## FANDOM

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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:

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