**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 \(J_8\) blog.

*I have created a new blog which updates the material below and provides the detailed rule-set on Fundamental Sequences used by my* Beta Function *blogs.*

**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

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

**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:

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

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 \(= \varphi^{\omega[n]}(\alpha_{[x-1]},\alpha_x-1,\varphi(\alpha_{[x]},\beta-1)+1_*)\)

- \(y > 0\)

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 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.