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The Veblen functions are a hierarchy of normal functions $$\varphi_\alpha: On \rightarrow On$$, proposed by american mathematician Oswald Veblen in his article "Continuous Increasing Functions of Finite and Transfinite Ordinals" in 1908.[1] This allows to obtain ordinals beyond limit of the Cantor normal form. The modern version of the Veblen function is described below.

## The Veblen hierarchy Edit

The hierarchy is defined as follows:

1) $$\varphi_0(\gamma)=\omega^\gamma$$

2) For $$\alpha>0$$, $$\varphi_\alpha(\gamma)=$$ the $$1+\gamma$$th common fixed point of the functions $$\varphi_\beta(\xi)=\xi$$ for all $$\beta<\alpha$$.

Thus $$\varphi_1(\gamma)=\varepsilon_\gamma$$, $$\varphi_2(\gamma)=\zeta_\gamma$$ and so on ($$\varepsilon_\gamma$$ enumerates the ordinals $$\xi$$ such that $$\xi\mapsto \omega^\xi$$ and $$\zeta_\gamma$$ enumerates the ordinals $$\xi$$ such that $$\xi\mapsto \varepsilon_\xi$$ ).

For example: $$\varphi_2(2)=\zeta_2$$ is common fixed point of the functions $$\varphi_0(\xi)=\xi$$ and $$\varphi_1(\xi)=\xi$$ so far as $$\zeta_2=\omega^{\zeta_2}$$ as well as $$\zeta_2=\varepsilon_{\zeta_2}$$ and this is third common fixed point for this functions after $$\zeta_0$$ and $$\zeta_1$$.

Every non-zero ordinal $$\alpha<\Gamma_0$$ can be uniquely written in normal form for the Veblen hierarchy:

$$\alpha=\varphi_{\beta_1}(\gamma_1) + \varphi_{\beta_2}(\gamma_2) + \cdots + \varphi_{\beta_k}(\gamma_k)$$,

where

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

Note: $$\Gamma_0$$ is the smallest ordinal $$\alpha$$ such that $$\varphi_\alpha(0)=\alpha$$.

### Fundamental sequences for the Veblen hierarchy Edit

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

Fundamental sequences for the Veblen's hierarchy are defined as follows: For limit ordinals $$\alpha<\Gamma_0$$, written in normal form for the Veblen hierarchy

1.1) $$(\varphi_{\beta_1}(\gamma_1) + \varphi_{\beta_2}(\gamma_2) + \cdots + \varphi_{\beta_k}(\gamma_k))[n]=\varphi_{\beta_1}(\gamma_1) + \cdots + \varphi_{\beta_{k-1}}(\gamma_{k-1}) + \varphi_{\beta_k}(\gamma_k) [n]$$,

1.2) $$\varphi_0(\gamma)=\omega^{\gamma}$$ and $$\varphi_0(\gamma+1) [n] = \omega^{\gamma} \cdot n$$,

1.3) $$\varphi_{\beta+1}(0)[n]=\varphi_{\beta}^n(0)$$,

1.4) $$\varphi_{\beta+1}(\gamma+1)[n]=\varphi_{\beta}^n(\varphi_{\beta+1}(\gamma)+1)$$,

1.5) $$\varphi_{\beta}(\gamma) [n] = \varphi_{\beta}(\gamma [n])$$ for a limit ordinal $$\gamma<\varphi_\beta(\gamma)$$,

1.6) $$\varphi_{\beta}(0) [n] = \varphi_{\beta [n]}(0)$$ for a limit ordinal $$\beta<\varphi_\beta(0)$$,

1.7) $$\varphi_{\beta}(\gamma+1) [n] = \varphi_{\beta [n]}(\varphi_{\beta}(\gamma)+1)$$ for a limit ordinal $$\beta$$.

In rules 1.3 and 1.4, $$\varphi^n$$ denotes function iteration: $$\varphi_{\beta}^0(\gamma)=\gamma$$ and $$\varphi_{\beta}^{m+1}(\gamma)=\varphi_{\beta}(\varphi_{\beta}^{m}(\gamma))$$.

