User blog:B1mb0w/Fundamental Sequences used by the Beta Function

Fundamental Sequences (used by The Beta Function)
This blog will cover the standard definitions on Fundamental Sequences for Ordinals. It will also provide a precise rule-set for the Fundamental Sequences used by in my Beta Function blogs.

This blog is a complete update of my previous blog on Fundamental Sequences. Please keep this in mind if you refer to that blog.

Basics (Cantor's Normal Form)
Let \(\gamma\) and \(\delta\) be two arbitrary transfinite ordinals, \(\lambda\) is an arbitrary limit ordinal, and \(n\) is a finite integer. Then:

\((\gamma + 1)[n] = \gamma\)

\((\gamma + \lambda)[n] = \gamma + \lambda[n]\) when \(\gamma >= \lambda\)

\(\lambda.(\delta + 1)[n] = \lambda.\delta + \lambda[n]\)

\(\gamma.\lambda[n] = \gamma.(\lambda[n])\) when \(\gamma >= \lambda\)

\(\lambda^{\delta + 1}[n] = \lambda^{\delta}.(\lambda[n])\)

and

\(\gamma^{\lambda}[n] = \gamma^{\lambda[n]}\)

Some Common Transfinite Ordinals
\(\omega[n] = n\)

\(\epsilon_0[n] = \omega\uparrow\uparrow n\)

\(\epsilon_1[n] = \epsilon_0\uparrow\uparrow n\)

\(\epsilon_{j+1}[n] = \epsilon_j\uparrow\uparrow n\)

and

\(\epsilon_{\omega}[n] = \epsilon_{\omega[n]} = \epsilon_n\)

Veblen Hierarchy
Continuing into Veblen Hierarchy and the \(\varphi\) function. Lets start with these equations which are equivalent to those in the Common Transfinite Ordinal section.

\(\varphi(1)[n] = \omega[n] = n\)

\(\varphi(1,0)[n] = \epsilon_0[n] = \varphi(n) = \omega\uparrow\uparrow n\)

\(\varphi(1,1)[n] = \epsilon_1[n] = \varphi(1,0)\uparrow\uparrow n\)

\(\varphi(1,j + 1)[n] = \epsilon_{j + 1}[n] = \varphi(1,j)\uparrow\uparrow n\)

and

\(\varphi(1,\omega)[n] = \varphi(1,\omega[n]) = \varphi(1,n)\)

The following extends the Veblen function definition for completeness:

\(\varphi = 0\)

\(\varphi(0) = 1\)

\(\varphi(1) = \omega\)

and

\(\varphi(n) = \varphi^n(1) = \omega\uparrow\uparrow n = \omega^{\varphi(n-1)}\)

Rule-set (used by The Beta Function)
The following rule-set is used by my Beta Function blogs and is intended to be clearly distinguishable from other rule-set definitions. Before we start, some notational conventions that will be used are:

\(k^2(n,p_*) = k(n,k(n,p))\)

\(k^2(n_*,p) = k(k(n,p),p)\)

and

\(k(a_{[2]},b_{[3]}) = k(a_1,a_2,b_1,b_2,b_3)\)

The rule-set starts with this arbitrary Veblen function:

\(\varphi(\alpha_{[x]},\beta,0_{[y]})[n]\)

where \(x >= 0\) and \(y >= 1\), \(\alpha\) and \(\beta\) can be any ordinal, but \(\beta >= 1\)

then


 * \(\beta\) is a limit ordinal \(= \varphi(\alpha_{[x]},\beta[n],0_{[y]})\)
 * else
 * \(y > 0\)
 * \(x = 0\) and \(\beta = 1\) then \(= \varphi(1,0_{[y]}) = \varphi^{\omega[n]}(1_*,0_{[y-1]})\)
 * else \(= \varphi^{\omega}(\alpha_{[x]},\beta-1,0_*,0_{[y-1]}) = \varphi^{\omega[n]}(\alpha_{[x]},\beta-1,0_*,0_{[y-1]})\)
 * else
 * \(\alpha_x\) is a limit ordinal \(= \varphi(\alpha_{[x-1]},\alpha_x[n],\varphi(\alpha_{[x]},\beta-1)+1)\)
 * else \(= \varphi^{\omega[n]}(\alpha_{[x-1]},\alpha_x-1,\varphi(\alpha_{[x]},\beta-1)+1_*)\)

with this additional rule thrown in for completeness:

\(\varphi^{\omega}(m,\varphi(n,p)+1_*)\)

\(= \varphi(m+1,\varphi(n,p)+1)\) when \(m+1 < n\)

\(= \varphi(n,p+1)\) when \(m+1 = n\)

The additional rule can be illustrated as follows:

\(\varphi^{\omega}(1,\varphi(2,0)+1_*) = \varphi(2,1)\)

\(\varphi^{\omega}(1,\varphi(3,0)+1_*) = \varphi(2,\varphi(3,0)+1)\)

\(\varphi^{\omega}(2,\varphi(3,0)+1_*) = \varphi(3,1)\)

Calculated Examples
What is the fundamental sequence for \(\zeta_0[2]\) ?

\(\zeta_0[2] = \varphi(2,0)[2] = \varphi^{\omega[2]}(1,0_*) = \varphi^2(1,0_*) = \varphi(1,\varphi(1,0))\)

or

\(\zeta_0[2] = \varphi(1,\varphi(1,0)) = \epsilon_{\epsilon_0}\)

I have created another blog to calculate \(f_{\zeta_0}(2)\) in detail.

\(\Gamma_0\)
Calculating \(\Gamma_0\) we get:

\(\varphi(1,0,0) = \Gamma_0\)

then

\(\Gamma_0[2] = \varphi(1,0,0)[2] = \varphi^{\omega[2]}(1_*,0) = \varphi^2(1_*,0) = \varphi(\varphi(1,0),0)\)

Small Veblen Ordinal (SVO)
SVO is defined as follows:

\(SVO = \varphi(1,0_{[\omega]})\)

Diagonalising SVO for n=2 produces this result:

\(SVO[2] = \varphi(1,0_{[\omega]})[2] = \varphi(1,0_{[2]}) = \varphi(1,0,0) = \Gamma_0\)