User blog:Denis Maksudov/Fundamental sequences for extended Veblen-function

Fundamental sequences for limit ordinals of the binary Veblen-function $$\varphi(\beta, \gamma)$$ up to $$\Gamma_0$$:

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

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

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

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

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

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

1.5) $$\varphi(\beta, \gamma)[n]=\varphi(\beta, \gamma [n])$$ for any $$\beta$$ iff $$\gamma$$ is a limit ordinal $$\gamma<\varphi(\beta, \gamma)$$,

1.6) $$\varphi(\beta,0)[n]=\varphi(\beta[n],0)$$ if $$\beta$$ is a limit ordinal $$\beta < \varphi(\beta,0)$$,

1.7) $$\varphi(\beta, \gamma+1)[n]=\varphi(\beta[n],\varphi(\beta,\gamma)+1)$$ if $$\beta$$ is a limit ordinal.

Fundamental sequences for limit ordinals of the extended Veblen-function $$\varphi(\alpha_1, \alpha_2,...,\alpha_k, \beta, 0,...,0,\gamma)$$ up to Small Veblen Ordinal $$\theta(\Omega^\omega,0)$$

Let's define:


 * $$s=\alpha_1, \alpha_2,...,\alpha_k$$ is a string of $$k$$ ordinal variables before $$\beta$$ (where $$k \geq 0$$ and $$ \alpha_m \geq 0$$ for  $$m \in \{1,2,...,k\}$$, if $$k=0$$ then no  variables  before $$\beta$$),


 * $$z=0,...,0$$ is a string of $$n$$ zeros in between $$\beta$$ and $$\gamma$$ (where $$n \geq 0$$, if $$n=0$$ no zeros in between $$\beta$$ and $$\gamma$$),


 * $$\varphi(\gamma)=\omega^\gamma$$ and $$\varphi(0,\alpha_1, \alpha_2,...,\alpha_k)=\varphi(\alpha_1, \alpha_2,...,\alpha_k)$$.

Then

2.1) $$\varphi(0)=1$$ and $$\varphi(\gamma+1)[n]=\omega^\gamma n$$,

2.2) $$(\varphi(s_1, \beta_1,z_1\gamma_1) + \varphi(s_2, \beta_2, z_2, \gamma_2) + \cdots + \varphi(s_k, \beta_k, z_k, \gamma_k))[n]=$$

$$=\varphi(s_1, \beta_1,z_1\gamma_1) + \varphi(s_2, \beta_2, z_2, \gamma_2) + \cdots + \varphi(s_k, \beta_k, z_k, \gamma_k) [n]$$,

where $$\varphi(s_1, \beta_1,z_1\gamma_1) \ge \varphi(s_2, \beta_2, z_2, \gamma_2) \ge \cdots \ge \varphi(s_k, \beta_k, z_k, \gamma_k) $$ and $$\gamma_m <\varphi(s_m, \beta_m, z_m, \gamma_m)$$ for $$m \in \{1,2,...,k\}$$,

2.3) $$\varphi(s,\beta+1,z,0)[0]=0$$ and $$\varphi(\beta+1,0)[n+1]=\varphi(s,\beta, \varphi(s,\beta+1,z,0)[n],z)$$,

2.4) $$\varphi(s,\beta+1,z,\gamma+1)[0]=\varphi(s,\beta+1,z,\gamma)+1$$ and $$\varphi(s,\beta+1,z,\gamma+1)[n+1]=\varphi(s,\beta, \varphi(\beta+1,\gamma+1)[n],z)$$,

2.5) $$\varphi(s,\beta, z, \gamma)[n]=\varphi(s,\beta, z,\gamma [n])$$ for any $$\beta$$ iff $$\gamma$$ is a limit ordinal $$\gamma<\varphi(s,\beta, z, \gamma)$$,

2.6) $$\varphi(s,\beta,z,0)[n]=\varphi(s,\beta[n],z,0)$$ if $$\beta$$ is a limit ordinal $$\beta < \varphi(s,\beta,z,0)$$,

2.7) $$\varphi(s,\beta, z,\gamma+1)[n]=\varphi(s,\beta[n],z,\varphi(s,\beta,z,\gamma)+1)$$ if $$\beta$$ is a limit ordinal.

Examples

$$\varphi(1,0)[n]=\underbrace{\varphi(0, \varphi (0, ... \varphi}_{n \quad \varphi's}(0,0)...))=\underbrace{\varphi(\varphi(...\varphi}_{n \quad \varphi's}(0)...))=\varepsilon_0$$

$$\varphi(1,0,0)[n]=\underbrace{\varphi(0, \varphi (0, ... \varphi}_{n \quad \varphi's}(0,0,0)...,0),0)=\underbrace{\varphi( \varphi ( ... \varphi}_{n \quad \varphi's}(0,0)...,0),0)=\Gamma_0$$

$$\varphi(1,1,1,0,0,0)[n]=\underbrace{\varphi(1,1,0 \varphi (\varphi(1,1,0 ... \varphi}_{n \quad \varphi's}(1,1,0,0,0,0)...,0,0),0,0)$$