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The Beta Function

The Beta Function is a re-designed version of the The Alpha Function which greatly expands the number of large ordinals that can be accessed using two simple input parameters.

The Beta Function has one additional parameter compared to The Alpha Function:

\(\beta(r,v)\) where

  • r is any real number and is the seed used to generate long sequences of Veblen ordinals and FGH functions
  • v is any finite integer and is the base used to calibrate the FGH functions initial input value.

The Beta Function is designed to access every Veblen ordinal between \(\varphi(1,0)\) and \(\varphi(1_{[\omega]}) =\) SVO Small Veblen Ordinal, and also, every finite number between any two FGH functions of the form: \(f_{\mu}^n(v)\) to \(f_{\mu}^{n+1}(v)\). In effect the Beta Function can be used to access and generate any number up to \(f_{SVO}(n)\) by using the depth of the real numbers to generate the relevant sequences.


What are the Alpha and Beta Functions

The Alpha and Beta Functions are notational and not computable functions. By this I mean they generate a literal string of Veblen ordinals and FGH functions equivalent to a very large finite number. They do not compute these numbers. It is a separate exercise to calculate the FGH function to calculate the large finite number.

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. Both the Alpha and Beta functions generate sequences of finite integers to represent unique Veblen ordinals and FGH functions equivalent to a unique large finite number. The functions are monotonically increasing and every input real \(a > b\), results in a larger output number, where \(\alpha(a) >= \alpha(b)\) 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 and Beta Functions have a growth rate of up to \(f_{SVO}(n)\).

The Beta Function

The Beta Function is designed to access every Veblen ordinal and every finite number in this range without any gaps (*). Refer to the Sequence Generating Code used by this function in the References section at the end of this blog. For more information about the code and logic used.

(*) Refer to my blog on Version 3 code for a clarification on this point.


Valid Sequence Counts \(C(n)\) Function

Separate to the Beta Function itself, I am interested in a function \(C(n)\) which represent all the valid sequences that can be generated by the Beta Function using integers of 0 to n only. Because the Beta Function can generate unique sequences for all finite integers up to \(f_{SVO}(n)\), it will have a growth rate of \(f_{SVO}(n)\).

This is interesting because while the Beta Function is 'notational', the Valid Sequence Count Function is 'computable'. A recursive program (but not primitive recursive) would be able to compute the number of valid sequences. I will investigate this further but appreciate any comments that may help clarify the status of this function.


Some Calculations

Refer to my other blogs for the Sequence generating code for the Beta Function for all definitions and explanations:

Base v = 2

\(\beta(0,2) = 0\)

\(\beta(1,2) = 2\)

\(\beta(2,2) = 7\)

\(\beta(e,2) = \beta(2.71828182845905,2) = f_{6}(f_{\omega}(2)).2 + 5\)

\(\beta(3,2) = f_{\omega + 1}(2).2 + 7\)

\(\beta(\pi,2) = \beta(3.14159265358979,2) = f_{3}^{6}(f_{\omega + 1}(2)).2\)

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

\(\beta(5,2) = f_{\varphi(1,\omega)^{\omega + 1}.(\omega) + \varphi(1,0)^{\omega} + \varphi(1,0)}(2) + 1\)

\(\beta(8,2) = f_{\varphi(1,0,0)}(2) = f_{\Gamma_0}(2) = f_{SVO}(2)\) by definition

Base v = 3

\(\beta(0,3) = 0\)

\(\beta(1,3) = 3\)

\(\beta(2,3) = f_{1}^{2}(3) + 6 = 12 + 6 = 18\)

\(\beta(e,2) = \beta(2.71828182845905,3) = f_{2}^{2}(3) + f_{2}(3).2 + 1\)

\(\beta(3,3) = f_{\omega}(3)\)

\(\beta(\pi,3) = \beta(3.14159265358979,3) = f_{\omega + 1}(3) + 1\)

\(\beta(4,3) = f_{1}^{f_{1}^{2}(3) + 9}(f_{\omega^2}^{2}(3)) + 2\)

