## FANDOM

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Definition

The extended (finitary) Veblen functions are defined as follows:

• $$\varphi(\gamma)=\omega^\gamma$$,
• $$\varphi(0,\alpha_1,...,\alpha_n,\gamma)=\varphi(\alpha_1,...,\alpha_n,\gamma)$$ for $$n \geq 0$$,
• if $$\alpha_1, \alpha_k>0$$, where $$1\le k \le n$$, then

$$\varphi(\alpha_1,...,\alpha_{k}, \underbrace{0,...,0}_{n-k \quad 0's}, \gamma)$$ denotes the $$\gamma$$-th common fixed point of the functions $$\xi \mapsto \varphi(\alpha_1,...\alpha_{k-1}, \beta, \xi,\underbrace{0,...,0}_{n-k \quad 0's})$$ for each $$\beta<\alpha_k$$.

The limit of this notation is the Small Veblen Ordinal (SVO).

Fundamental sequences for limit ordinals of single-argument Veblen function:

if a limit ordinal $$\alpha<\varepsilon_0$$ written in next normal form

$$\alpha=\varphi(\gamma_1)+\cdots+\varphi(\gamma_{k-1})+\varphi(\gamma_{k})$$,

where

• $$\gamma_1\geq \cdots \gamma_{k-1} \geq \gamma_k \geq 1$$,
• $$k$$ is a positive integer,

then

$$\alpha[n]=\varphi(\gamma_1)+\cdots+\varphi(\gamma_{k-1})+\left\{\begin{array}{lcr} \varphi(\gamma_k-1)\cdot n \quad if \quad \gamma_k \quad is \quad a \quad successor \quad ordinal\\ \varphi(\gamma_k[n]) \quad if \quad \gamma_k \quad is \quad a \quad limit \quad ordinal\\ n \quad if \quad \gamma_k=1\\ \end{array}\right.$$,

where $$\alpha[n]$$ denotes the n-th element of the fundamental sequence assigned to the limit ordinal $$\alpha$$ (strictly increasing sequence which has the ordinal $$\alpha$$ as its limit) and $$n$$ is a non-negative integer.

Fundamental sequences for limit ordinals of finitary Veblen function:

1) If a limit ordinal $$\alpha < SVO$$ written in next normal form

$$\alpha=\varphi(\alpha_{1,1},\alpha_{1,2},...,\alpha_{1,n_1})+\varphi(\alpha_{2,1},\alpha_{2,2},...,\alpha_{2,n_2})+\cdots+\varphi(\alpha_{k,1},\alpha_{k,2},...,\alpha_{k,n_k})$$

where

• $$\varphi(\alpha_{1,1},\alpha_{1,2},...,\alpha_{1,n_1})\geq\varphi(\alpha_{2,1},\alpha_{2,2},...,\alpha_{2,n_2})\geq\cdots\geq\varphi(\alpha_{k,1},\alpha_{k,2},...,\alpha_{k,n_k})$$,
• $$\alpha_{m,i} <\varphi(\alpha_{m,1},\alpha_{m,2},...,\alpha_{m,n_m})$$ for i-th argument of m-th function, $$m \in \{1,...,k\}$$ and $$i \in \{1,..,n_m\}$$,
• $$\alpha_{m,1}>0$$ for all $$m \in \{1,...,k\}$$,
• $$\varphi(\alpha_{k,1},\alpha_{k,2},...,\alpha_{k,n_k})$$ is a limit ordinal,
• $$k, n_1,...,n_k$$ are positive integers,

then

$$\alpha[n]=\varphi(\alpha_{1,1},\alpha_{1,2},...,\alpha_{1,n_1})+\varphi(\alpha_{2,1},\alpha_{2,2},...,\alpha_{2,n_2})+\cdots+\varphi(\alpha_{k,1},\alpha_{k,2},...,\alpha_{k,n_k})[n]$$ (1).

If $$n_k=1$$ then use rule for single-argument form to assign fundamental sequences (FS) for last term of expression (1), otherwise use rules 2.1-2.5 to assign FS for last term.

2) Let's write a Veblen function as $$\varphi(s,\beta,z,\gamma)$$, where

• $$s=\alpha_1,...,\alpha_c$$ and $$z=\underbrace{0,...,0}_{b \quad 0's}$$ and $$b,c$$ are non-negative integers (if $$b=c=0$$ then it is binary Veblen function $$\varphi(\beta, \gamma)$$, if $$c>0$$ then $$\alpha_1>0$$),
• $$\beta>0$$,

then

2.1) $$\varphi(s,\beta,z,\gamma)[0]=0$$ and $$\varphi(s,\beta,z,\gamma)[n+1]=\varphi(s,\beta-1,\varphi(s,\beta,z,\gamma)[n],z)$$ if $$\gamma=0$$ and $$\beta$$ is a successor ordinal,

2.2) $$\varphi(s,\beta,z,\gamma)[0]=\varphi(s,\beta,z,\gamma-1)+1$$ and $$\varphi(s,\beta,z,\gamma)[n+1]=\varphi(s,\beta-1,\varphi(s,\beta,z,\gamma)[n],z)$$ if $$\gamma$$ and $$\beta$$ are successor ordinals,

