Veblen function

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. This allows to obtain ordinals beyond limit of the Cantor normal form. Below the modern version of Veblen function is described.

The Veblen hierarchy
The hierarchy is defined as follows:

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

2) for \(\alpha>0\), \(\varphi_\alpha(\gamma)=1+\gamma\)-th common fixed points for 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 \(\alpha\) such that \(\varphi_\alpha(0)=\alpha\).

Fundamental sequences for the Veblen's hierarchy

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) \(\varphi_{\beta+1}(0)[n]=\varphi_{\beta}^n(0)\), where \(\varphi^n\) denotes function iteration,

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 \(\beta\).

The extended (finitary) Veblen function
For the building of Veblen function with arbitrary amount of arguments, let's consider \(\varphi_\alpha(\gamma)\) as binary function \(\varphi(\alpha, \gamma)\). In general case it is possible to write Veblen function in next form

\(\varphi(\alpha_1, \alpha_2, ..., \alpha_n, \gamma)=\varphi(s,\alpha_k,z,\gamma)\)

where


 * \(z\) denotes a string of \(n-k\) zeros (or no one zero if \(k=n\)),


 * \(\alpha_k\) is first non-zero argument before string of zeros \(z\) prior gamma,


 * \(s\) denotes a string of arguments before \(\alpha_k\) with \(\alpha_1>0\) (or no one arguments before \(\alpha_k\) if \(k=1\)).

If \(n=k=1\) then this is just binary Veblen function.

Then generalized definition:


 * \(\varphi(\gamma)=\omega^\gamma\),


 * \(\varphi(0,\alpha_1,...,\alpha_n,\gamma)=\varphi(\alpha_1,...,\alpha_n,\gamma)\) for \(n \geq 0\),


 * if \(\alpha_1, \alpha_k>0\), where \(1\le k \le n\), then

\(\varphi(s,\alpha_{k}, z, \gamma)\) denotes the \(1+\gamma\)-th common fixed point of the functions \(\xi \mapsto \varphi(s, \beta, \xi,z)\) for each \(\beta<\alpha_k\).

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

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

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 \(m \in \{1,...,k\}\),


 * \(k, n_1,...,n_k\) are positive integers.

Fundamental sequences for limit ordinals of finitary Veblen function

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

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

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

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

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

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

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

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

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

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

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