\(\beta(5,3) = f_{2}^{f_{1}^{2}(3) + 2}(f_{4}^{f_{1}^{f_{2}(3).2 + 2}(f_{2}^{f_{1}^{2}(3)}(f_{\omega + 2}(3))) + 1}(f_{\omega^2.2 + \omega + 2}^{2}(3))).2 + f_{3}(f_{\omega}^{f_{\omega}^{2}(3) + 2}(f_{\omega + 1}(3)))\) corrected using Version 2 code

\(\beta(9,3) = f_{\varphi(1,0)}(3)\)

\(\beta(27,3) = f_{\varphi(1,0,0)}(3) = f_{\Gamma_0}(3)\)

\(\beta(81,3) = f_{\varphi(1,0,0,0)}(3) = f_{SVO}(3)\) by definition

Base v = 4

\(\beta(0,4) = 0\)

\(\beta(1,4) = 4\)

\(\beta(2,4) = f_{2}(4) = 64\)

\(\beta(e,4) = \beta(2.71828182845905,4) = f_{1}^{2}(f_{2}^{3}(4)) + f_{1}^{3}(f_{2}(4)) + f_{1}^{3}(4) + 2\)

\(\beta(3,4) = f_{1}^{4}(f_{3}(4)) + f_{1}^{2}(4) + 12\)

\(\beta(\pi,4) = \beta(3.14159265358979,4) = f_{1}^{3}(f_{2}^{f_{1}^{2}(4)}(f_{3}(4))) + 1\)

\(\beta(4,4) = f_{2}^{f_{1}^{4}(f_{2}^{f_{2}^{f_{1}^{f_{1}^{f_{1}^{f_{1}^{3}(4) + f_{1}^{2}(4) + 15}(f_{2}(4)) + f_{1}^{f_{1}^{3}(4) + f_{1}^{2}(4) + 6}(f_{2}(4)) + 6}(f_{2}^{2}(4)) + 2}(f_{2}^{3}(4))}(f_{3}(4)) + 3}(f_{3}^{2}(4)))}(f_{3}^{3}(4))\)

\(\beta(5,4) = f_{f_{1}^{3}(4) + 2}(f_{\omega^2.2 + 2}^{3}(4)) + 7\)

\(\beta(16,4) = f_{\varphi(1,0)}(4)\)

\(\beta(64,4) = f_{\varphi(1,0,0)}(4) = f_{\Gamma_0}(4)\)

\(\beta(256,4) = f_{\varphi(1,0,0,0)}(4)\)

\(\beta(1024,4) = f_{\varphi(1,0,0,0,0)}(4) = f_{SVO}(4)\) by definition

Googol

\(\beta(2.99046262281061,3)\)

\(= f_{1}^{f_{1}^{3}(f_{2}(3)) + f_{1}^{2}(f_{2}(3)) + f_{1}^{2}(3) + 3}(f_{2}^{2}(3)) + f_{1}^{f_{1}^{3}(f_{2}(3)) + f_{1}^{2}(f_{2}(3)) + f_{1}^{2}(3) + 2}(f_{2}^{2}(3)) + f_{1}^{f_{1}^{3}(f_{2}(3)) + 1}(f_{2}^{2}(3)) + 1\)

\(<\) Googol \(= 10^{100}\)

\(< \beta(2.99046262281062,3) = f_{1}^{f_{1}^{3}(f_{2}(3)) + f_{1}^{2}(f_{2}(3)) + f_{1}^{2}(3) + 4}(f_{2}^{2}(3))\)


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, corrected using Version 3 code, big numbers can be accessed around a large ordinal by slightly changing the precision of the input real number.

\(\beta(12.02,3) = f_{f_{\omega + 2}^{2}(f_{\omega.2 + 2}^{2}(3)) + 5}^{2}(f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0)^2 + \varphi(1,0)^{(\omega\uparrow\uparrow 2)^{\omega + 2}.2 + 1} + 1}.(\varphi(2,0).2 + \omega^2) + (\omega\uparrow\uparrow 2)^{\omega}.2}^{2}(3))\)

\(\beta(12.026,3) = f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0)^2.((\varphi(1,\varphi(1,0)^{(\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega) + \omega + 2}.2 + 2)\uparrow\uparrow 2).(\varphi(1,2)^{\omega.2}.((\omega\uparrow\uparrow 2)^{\omega}.(\omega + 2) + (\omega\uparrow\uparrow 2)^2.(\omega.2 + 1) + (\omega\uparrow\uparrow 2).(\omega + 2) + \omega.2 + 1) + 1) + 1) + 1} + \varphi(1,(\omega\uparrow\uparrow 2))}(3)\)

