## FANDOM

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Below you can see rules to assign fundamental sequences for the Feferman theta-function at least up to Large Veblen ordinal (they are same as rules for finitary/transfinitary Veblen function from previous post, but I rewrote them for the application for theta-function). Here theta-function is considered as a two-argument function with $$\theta_\xi(\gamma)$$ written as $$\theta(\xi,\gamma)$$.

If a limit ordinal $$\alpha$$ is written in next normal form

$$\alpha=\theta(\xi_1,\gamma_1)+\theta(\xi_2,\gamma_2)+\cdots+\theta(\xi_k,\gamma_k)$$,

where

• $$\theta(\xi_1,\gamma_1)\geq \theta(\xi_2,\gamma_2)\geq\cdots\geq\theta(\xi_k,\gamma_k)$$,
• $$\xi_i=\Omega^{\beta_{i,1}}\cdot \alpha_{i,1}+\Omega^{\beta_{i,2}}\cdot \alpha_{i,2}+\cdots+\Omega^{\beta_{i,n_i}}\cdot \alpha_{i,n_i}$$ for all $$i \in \{1,...,k\}$$ where
• $$\beta_{i,1}>\beta_{i,2}>\cdots>\beta_{i,n_i} \geq 0$$ ,
• $$\alpha_{i,j}\geq 1$$ for all $$j \in \{1,...,n_i \}$$,
• $$n_i$$ is a non-negative integer,
• $$\theta(\xi_k,\gamma_k)$$ is a limit ordinal,
• $$\beta_{i,j},\alpha_{i,j},\gamma_i < \theta(\xi_i, \gamma_i)$$ for all $$i \in \{1,...,k\}$$, $$j \in \{1,...,n_i \}$$,
• $$k$$ is a positive integer,

then $$\alpha[n]=\theta(\xi_1,\gamma_1)+\theta(\xi_2,\gamma_2)+\cdots+\theta(\xi_k,\gamma_k)[n]$$

If write a limit ordinal as $$\theta(\cdots+\Omega^{\beta_k} \cdot \alpha_k,\gamma)$$ where dots $$\cdots$$ denote $$\sum_{i=1}^{k-1}\Omega^{\beta_{i}}\cdot \alpha_{i}$$,

then

1)if $$k=0$$ then $$\theta(\cdots+\Omega^{\beta_k} \cdot \alpha_k,\gamma)=\theta(0,\gamma)$$ and in this case:

1.1) $$\theta(0,\gamma)=\omega^\gamma$$,

1.2) $$\theta(0,0)=\omega^0=1$$,

1.3) $$\theta(0,\gamma)[n]=\theta(0,\gamma-1)\cdot n=\omega^{\gamma-1} n$$ if $$\gamma$$ is a successor ordinal,

1.4) $$\theta(0,\gamma)[n]=\theta(0,\gamma[n])=\omega^{\gamma[n]}$$ if $$\gamma$$ is a limit ordinal,

1.5) $$(\theta(0,\gamma_1)+\cdots+\theta(0,\gamma_k))[n]=\theta(0,\gamma_1)+\cdots+\theta(0,\gamma_k)[n]$$, where

• $$\gamma_1 \geq \cdots \geq \gamma_k \geq 1$$,
• $$\gamma_m<\theta(0,\gamma_m)$$ for all $$m \in \{1,...,k\}$$,

2) if $$\beta_k=0$$ then $$\Omega^{\beta_k} \cdot \alpha_k=\alpha_k$$ and in this case:

2.1) $$\theta(\cdots+\alpha_k,0)[0]=0$$

and $$\theta(\cdots+\alpha_k,0)[n+1]=\theta(\cdots+\alpha_k-1,\theta(\cdots+\alpha_k,0)[n])$$ if $$\alpha_k$$ is a successor ordinal,

2.2) $$\theta(\cdots+\alpha_k,\gamma+1)[0]=\theta(\cdots+\alpha_k,\gamma)+1$$

and $$\theta(\cdots+\alpha_k,\gamma+1)[n+1]=\theta(\cdots+\alpha_k-1,\theta(\cdots+\alpha_k,\gamma+1)[n])$$ if $$\alpha_k$$ is a successor ordinal,

2.3) $$\theta(\cdots+\alpha_k,\gamma)[n]=\theta(\cdots+\alpha_k,\gamma[n])$$ if $$\gamma$$ is a limit ordinal,

2.4) $$\theta(\cdots+\alpha_k,0)[n]=\theta(\cdots+\alpha_k[n],0)$$ if $$\alpha_k$$ is a limit ordinal,

2.5) $$\theta(\cdots+\alpha_k,\gamma+1)[n]=\theta(\cdots+\alpha_k[n],\theta(\cdots+\alpha_k,\gamma))$$ if $$\alpha_k$$ is a limit ordinal,

3) if $$\beta_k > 0$$ then:

3.1) $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,0)[0]=0$$

and $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,0)[n+1]=\theta(\cdots+\Omega^{\beta_k}\cdot(\alpha_k-1)+\Omega^{\beta_k-1}\cdot(\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,0)[n]),0)$$

if $$\alpha_k$$ and $$\beta_k$$ are successor ordinals,

3.2) $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma)[0]=\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma-1)+1$$

and $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma)[n+1]=\theta(\cdots+\Omega^{\beta_k}\cdot(\alpha_k-1)+\Omega^{\beta_k-1}\cdot(\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma)[n]),0)$$

if $$\alpha_k$$ and $$\beta_k$$ are successor ordinals,

3.3) $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma)[n]=\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma[n])$$ if $$\gamma$$ is a limit ordinal,

3.4) $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,0)[n]=\theta(\cdots+\Omega^{\beta_k}\cdot(\alpha_k[n]),0)$$ if $$\alpha_k$$ is a limit ordinal and $$\beta_k$$ is a successor ordinal,

3.5) $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma)[n]=\theta(\cdots+\Omega^{\beta_k}\cdot(\alpha_k[n])+\Omega^{\beta_k-1}\cdot(\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma-1)+1),0)$$

if $$\alpha_k$$ is a limit ordinal, $$\beta_k$$ and $$\gamma$$ are successor ordinals,

3.6) $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,0)[n]=\theta(\cdots+\Omega^{\beta_k}\cdot(\alpha_k-1)+\Omega^{\beta_k[n]},0)$$ if $$\beta_k$$ is a limit ordinal and $$\alpha_k$$ is a successor ordinal,

3.7) $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma)[n]=\theta(\cdots+\Omega^{\beta_k}\cdot(\alpha_k-1)+\Omega^{\beta_k[n]}\cdot(\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma-1)+1),0)$$

if $$\beta_k$$ is a limit ordinal, $$\alpha_k$$ and $$\gamma$$ are successor ordinals,

3.8) $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,0)[n]=\theta(\cdots+\Omega^{\beta_k}\cdot(\alpha_k[n]),0)$$ if $$\beta_k$$ and $$\alpha_k$$ are limit ordinals,

3.9) $$\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma)[n]=\theta(\cdots+\Omega^{\beta_k}\cdot(\alpha_k[n])+\Omega^{\beta_k[n]}\cdot(\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,\gamma-1)+1),0)$$

if $$\beta_k$$ and $$\alpha_k$$ are limit ordinals and $$\gamma$$ is a successor ordinal.

Large Veblen ordinal $$\theta(\Omega^\Omega,0)[0]=0$$ and $$\theta(\Omega^\Omega,0)[n+1]=\theta(\Omega^{\theta(\Omega^\Omega,0)[n]},0)$$.

Note: $$\theta(\xi,0)$$ can be abbriviated as $$\theta(\xi)$$.