**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