2.3) $$\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta,z,\gamma[n])$$ if $$\gamma$$ is a limit ordinal,

2.4) $$\varphi(s,\beta,z,\gamma)[n]=\varphi(s,\beta[n],z,\gamma)$$ if $$\gamma=0$$ and $$\beta$$ is a limit ordinal,

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

Examples

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

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

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

The extended Veblen function and Bird/Feferman theta-functions up to SVO are connected by the next expression:

$$\theta(\Omega^{n - 1}\alpha_{1} + \cdots + \Omega^2\alpha_{n-2} + \Omega \alpha_{n-1} + \alpha_n, \gamma) = \varphi(\alpha_{1}, \ldots, \alpha_{n-2}, \alpha_{n-1}, \alpha_n, \gamma)$$

and $$\theta(\alpha, 0)$$ can be abbreviated as $$\theta(\alpha)$$. In this terms $$SVO=\theta(\Omega^\omega)$$.

3) Fundamental sequences for transfinitary Veblen function

For definition of fundamental sequences of Veblen function with ordinal number of variables it is possible to use Schutte Klammersymbolen in form of two-row matrix where a k-th ordinal of second row $$\beta_k \geq 0$$ defines position of a k-th ordinal of the first row $$\alpha_k>0$$ in string of arguments of the Veblen function.

For example: $$\begin{pmatrix}\alpha_1 & \alpha_2 & \alpha_3 \\8 & 5 & 0 \end{pmatrix}=\varphi(\alpha_1,0,0,\alpha_2,0,0,0,0,\alpha_3)$$.

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

$$\begin{pmatrix} \alpha_{1,1} & \cdots &\alpha_{1,n_1} \\ \beta_{1,1} & \cdots & \beta_{1,n_1} \end{pmatrix}+\begin{pmatrix} \alpha_{2,1} & \cdots &\alpha_{2,n_2} \\ \beta_{2,1} & \cdots & \beta_{2,n_2} \end{pmatrix}+\cdots+\begin{pmatrix} \alpha_{k,1} & \cdots &\alpha_{k,n_k} \\ \beta_{k,1} & \cdots & \beta_{k,n_k} \end{pmatrix}$$,

where

• $$\begin{pmatrix} \alpha_{1,1} & \cdots &\alpha_{1,n_1} \\ \beta_{1,1} & \cdots & \beta_{1,n_1} \end{pmatrix} \geq \begin{pmatrix} \alpha_{2,1} & \cdots &\alpha_{2,n_2} \\ \beta_{2,1} & \cdots & \beta_{2,n_2} \end{pmatrix} \geq \cdots \geq \begin{pmatrix} \alpha_{k,1} & \cdots &\alpha_{k,n_k} \\ \beta_{k,1} & \cdots & \beta_{k,n_k} \end{pmatrix}$$,
• $$\alpha_{m,i}<\begin{pmatrix} \alpha_{m,1} & \cdots &\alpha_{m,n_m} \\ \beta_{m,1} & \cdots & \beta_{m,n_m} \end{pmatrix}$$ for all $$i \in \{1,...,n_m\}$$, $$m \in \{1,...,k\}$$,
• $$\beta_{m,i}<\begin{pmatrix} \alpha_{m,1} & \cdots &\alpha_{m,n_m} \\ \beta_{m,1} & \cdots & \beta_{m,n_m} \end{pmatrix}$$ for all $$i \in \{1,...,n_m\}$$, $$m \in \{1,...,k\}$$,
• $$\begin{pmatrix} \alpha_{k,1} & \cdots &\alpha_{k,n_k} \\ \beta_{k,1} & \cdots & \beta_{k,n_k} \end{pmatrix}$$ is a limit ordinal,
• $$k,n_1,...,n_k$$ are positive integers,

then

$$\alpha[n]=\begin{pmatrix} \alpha_{1,1} & \cdots &\alpha_{1,n_1} \\ \beta_{1,1} & \cdots & \beta_{1,n_1} \end{pmatrix}+\begin{pmatrix} \alpha_{2,1} & \cdots &\alpha_{2,n_2} \\ \beta_{2,1} & \cdots & \beta_{2,n_2} \end{pmatrix}+\cdots+\begin{pmatrix} \alpha_{k,1} & \cdots &\alpha_{k,n_k} \\ \beta_{k,1} & \cdots & \beta_{k,n_k} \end{pmatrix}[n]$$ (2).

