User blog:Denis Maksudov/Fundamental sequences for the theta-function

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 \(\theta(\cdots+\Omega^{\beta_k} \cdot \alpha_k,\gamma)=\theta(\gamma)\) and in this case:

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

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

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

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

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


 * \(\gamma_1 \geq \cdots \geq \gamma_k \geq 1\),


 * \(\gamma_m<\theta(\gamma_m)\) 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) \(\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,0)[n]=\theta(\cdots+\Omega^{\beta_k[n]}\cdot\alpha_k,0)\) 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) \(\theta(\cdots+\Omega^{\beta_k}\cdot\alpha_k,0)[n]=\theta(\cdots+\Omega^{\beta_k[n]}\cdot(\alpha_k[n]),0)\) 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.

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

Example: \(f_{\theta(\Omega^\omega \cdot \omega)}(3)=f_{\theta(\Omega^\omega \cdot \omega)}[3](3)=f_{\theta(\Omega^{\omega[3]} \cdot \omega[3])}(3)=f_{\theta(\Omega^3 \cdot 3)}(3)\approx TREE(3)\) where TREE (3) is the famous Harvey Friedman's number.