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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)

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