If $$n_k=1$$ and $$\beta_{k,n_k}=0$$ then the last term (LT) in expression (2) is equal to $$\begin{pmatrix}\alpha_{k,1} \\ 0 \end{pmatrix}=\varphi(\alpha_{k,1})=\omega^{\alpha_{k,1}}$$ and should use rule for single-argument form to assign fundamental sequences (FS) for LT, otherwise use rules 3.1-3.9 to assign FS for LT:

3.1) $$\begin{pmatrix}\cdots & \alpha+1 \\ \cdots & \beta+1 \end{pmatrix}[0]=0$$

and $$\begin{pmatrix}\cdots & \alpha+1 \\ \cdots & \beta+1 \end{pmatrix}[n+1]=\begin{pmatrix}\cdots & \alpha & \begin{pmatrix}\cdots & \alpha+1 \\ \cdots & \beta+1 \end{pmatrix}[n] \\ \cdots & \beta+1 & \beta \end{pmatrix}$$,

3.2) $$\begin{pmatrix}\cdots & \alpha+1 & \gamma+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}[0]=\begin{pmatrix}\cdots & \alpha+1 & \gamma \\ \cdots & \beta+1 & 0 \end{pmatrix}+1$$

and $$\begin{pmatrix}\cdots & \alpha+1 & \gamma+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}[n+1]=\begin{pmatrix}\cdots & \alpha & \begin{pmatrix}\cdots & \alpha+1 & \gamma+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}[n] \\ \cdots & \beta+1 & \beta \end{pmatrix}$$,

3.3) $$\begin{pmatrix}\cdots & \alpha & \gamma \\ \cdots & \beta & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha & \gamma [n] \\ \cdots & \beta & 0 \end{pmatrix}$$ if $$\gamma$$ is a limit ordinal,

3.4) $$\begin{pmatrix}\cdots & \alpha & \\ \cdots & \beta+1 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n] & \\ \cdots & \beta+1 \end{pmatrix}$$ if $$\alpha$$ is a limit ordinal,

3.5) $$\begin{pmatrix}\cdots & \alpha & \gamma+1 \\ \cdots & \beta+1 & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n] & \begin{pmatrix}\cdots & \alpha & \gamma \\ \cdots & \beta+1 & 0 \end{pmatrix}+1 \\ \cdots & \beta+1 & \beta \end{pmatrix}$$ if $$\alpha$$ is a limit ordinal,

3.6) $$\begin{pmatrix}\cdots & \alpha+1\\ \cdots & \beta\end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha & 1 \\ \cdots & \beta& \beta [n]\end{pmatrix}$$ if $$\beta$$ is a limit ordinal,

3.7) $$\begin{pmatrix}\cdots & \alpha+1 & \gamma+1 \\ \cdots & \beta & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha & \begin{pmatrix}\cdots & \alpha+1 & \gamma \\ \cdots & \beta & 0 \end{pmatrix}+1 \\ \cdots & \beta & \beta[n] \end{pmatrix}$$ if $$\beta$$ is a limit ordinal,

3.8) $$\begin{pmatrix}\cdots & \alpha\\ \cdots & \beta\end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n] \\ \cdots & \beta \end{pmatrix}$$ if $$\alpha$$ and $$\beta$$ are limit ordinals,

3.9) $$\begin{pmatrix}\cdots & \alpha & \gamma+1 \\ \cdots & \beta & 0 \end{pmatrix}[n]=\begin{pmatrix}\cdots & \alpha [n]& \begin{pmatrix}\cdots & \alpha & \gamma \\ \cdots & \beta & 0 \end{pmatrix}+1 \\ \cdots & \beta & \beta [n] \end{pmatrix}$$ if $$\alpha$$ and $$\beta$$ are limit ordinals.

The limit of this notation is Large Veblen ordinal (LVO):

• $$LVO[0]=0$$,
• $$LVO[n+1]=\begin{pmatrix}1 \\ LVO[n] \end{pmatrix}$$.

The interconnection with theta-function

The extended Veblen function and Bird/Feferman theta-functions up to SVO are connected by the next expression:

$$\theta(\Omega^{n - 1}\alpha_{1} + \cdots + \Omega^2\alpha_{n-2} + \Omega \alpha_{n-1} + \alpha_n, \gamma) = \varphi(\alpha_{1}, \ldots, \alpha_{n-2}, \alpha_{n-1}, \alpha_n, \gamma)$$

and $$\theta(\alpha, 0)$$ can be abbreviated as $$\theta(\alpha)$$. In this terms $$SVO=\theta(\Omega^\omega)$$.

For transfinary Veblen function for example:

$$\begin{pmatrix} 1\\ \varphi(\alpha_{1}, \ldots, \alpha_{n-2}, \alpha_{n-1}, \alpha_n, \gamma)\end{pmatrix}=\theta(\Omega^{\theta(\Omega^{n - 1}\alpha_{1} + \cdots + \Omega^2\alpha_{n-2} + \Omega \alpha_{n-1} + \alpha_n, \gamma)})$$,

$$\begin{pmatrix} 1\\\begin{pmatrix} 1\\ \varphi(\alpha_{1}, \ldots, \alpha_{n-2}, \alpha_{n-1}, \alpha_n, \gamma)\end{pmatrix}\end{pmatrix}=\theta(\Omega^{\theta(\Omega^{\theta(\Omega^{n - 1}\alpha_{1} + \cdots + \Omega^2\alpha_{n-2} + \Omega \alpha_{n-1} + \alpha_n, \gamma)})})$$

and so on.

Then $$LVO=\theta(\Omega^\Omega)$$.