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## 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

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.