## FANDOM

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This is my attempt to create system, which gives possibility to define fundamental sequences for all limit ordinals up to Large Veblen Ordinal (LVO). Unfortunately I found no one sourse where somebody gives detailed definition for fundamental sequences for all limit ordinals between Gamma_0 and LVO. And that is why I created this FS-system, using only the system of fundamental sequences for binary Veblen function as the prototype. Previously I published part of this system on this page of wikia as well as on my site.

Let $0 \le\beta_m < \Omega$, $0 \le\gamma_m < \Omega$, $\lambda$ is a limit ordinal, $\Omega$ is first uncountable ordinal.

Fundamental sequences for limit ordinals of the theta-function $\theta(\beta, \gamma)$ up to $\theta(\Omega,0)=\varphi(1,0,0)=\Gamma_0$:

1.1) $(\theta(\beta_1,\gamma_1) + \theta(\beta_2,\gamma_2) + \cdots + \theta(\beta_k,\gamma_k))[n]=$

$=\theta(\beta_1,\gamma_1) + \theta(\beta_2,\gamma_2) + \cdots + \theta(\beta_k,\gamma_k) [n]$,

where $\theta(\beta_1,\gamma_1) \ge \theta(\beta_2,\gamma_2) \ge \cdots \ge \theta(\beta_k,\gamma_k)$ and $\gamma_m <\theta(\beta_m, \gamma_m)$ for $m \in \{1,2,...,k\}$

1.2) $\theta(0,\gamma+1)[n]=\theta(0,\gamma)\cdot n=\omega^\gamma n$,

1.3) $\theta(\beta+1,0)[0]=0$ and $\theta(\beta+1,0)[n+1]=\theta(\beta, \theta(\beta+1,0)[n])$,

1.4) $\theta(\beta+1,\gamma+1)[0]=\theta(\beta+1,\gamma)+1$ and $\theta(\beta+1,\gamma+1)[n+1]=\theta(\beta, \theta(\beta+1,\gamma+1)[n])$,

1.5) $\theta(\beta, \lambda)[n]=\theta(\beta, \lambda [n])$,

1.6) $\theta(\lambda,0)[n]=\theta(\lambda[n],0)$,

1.7) $\theta(\lambda,\gamma+1)[n]=\theta(\lambda[n],\theta(\lambda,\gamma)+1)$.

Note: The theta-function is shown in the two-argument version $\theta(\beta, \gamma)=\theta_\beta(\gamma)$, if $\gamma=0$ it can be abbreviated as $\theta(\beta)=\theta(\beta,0)$. The theta-function is an extension of the two-argument Veblen function. For countable arguments theta-function is equal to Veblen function $\theta(\beta, \gamma)=\varphi(\beta, \gamma)$ and has same fundamental sequences.

Fundamental sequences for limit ordinals of the theta-function $\theta(\Omega+\beta, \gamma)$ up to $\theta(\Omega+\Omega, 0)=\varphi(2,0,0)$ (as analogy of fundamental sequences for Veblen function):

2.1) $\theta(\Omega,0)[0]=0$ and $\theta(\Omega,0)[n+1]=\theta(\theta(\Omega,0)[n],0)=\Gamma_0[n+1]=\varphi(1,0,0)[n+1]$,

2.2) $\theta(\Omega,\gamma+1)[0]=\theta(\Omega,\gamma)+1$ and $\theta(\Omega,\gamma+1)[n+1]=\theta(\theta(\Omega,\gamma+1)[n],0)=\Gamma_{\gamma+1}[n+1]=\varphi(1,0,\gamma+1)[n+1]$,

2.3) $\theta(\Omega, \lambda)[n]=\theta(\Omega, \lambda[n])=\Gamma_\lambda[n]=\varphi(1,0,\lambda)[n]$,

2.4) $\theta(\Omega+\beta+1,0)[0]=0$ and $\theta(\Omega+\beta+1,0)[n+1]=\theta(\Omega+\beta,\theta(\Omega+\beta+1,0)[n])$,

2.5) $\theta(\Omega+\beta+1,\gamma+1)[0]=\theta(\Omega+\beta+1,\gamma)+1$ and $\theta(\Omega+\beta+1,\gamma+1)[n+1]=\theta(\Omega+\beta, \theta(\Omega+\beta+1, \gamma+1)[n])$,

2.6) $\theta(\Omega+\beta, \lambda)[n]=\theta(\Omega+\beta, \lambda [n])$,

2.7) $\theta(\Omega+\lambda,0)[n]=\theta(\Omega+\lambda[n],0)$,

2.8) $\theta(\Omega+\lambda,\gamma+1)[n]=\theta(\Omega+\lambda[n],\theta(\Omega+\lambda,\gamma)+1)$,

2.9) $\theta(\Omega+\Omega,0)[0]=0$ and $\theta(\Omega+\Omega,0)[n+1]=\theta(\Omega+\theta(\Omega+\Omega,0)[n],0)$.

