User blog:B1mb0w/The Beta Function

The Beta Function
The Beta Function has one parameter: \(\beta(r,v)\) where r is any real number. It is derived from the The Alpha 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 are the Alpha and Beta Functions
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 my other blogs for the Sequence generating code for the Beta Function for all definitions and explanations:

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

WORK IN PROGRESS

\(>> g_{64} = G\) 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)\)

WORK IN PROGRESS

Granularity Examples of this Function
The Beta 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 \(\omega^{\omega.5}\)

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

WORK IN PROGRESS

Sequence Generating Code (Program Code)
The Beta Function uses Sequence Generating Code to create long finite integer strings to define large Veblen ordinals and FGH functions (up to the size of SVO). Refer to my other blogs on Unique Ordinal Representation and Version 1 Code for more information.

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.