User blog:Denis Maksudov/Fundamental sequences system for limit ordinals up to Large Veblen Ordinal

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 can't find 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 functions):

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.