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Alpha Function Code - Version 6

The Alpha Function has been defined using Sequence Generator code shown below. A separate blog has been written to explain how to read Sequence Generator Code and how it works.


Changes in Version 6

Version 6 has been completely modified from Version 5 to align it completely to my my other work on Beta Function blogs.

The key change has been to use the power of the The Beta Function to access every Veblen ordinal and every FGH function and therefore every finite integer (up to the size of \(f_{SVO}(v)\) for a given base \(v\)).

The Alpha Function is a one input parameter version of the Beta Function that can access every finite integer up to \(f_{SVO}(v)\) for any n.

The Alpha Function has one parameter: \(\alpha(r)\) where r is any real number. The real number is manipulated by Sequence Generating Code (see below) to create a finite sequence of finite integers that represents a unique combination of Verben ordinals and FGH functions and therefore a unique finite integer (up to the size of \(f_{SVO}(v)\) for any n).

A high level mathematical description of the Alpha Function is:

\(\alpha(2) = f_{\omega}(2) = 8\)

\(\alpha(4) = \alpha(2\uparrow\uparrow 2) = f_{\varphi(1,0)}(2)\)

\(\alpha(2\uparrow\uparrow 3) = f_{SVO}(2)\)

\(\alpha(2\uparrow\uparrow 4) = f_{SVO}(3) + f_{SVO}(2)\)

and

\(\alpha(2\uparrow\uparrow n) = \sum^{n-1}_{i=2} f_{SVO}(i) > f_{SVO}(n-1)\)


Example of Changes in Version 6

Some examples of the above changes are as follows:

\(\alpha(10) = \beta(5.9,2) = f_{\varphi(\omega,\omega)}(2)\)

\(\alpha(100) = \beta(36,3) + f_{SVO}(2) = f_{\varphi(2,(\omega\uparrow\uparrow 2)^{\omega^2.2 + \omega.2 + 1},0)}(3) + f_{SVO}(2)\)

\(\alpha(1000) = \beta(59,3) + f_{SVO}(2) = f_{\varphi((\omega\uparrow\uparrow 2)^{\omega.2},0,0)}(3) + f_{SVO}(2)\)

WORK IN PROGRESS


Version 6 Code

The code for the Alpha Function sits on top of the code created for the Beta Function. Refer to my Version 4 blog for a complete description of that code. In this code, the reference to \(h_x\) is a reference to the \(h\) sequence type defined in Beta Function Version 4 code.

a_x = (C_v,V_v=C_v+2,h_x,r_0,[[h_x][r_0]])

r_x = (V_v>(0:,1:,2:,(V_v=V_v-1,r_x+1,[ + f_{SVO}([V_v])[r_x+1]])))


Valid Sequence Counts

WORK IN PROGRESS


Test Bed for Version 6

Below is the test bed and various results using version 6.

WORK IN PROGRESS

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