\(\beta(12.0267,3) = f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0)^2.(\varphi(1,\varphi(1,(\omega\uparrow\uparrow 2)^{\omega + 1}.2)^{\omega^2.2 + \omega.2 + 1}.(\omega^2.2) + 2)^{(\omega\uparrow\uparrow 2)^{\omega + 1}.2 + (\omega\uparrow\uparrow 2)^2.(\omega + 2)}.((\varphi(1,(\omega\uparrow\uparrow 2)^2.(\omega.2) + \omega)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)}))}}(3)\)

\(\beta(12.02678,3) = f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0)^2.(\varphi(1,(\varphi(1,(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2 + 1) + 2)\uparrow\uparrow 2)^{\omega.2} + \omega.2)^{(\varphi(1,(\omega\uparrow\uparrow 2)^2.(\omega^2.2 + 1) + (\omega\uparrow\uparrow 2) + \omega.2)\uparrow\uparrow 2)^2.((\omega\uparrow\uparrow 2)^2.2 + \omega^2)})}}(3)\)

\(\beta(12.026783,3) = f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0)^2.((\varphi(1,\varphi(1,(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega + 1) + 1) + 2)\uparrow\uparrow 2)^{(\varphi(1,\varphi(1,0)^2.(\omega + 1) + \omega)\uparrow\uparrow 2).(\varphi(1,(\omega\uparrow\uparrow 2)^2.(\omega + 1) + (\omega\uparrow\uparrow 2) + 2)^{\varphi(1,\omega^2)})})}}(3)\)

\(\beta(12.0267838,3) = f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0)^2.(\varphi(1,\varphi(1,(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + \omega)^{(\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega^2) + 2}.((\omega\uparrow\uparrow 2)^2.2 + \omega^2.2 + \omega + 2))^{\varphi(1,(\omega\uparrow\uparrow 2)^{\omega} + 1)^2.((\omega\uparrow\uparrow 2))})}}(3)\)

\(\beta(12.02678389,3) = f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0)^2.((\varphi(1,(\varphi(1,(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2 + 2) + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega^2 + \omega + 1) + 2)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega.2} + \omega^2 + \omega + 1}.2 + (\omega\uparrow\uparrow 2)^{\omega.2 + 1}.(\omega + 1))\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)})}}(3)\)

\(\beta(12.026783892,3) = f_{\varphi(1,\varphi(2,0) + 1)}(3)\)

\(\beta(12.026783893,3) = f_{\varphi(1,\varphi(2,0) + 1)}(3) + 1\)

\(\beta(12.0267839,3) = f_{\varphi(1,\varphi(2,0) + 1)}(3) + f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(1,\omega^2.2 + 1)^{\omega^2 + \omega + 2}.(\omega^2.2 + 2) + 1}.(\varphi(2,0).2 + (\omega\uparrow\uparrow 2)^{\omega + 2}) + (\omega\uparrow\uparrow 2).(\omega^2 + 2) + 1}^{2}(3)\)

\(\beta(12.026784,3) = f_{(\omega\uparrow\uparrow (f_{(\varphi(2,0)\uparrow\uparrow 2)^{\varphi(2,0).(\varphi(1,0)^{(\omega\uparrow\uparrow 2)^{\omega + 1}.(\omega^2.2 + 1) + 1}.(\omega^2 + 2) + \varphi(1,0)^2 + \varphi(1,0).((\omega\uparrow\uparrow 2)^{\omega + 2}.(\omega^2.2 + \omega.2 + 1) + \omega.2 + 1) + \omega)}}(3)))}(f_{\varphi(1,\varphi(2,0) + 1)}(3))\)

\(\beta(12.02679,3) = f_{(\omega\uparrow\uparrow 2)^{f_{(\omega\uparrow\uparrow 2)^2 + \omega.2 + 1}^{2}(3) + 3}.(\omega + 1) + (\omega\uparrow\uparrow 2)^3}(f_{\varphi(1,\varphi(2,0) + 1).(\omega) + \varphi(2,0)^2.2 + \omega^2 + \omega.2 + 1}(3))\)