## The extended (finitary) Veblen function Edit

For the building of Veblen function with arbitrary amount of arguments, let's consider $$\varphi_\alpha(\gamma)$$ as binary function $$\varphi(\alpha, \gamma)$$.

Let z be an empty string or a string with one or more zeros $$0,0,...,0$$ and s be an empty string or an arbitrary string of ordinal variables $$\alpha_1, \alpha_2,...,\alpha_n$$ with $$\alpha_1>0$$. The binary function $$\varphi(\alpha, \gamma)$$ can be written as $$\varphi(s,\alpha, z,\gamma)$$ where both s and z are empty strings.

The extended Veblen functions are defined as follows:[2]

• $$\varphi(\gamma)=\omega^\gamma$$,
• $$\varphi(z,s,\gamma)=\varphi(s,\gamma)$$,
• if $$\alpha_{n+1}>0$$, where $$n\geq 0$$, then $$\varphi(s,\alpha_{n+1}, z, \gamma)$$ denotes the $$\gamma$$th common fixed point of the functions $$\xi \mapsto \varphi(s, \beta, \xi,z)$$ for each $$\beta<\alpha_{n+1}$$.

Every non-zero ordinal $$\alpha$$ less than the small Veblen ordinal (SVO) can be uniquely written in normal form for the finitary Veblen function:

$$\alpha=\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k)$$

where

• $$\varphi(s_1)\geq\varphi(s_2)\geq\cdots\geq\varphi(s_k)$$,
• $$s_m$$ is an arbitrary string of ordinal variables $$\alpha_{m,1}, \alpha_{m,2},...,\alpha_{m,n_m}$$ where $$m \in \{1,...,k\}$$
• $$\alpha_{m,1}>0$$ and $$\alpha_{m,i} <\varphi(s_m)$$ for $$m \in \{1,...,k\}$$ and $$i \in \{1,..,n_m\}$$,
• $$k, n_1,...,n_k$$ are positive integers.

### Fundamental sequences for limit ordinals of finitary Veblen function Edit

For limit ordinals $$\alpha<SVO$$, written in normal form for the finitary Veblen function

2.1) $$(\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k))[n]=\varphi(s_1)+\varphi(s_2)+\cdots+\varphi(s_k)[n]$$,

2.2) $$\varphi(\gamma)[n]=\left\{\begin{array}{lcr} n \quad \text{if} \quad \gamma=1\\ \varphi(\gamma-1)\cdot n \quad \text{if} \quad \gamma \quad \text{is a successor ordinal}\\ \varphi(\gamma[n]) \quad \text{if} \quad \gamma \quad \text{is a limit ordinal}\\ \end{array}\right.$$,

2.3) $$\varphi(s,\beta,z,\gamma)[0]=0$$ 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,

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

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

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

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

### Examples Edit

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

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

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

### Gamma-function Edit

Gamma-function is the function enumerates the ordinals $$\alpha$$ such that $$\varphi(\alpha,0)=\alpha$$ or in other words $$\Gamma_\beta=\varphi(1,0,\beta)=$$the $$(1+\beta)$$-th ordinal in the set $$\{\gamma|\varphi(\gamma,0)=\gamma\}$$. Same way we assign $$\Gamma_\beta[n]=\varphi(1,0,\beta)[n]$$.

## Transfinitary Veblen function Edit

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):

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

### Comparison with theta function Edit

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. In this terms $$LVO=\theta(\Omega^\Omega)$$.

According to Hyp cos' estimation, the growth rate of the function of fast-growing hierarchy $$f_{\begin{pmatrix}\omega \\ \omega \end{pmatrix}}(n)$$ is less than or equal to the growth rate of Harvey Friedman's TREE(n) function.

The fast-growing hierarchy function indexed by the Large Veblen ordinal, $$f_{LVO}(10)$$, is near to the lowest estimation of Bowers' meameamealokkapoowa oompa.

## Sources Edit

1. Veblen, Oswald. Continuous Increasing Functions of Finite and Transfinite Ordinals. Retrieved 2017-03-16.
2. Maksudov, Denis. Fundamental sequences for extended Veblen functionTraveling To The Infinity. Retrieved 2017-10-02.