User blog:B1mb0w/Rule 2B

Fundamental Sequences
This blog will map out much of the standard definitions on Fundamental Sequences for Ordinals. It will attempt to simplify the rule-set where possible. The alternative rule-set presented here will be used in my other blog for the J Function and in particular in the Sandpit \(J_4\) 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

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

I have written another blog to further extend Normal Form to provide detailed definitions for ordinals of arbitrary complexity.

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

\(\varphi(n+1) = \varphi^{n+1}(1) = \omega^{\varphi(n)}\)

Rule-set (The Aristo Sequence)
The following rule-set has been given a name of the "Aristo Sequence", so that it is 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:

\(\varphi(a_{[x]},0_{[y]})\)

Rule 1: if \(a_x\) is a limit ordinal then:

\(\varphi(a_{[x]},0_{[y]})[n] = \varphi(a_{[x-1]},a_x[n],0_{[y]})\)

Rule 2a: if \(a_x\) is not a limit ordinal and \(y>0\) then:

\(\varphi(a_{[x]},0_{[y]})[n] = \varphi^{\omega[n]}(a_{[x-1]},a_x-1,0_*,0_{[y-1]}) = \varphi^n(a_{[x-1]},a_x-1,0_*,0_{[y-1]})\)

Rule 2b: if \(a_x\) is not a limit ordinal and \(y=0\) then:

\(\varphi(a_{[x]},0_{[y]})[n] = \varphi(a_{[x]})[n] = \varphi(a_{[x-1]},a_x-1)\uparrow\uparrow\omega[n] = \varphi(a_{[x-1]},a_x-1)\uparrow\uparrow n\)

A detailed explanation of of this rule is given on another blog.

Rule 3: if \(x=1\) and \(a_x=1\) and \(y>0\) then:

\(\varphi(a_{[x]},0_{[y]})[n] = \varphi(1,0_{[y]})[n] = \varphi^{\omega[n]}(1_*,0_{[y-1]}) = \varphi^n(1_*,0_{[y-1]})\)

Calculated Example
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:

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

Appreciate any comments on this blog.