Fundamental sequences for limit ordinals of the theta-function up to Large Veblen Ordinal $\theta(\Omega^\Omega, 0)$:

let $\alpha=\Omega^{\delta_1} \cdot \xi_1+\Omega^{\delta_2} \cdot \xi_2+\cdots+\Omega^{\delta_k} \cdot \xi_k$ and $\delta_1\geq\delta_2\geq \cdots\delta_k\geq 1$ and $\xi_m \geq 0$ for $m \in \{1,2,...,k\}$ then

3.1) $\theta(\alpha+\beta+1,0)[0]=0$ and $\theta(\alpha+\beta+1,0)[n+1]=\theta(\alpha+\beta,\theta(\alpha+\beta+1,0)[n])$,

3.2) $\theta(\alpha+\beta+1,\gamma+1)[0]=\theta(\alpha+\beta+1,\gamma)+1$ and $\theta(\alpha+\beta+1,\gamma+1)[n+1]=\theta(\alpha+\beta, \theta(\alpha+\beta+1, \gamma+1)[n])$,

3.3) $\theta(\alpha+\beta, \lambda)[n]=\theta(\alpha+\beta, \lambda [n])$,

3.4) $\theta(\alpha+\lambda,0)[n]=\theta(\alpha+\lambda[n],0)$,

3.5) $\theta(\alpha+\lambda,\gamma+1)[n]=\theta(\alpha+\lambda[n],\theta(\alpha+\lambda,\gamma)+1)$,

3.6)$\theta(\alpha+\Omega,0)[0]=0$ and $\theta(\alpha+\Omega,0)[n+1]=\theta(\alpha+\theta(\alpha+\Omega,0)[n],0)$,

3.7) $\theta(\alpha+\Omega,\gamma+1)[0]=\theta(\alpha+\Omega,\gamma)+1$ and $\theta(\alpha+\Omega,\gamma+1)[n+1]=\theta(\alpha+\theta(\alpha+\Omega,\gamma+1)[n],0)$.

3.8) $\theta(\alpha \cdot\Omega,0)[0]=1$ and $\theta(\alpha \cdot\Omega,0)[n+1]=\theta(\alpha \cdot\theta(\alpha \cdot \Omega,0)[n],0)[n+1]$

3.9) $\theta(\alpha \cdot\Omega,\gamma+1)[0]=\theta(\alpha \cdot\Omega,\gamma)+1$ and $\theta(\alpha \cdot\Omega,\gamma+1)[n+1]=\theta(\alpha \cdot\theta(\alpha \cdot\Omega,\gamma+1)[n],0)$

3.10) $\theta(\alpha^\Omega,0)[0]=0$ and $\theta(\alpha^\Omega,0)[n+1]=\theta(\alpha^{\theta(\alpha^\Omega,0)[n]},0)[n+1]$

3.11) $\theta(\alpha^\Omega,\gamma+1)[0]=\theta(\alpha^\Omega,\gamma)+1$ and $\theta(\alpha^\Omega,\gamma+1)[n+1]=\theta(\alpha^{\theta(\alpha^\Omega,\gamma+1)[n]},0)$

Note: if $\alpha=0$ then expressions 3.1-3.5 should rewrite as 1.3-1.7 and expressions 3.6-3.7 should rewrite as 2.1-2.2.

Up to Bachmann-Howard ordinal $\theta(\varepsilon_{\Omega+1},0)$

3.12) $\theta(\alpha_1^{\alpha_2^{\cdots^{\alpha_k^{\Omega}}}},0)[0]=0$ and $\theta(\alpha_1^{\alpha_2^{\cdots^{\alpha_k^{\Omega}}}},0)[n+1]=\theta(\alpha_1^{\alpha_2^{\cdots^{\alpha_k^{\theta(\alpha_1^{\alpha_2^{\cdots^{\alpha_k^{\Omega}}}},0)[n]}}}},0)$

3.13) $\theta(\varepsilon_{\Omega+1},0)[0]=0$ and $\theta(\varepsilon_{\Omega+1},0)[n+1]=\theta(\Omega^{\theta(\varepsilon_{\Omega+1},0)[n]},0)$