\(\beta(12.0268,3) = f_{\varphi(1,\varphi(2,0) + 1).(\varphi(2,0)^{(\varphi(1,(\varphi(1,1)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2).(\omega + 2) + 1}.2 + (\omega\uparrow\uparrow 2)^{\omega} + (\omega\uparrow\uparrow 2)^2.(\omega^2 + 1) + (\omega\uparrow\uparrow 2).2 + \omega^2.2 + 2)\uparrow\uparrow 2).((\omega\uparrow\uparrow 2)^{\omega.2 + 2}.2)})}(3)\)

\(\beta(12.027,3) = f_{\varphi(1,(\omega\uparrow\uparrow (f_{1}^{2}(f_{\omega^2.2 + 2}(3)) + 7))^{f_{f_{\omega}(3)}(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega.2) + 1}^{2}(3))})}(f_{\varphi(2,\varphi(2,0) + 1).(\omega^2) + \omega + 2}^{2}(3))\)

\(\beta(12.03,3) = f_{2}^{f_{2}^{2}(3) + f_{1}^{2}(3)}(f_{(\varphi(1,\varphi(2,0).2 + 2)\uparrow\uparrow 2)^{\omega^2}.(\omega^2.2 + 1) + 1}^{2}(3))\)

\(\beta(12.1,3) = f_{\varphi(1,\varphi(2,0)^{(\varphi(1,\omega.2)\uparrow\uparrow 2)^{\varphi(1,0).2 + 1}.(\varphi(1,\omega.2)^{\varphi(1,0)^2.((\omega\uparrow\uparrow 2)^2.(\omega.2 + 2) + 2) + (\omega\uparrow\uparrow 2).(\omega.2 + 1) + \omega}.(\varphi(1,\omega)^2.2 + \varphi(1,2)^{(\varphi(1,1)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega.2}}}))})}(3)\)

\(\beta(13,3) = f_{(\omega\uparrow\uparrow (f_{3}(f_{(\omega\uparrow\uparrow 2)^{\omega.2 + 2}.(\omega^2.2 + \omega.2) + (\omega\uparrow\uparrow 2)^{\omega.2}.2 + (\omega\uparrow\uparrow 2)^{\omega}.(\omega.2) + 1}(3))))}(f_{(\varphi(2,\omega)\uparrow\uparrow 2)^{(\omega\uparrow\uparrow 2)^{\omega}.(\omega^2 + 2) + (\omega\uparrow\uparrow 2)^2}.(\omega + 2)}^{2}(3))\)


Comparing Outputs Using Different Base \(v\) Values

Typically using larger base input values like \(v = 3, 4, 5\) etc. will generate larger ordinals and big numbers. This is clear when using the maximum allowable inputs for each base:

\(\beta(8,2) = f_{\varphi(1,0,0)}(2) = f_{\Gamma_0}(2) = f_{SVO}(2)\) by definition

\(\beta(81,3) = f_{\varphi(1,0,0,0)}(3) = f_{SVO}(3)\) by definition

\(\beta(1024,4) = f_{\varphi(1,0,0,0,0)}(4) = f_{SVO}(4)\) by definition

When using equivalent real number inputs with different bases, the same effect can be seen:

\(\beta(2.00001,2) = f_{\omega}(2) = 8\)

\(\beta(2.00001,3) = f_{1}^{2}(3) + 6 = 18\)

\(\beta(2.00001,4) = f_{2}(4) = 4.2^4 = 64\)

Predicting the real number input required to access the same big number using different base values is not easy:

\(\beta(2.305,2) = f_{1}^{3}(f_{\omega}(2)) = 8.2^3 = 64\)

\(\beta(2.329,3) = f_{2}(3).2 + f_{2}(3) + 4 = 24.2 + 12 + 4 = 64\)

\(\beta(2.00001,4) = f_{2}(4) = 4.2^4 = 64\)


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.

The syntax and example code for my Sequence Generator Code is available on another blog. See the references section at the end of this blog.


Comments and Questions

Look forward to comments and questions.

Cheers B1mb0w.


References

The Beta Function

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