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## 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.

## 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 definition of an arbitrary Veblen function:

$$\varphi(\lambda_{[b]})$$

We can unpack the arbitrary Veblen function if we let:

$$b = a + 1 + z$$

Then

$$\varphi(\lambda_{[b]}) = \varphi(\lambda_{[a + 1 + z]})$$

And

$$\varphi(\lambda_{[b]}) = \varphi(\lambda_{[a + 1 + z]}) = \varphi(\delta_{[a]},\gamma,0_{[z]})$$

Where

$$\delta_1 > 0$$ if $$a > 0$$

We can continue to unpack the arbitrary Veblen function if we let:

$$a = x + 1 + y$$

Then

$$\varphi(\delta_{[a]},\gamma,0_{[z]}) = \varphi(\delta_{[x + 1 + y]},\gamma,0_{[z]})$$

And

$$\varphi(\delta_{[a]},\gamma,0_{[z]}) = \varphi(\delta_{[x + 1 + y]},\gamma,0_{[z]}) = \varphi(\alpha_{[x]},\beta,0_{[y]},\gamma,0_{[z]})$$

Where

$$\alpha_1 > 0$$ if $$x > 0$$

We can now begin the rule-set:

$$\gamma$$ $$z$$ $$a$$ or $$x$$ $$\beta$$ $$\varphi(\lambda_{[b]})[n] = \varphi(\delta_{[a]},\gamma,0_{[z]})[n]$$
limit any any any $$= \varphi(\delta_{[a]},\gamma[n],0_{[z]})$$
$$= 1$$ $$> 0$$ $$a = 0$$ any $$= \varphi(\gamma,0_{[z]})[n] = \varphi(1,0_{[z]})[n]$$

$$= \varphi^{\omega[n]}(1_*,0_{[z-1]})$$

successor $$> 0$$ $$a > 0$$ any $$= \varphi^{\omega[n]}(\delta_{[a]},\gamma-1,0_*,0_{[z-1]})$$
$$= 0$$ any limit $$= \varphi(\alpha_{[x]},\beta,0_{[y]},\gamma)[n]$$

$$= \varphi^{\omega[n]}(\alpha_{[x]},\beta[n],0_{[y]},\varphi(\alpha_{[x]},\beta,0_{[y]},\gamma-1) + 1_*)$$

$$= 0$$ $$x = 0$$ $$= 1$$ $$= \varphi(\beta,0_{[y]},\gamma)[n] = \varphi(1,0_{[y]},\gamma)[n]$$

$$= \varphi^{\omega[n]}(1,0_{[y-1]},\varphi(1,0_{[y]},\gamma-1) + 1_*)$$

$$= 0$$ $$x > 0$$ successor $$= \varphi^{\omega[n]}(\alpha_{[x]},\beta-1,0_{[y]},\varphi(\alpha_{[x]},\beta,0_{[y]},\gamma-1) + 1_*)$$

## Calculated Example for $$\zeta_0[n]$$

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

$$\zeta_0[3] = \varphi(2,0)[3] = \varphi^{\omega[3]}(1,0_*) = \varphi^3(1,0_*) = \varphi(1,\varphi(1,\varphi(1,0)))$$

## Calculated Example for $$\Gamma_0[n]$$

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

$$\Gamma_0[3] = \varphi(1,0,0)[3] = \varphi^{\omega[3]}(1_*,0) = \varphi^3(1_*,0) = \varphi(\varphi(\varphi(1,0),0),0)$$

## Calculated Example for Small Veblen Ordinal $$SVO[n]$$

SVO is defined as follows:

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

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

$$SVO[3] = \varphi(1,0_{[\omega]})[3] = \varphi(1,0_{[\omega[3]]}) = \varphi(1,0_{[3]}) = \varphi(1,0,0,0)$$