{4}(8)\)

\(\alpha(17) = f_{(\omega\uparrow\uparrow 2)^{5}.5 + 1}^{3}(9)\)

The ordinals quickly scale the Veblen Hierarchy for Alpha input values above 100.

Alpha(115)

\(\alpha(115) = f_{(\varphi^{5}(3,\varphi^{2}(4,(\omega\uparrow\uparrow 5)^{6}.((\omega\uparrow\uparrow 3)^{5}.4 + (\omega\uparrow\uparrow 2)^{3}.5 + \omega^{5}.2 + 8)_*) + (\omega\uparrow\uparrow 12)^{4}.((\omega\uparrow\uparrow 8).((\omega\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 3)^{11}.(\omega.2 + 2) + 1) + (\omega\uparrow\uparrow 4)^{2}.((\omega\uparrow\uparrow 2).7 + \omega^{2}.5) + 2) + (\omega\uparrow\uparrow 4)^{6}.(\omega^{11}.5 + \omega^{8}.7 + 9) + (\omega\uparrow\uparrow 3)^{7}.3 + 9) + 5_*)^{(\omega\uparrow\uparrow 6)^{11}.((\omega\uparrow\uparrow 4)^{3}.7 + 2) + 2}.(\varphi((\omega\uparrow\uparrow 6)^{5} + (\omega\uparrow\uparrow 2)^{7}.(\omega^{6}.2 + \omega^{3}.4 + \omega),0))}^2(12)\)

Where the ordinal is:

\((\varphi^{5}(3,\varphi^{2}(4,(\omega\uparrow\uparrow 5)^{6}.((\omega\uparrow\uparrow 3)^{5}.4 + (\omega\uparrow\uparrow 2)^{3}.5 + \omega^{5}.2 + 8)_*) + (\omega\uparrow\uparrow 12)^{4}.((\omega\uparrow\uparrow 8).((\omega\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 3)^{11}.(\omega.2 + 2) + 1) + (\omega\uparrow\uparrow 4)^{2}.((\omega\uparrow\uparrow 2).7 + \omega^{2}.5) + 2) + (\omega\uparrow\uparrow 4)^{6}.(\omega^{11}.5 + \omega^{8}.7 + 9) + (\omega\uparrow\uparrow 3)^{7}.3 + 9) + 5_*)^{(\omega\uparrow\uparrow 6)^{11}.((\omega\uparrow\uparrow 4)^{3}.7 + 2) + 2}.(\varphi((\omega\uparrow\uparrow 6)^{5} + (\omega\uparrow\uparrow 2)^{7}.(\omega^{6}.2 + \omega^{3}.4 + \omega),0))\)

And the root ordinal is:

\(\varphi^{2}(4,(\omega\uparrow\uparrow 5)^{6}.((\omega\uparrow\uparrow 3)^{5}.4 + (\omega\uparrow\uparrow 2)^{3}.5 + \omega^{5}.2 + 8)_*)\)

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

Alpha(116)

\(\alpha(116) = f_{(\varphi^{2}(5,(\omega\uparrow\uparrow 9)^{2}.3_*)\uparrow\uparrow 8)^{2}.((\omega\uparrow\uparrow 2)^{2}.11) + 14}^4(21)\)

Where the root ordinal is:

\(\varphi^{2}(5,(\omega\uparrow\uparrow 9)^{2}.3_*) = \varphi(5,\varphi(5,(\omega\uparrow\uparrow 9)^{2}.3))\)

Alpha(117)

\(\alpha(117) = f_{\varphi^{6}(1,\varphi^{2}(7,4_*) + (\varphi^{2}(6,(\omega\uparrow\uparrow 3)^{2}.(\omega^{4}.6 + \omega^{3}.9 + 3) + \omega^{5}.5 + \omega^{2}.3 + 3_*)\uparrow\uparrow 6)^{3}.((\varphi((\omega\uparrow\uparrow 7)^{5}.((\omega\uparrow\uparrow 4)^{3}.(\omega^{15}.11 + 6) + 7) + (\omega\uparrow\uparrow 5).((\omega\uparrow\uparrow 4)^{5}.5 + 4) + (\omega\uparrow\uparrow 3)^{9} + \omega.6 + 1,(\omega\uparrow\uparrow 4)^{2}.((\omega\uparrow\uparrow 3).((\omega\uparrow\uparrow 2).3 + \omega^{6} + 2) + \omega^{9}.5 + 2) + \omega^{4}.10 + \omega^{3}.5 + 5)\uparrow\uparrow 4)^{4}.((\omega\uparrow\uparrow 4)^{4}.(\omega)))_*)}^2(15)\)

Where the root ordinal is:

\(\varphi^{2}(7,4_*) = \varphi(7,\varphi(7,4))\)

Program Code and Description
Program code using Microsoft VBA and descriptions of the logic is available on other blogs. See the references section.

I'd like to write a javascript version of the program and make if available either on wikia or another site for live interaction with user entered real numbers. Does anybody know the simplest tools I can use to do this ?

Comments and Questions
Look forward to comments and questions. I am learning heaps by writing these blogs and correcting all the mistakes the community finds in them !

Cheers B1mb0w.