FANDOM


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 \(\theta(\xi)\).

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