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Introducing some low level ordinal notations. Using \lambda [n]  to denote the nth member of the fundamental sequence of \lambda

Omega exponentiation


\omega^{\alpha}[n] = 
\begin{cases}
\alpha = 0 \ \rightarrow \ 1\\ 
\alpha = \beta+1 \rightarrow \ \omega^{\beta}*\omega[n] \\
\alpha \ \text{is a fixed point of}\ \omega^{\alpha} \rightarrow \alpha[n] \\
\alpha \ \text{is a limit ordinal} \rightarrow \omega^{\alpha[n]} 
\end{cases}

Epsilon numbers

These are the fixed points of the \omega^\alpha function.

Using ^n\omega^\lambda to indicate \underbrace{\omega^{...^{\omega^\lambda}}}_\text{n}


\varepsilon_\alpha [n] = 
\begin{cases}
\alpha = 0 \rightarrow \ ^n\omega^1 \\
\alpha = \beta+1 \rightarrow \ ^n\omega^{\varepsilon_\beta+1} \\
\alpha \ \text{is a fixed point of}\ \varepsilon_\alpha \rightarrow \ \alpha[n]\\ 
\alpha \ \text{is a limit ordinal} \rightarrow \ \varepsilon_{\alpha[n]}\\

\end{cases}

Zeta Numbers

These are the fixed points of the \varepsilon_\alpha function.

Using \varepsilon^n_\lambda to indicate \underbrace{\varepsilon_{..._{\varepsilon_\lambda}}}_\text{n}


\zeta_\alpha [n] = 
\begin{cases}
\alpha = 0 \rightarrow \ \varepsilon^n_0 \\
\alpha = \beta+1 \rightarrow \ \varepsilon^n_{\zeta_{_\beta+1}} \\
\alpha \ \text{is a fixed point of}\ \zeta_\alpha \rightarrow \ \alpha[n]\\ 
\alpha \ \text{is a limit ordinal}  \rightarrow \ \zeta_{\alpha[n]}\\
\end{cases}

Eta Numbers

These are the fixed points of the \zeta_\alpha function.

Using \zeta^n_\lambda to indicate \underbrace{\zeta_{..._{\zeta_\lambda}}}_\text{n}


\eta_\alpha [n] = 
\begin{cases}
\alpha = 0 \rightarrow \ \zeta^n_0 \\
\alpha = \beta+1 \rightarrow \ \zeta^n_{\eta_{_\beta+1}} \\
\alpha \ \text{is a fixed point of}\ \eta_\alpha \rightarrow \ \alpha[n]\\ 
\alpha \ \text{is a limit ordinal}  \rightarrow \ \eta_{\alpha[n]}\\
\end{cases}

Single argument Veblen function

Using \varphi^n_\alpha(\lambda) to indicate \underbrace{\varphi_\alpha(\varphi_\alpha(...\varphi_\alpha(\lambda))}_\text{n}


\varphi_\alpha(\lambda) [n] = 
\begin{cases}
\alpha \ \text{is a fixed point of}\ \varphi_\alpha(\lambda) \rightarrow \ \alpha[n]\\
\lambda \ \text{is a fixed point of}\ \varphi_\alpha(\lambda) \rightarrow \ \lambda[n]\\
\alpha = 0  \rightarrow \ \omega^{\lambda}[n] \\
\lambda = 0 \begin{cases} & \alpha = \delta+1  \rightarrow \varphi^n_\delta(0) \\ & \alpha \ \text{is a limit ordinal}\ \rightarrow \varphi_{\alpha[n]}(\lambda) \end{cases} \\
\lambda = \beta+1  \begin{cases} & \alpha = \delta+1 \rightarrow \varphi^n_\delta(\varphi_{\delta+1}(\beta)+1)) \\ & \alpha \ \text{is a limit ordinal}\ \rightarrow \varphi_{\alpha[n]}(\varphi_\alpha(\beta)+1) \end{cases}  \\
\lambda \ \text{is a limit ordinal}\ \rightarrow \ \varphi_\alpha{(\lambda[n])}\\ 
\end{cases}

Relationship to the previous functions

\varphi_0(\alpha) = \omega^\alpha

\varphi_1(\alpha) = \varepsilon_\alpha

\varphi_2(\alpha) = \zeta_\alpha

\varphi_3(\alpha) = \eta_\alpha

Relationships of the fixed points

If \alpha is a fixed point of \varphi_\beta and \beta > \lambda then \alpha is also a fixed point of \varphi_\lambda (for example \omega^{\zeta_0} \rightarrow \ \zeta_0)


Gamma Numbers

These are the fixed points of the \varphi_\alpha(\lambda) function.


\Gamma_\alpha [n] = 
\begin{cases}
\alpha = 0 \rightarrow \ \underbrace{\varphi_{..._{\varphi_0(0)}}}_\text{n}(0) \\
\alpha = \beta+1 \rightarrow \ \underbrace{\varphi_{...\varphi_{_{\Gamma_{\beta}+1}}(0)}(0)}_\text{n}\\
\alpha \ \text{is a fixed point of}\ \Gamma_\alpha \rightarrow \ \alpha[n]\\ 
\alpha \ \text{is a limit ordinal}  \rightarrow \ \Gamma_{\alpha[n]}\\
\end{cases}

Relationship between the gamma and phi functions.

Note: Since \Gamma_0 goes beyond the basic veblen function, the gamma function represents ordinals in the three argument veblen function(also called the extended veblen function), which it relates to in the following way: \Gamma_\alpha =  \varphi(1,0,\alpha)


Gamma fixed point.

\varphi(1,1,0) , is the ordinal i refer to as "gamma fixed point". It is the first fixed point such that \alpha \rightarrow \Gamma_\alpha and thus marks the limit to the usefulness of the gamma function, which is illustrated by listing its fundamental sequence:

\varphi(1,1,0)[n] = \underbrace{\Gamma_{..._{\Gamma_0}}}_\text{n}

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