User blog:B1mb0w/Calculating Alpha Numbers

Calculating Alpha Numbers
The Alpha Function is reasonably complicated so it is usually the case that the ordinal and FGH function cannot be predicted for an arbitrary real number input to the function.

This blog will illustrate how to approach calculating Alpha numbers and will partially explain the pseudo code algorithm for how it works.

It will be useful to refer to the blog on Sequence Generating Code for more background information.


 * \(\alpha(r)\) 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.