User blog:Chronolegends/Ordinal Notations 0 : Up to the Gamma Fixed Point

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

Omega exponentiation
$$ \omega^{\alpha}[n] = \begin{cases} \alpha = 0 \ \rightarrow \ 1\\ \alpha = \beta+1 \rightarrow \ \omega^{\beta}*\omega[n] \\ \omega < \alpha < \varepsilon_0 \rightarrow \omega^{\alpha[n]}\\ \alpha \ \text{is a fixed point of}\ \omega^{\alpha} \rightarrow \alpha[n] \end{cases} $$

Epsilon numbers
Using $$^n\lambda$$ to indicate $$\lambda^{\lambda^{\lambda^{...}}} \big\}n$$

Epsilon functions enumerate the fixed points of such that $$\varepsilon_{\alpha} \rightarrow \ \omega^{\varepsilon_{\alpha}} $$

$$ \varepsilon_\alpha [n] = \begin{cases} \alpha = 0 \rightarrow \ ^n\omega \\ \alpha = \beta+1 \rightarrow \ ^n\varepsilon_\beta \\ \alpha < \zeta_0 \rightarrow \ \varepsilon_{\alpha[n]}\\ \alpha = \text{is a fixed point of}\ \varepsilon_\alpha \rightarrow \ \alpha[n]\\ \end{cases} $$

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

$$ \zeta_\alpha [n] = \begin{cases} \alpha = 0 \rightarrow \ \varepsilon^n_{\varepsilon_0} \\ \alpha = \beta+1 \rightarrow \ \varepsilon^n_{\zeta_\beta+1} \\ \alpha < \eta_0 \rightarrow \ \zeta_{\alpha[n]}\\ \alpha = \text{is a fixed point of}\ \zeta_\alpha \rightarrow \ \alpha[n]\\ \end{cases} $$

Single argument Veblen function
$$\varphi_0(\alpha) = \omega^\alpha$$
 * Equivalences with previous notations

$$\varphi_1(\alpha) = \varepsilon_\alpha$$

$$\varphi_2(\alpha) = \zeta_\alpha$$ If $$\alpha$$ is a fixed point of $$\varphi_\beta$$ and $$\beta > \lambda$$ then $$\alpha$$ is also a fixed point of $$\varphi_\lambda$$ (for example $$\omega^{\zeta_0} \rightarrow \ \zeta_0$$) Using $$\varphi^n_\alpha(\lambda)$$ to indicate $$\underbrace{\varphi_\alpha(\varphi_\alpha(...\varphi_\alpha(\lambda))}_\text{n} $$
 * Relationships of the fixed points
 * Sequences

$$ \varphi_\alpha(\lambda) [n] = \begin{cases} \alpha = 0 \rightarrow \ \omega^\lambda \\ \lambda = 0 \begin{cases} & \alpha = \delta+1 \rightarrow \varphi^n_\delta(\varphi_\delta(0)) \\ & \alpha < \Gamma_0 \rightarrow \varphi_{\alpha[n]} \end{cases} \\ \lambda = \beta+1 \begin{cases} & \alpha = \delta+1 \rightarrow \varphi^n_\delta(\varphi_{\delta+1}(\beta+1)+1)) \\ & \alpha < \Gamma_0 \rightarrow \varphi_{\alpha[n]}(\varphi_\alpha(\beta)) \end{cases}  \\ \lambda < \Gamma_0  \rightarrow \ \varphi_\alpha{\lambda[n]}\\ \alpha \ \text{is a fixed point of}\ \varphi_\alpha(\lambda) \rightarrow \ \alpha\\ \lambda \ \text{is a fixed point of}\ \varphi_\alpha(\lambda) \rightarrow \ \lambda\\ \end{cases} $$

Gamma Numbers
Define $$\varphi(1,1,0)$$ to be the first fixed point such that $$ \alpha \rightarrow \Gamma_\alpha$$

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

$$ \Gamma_\alpha [n] = \begin{cases} \alpha = 0 \rightarrow \ \underbrace{\varphi_{..._{\varphi_0}}}_\text{n}(0) \\ \alpha = \beta+1 \rightarrow \ \varphi^n_{\Gamma_{\beta}+1}(0) \\ \alpha < \varphi(1,1,0) \rightarrow \ \Gamma_{\alpha[n]}\\ \alpha = \text{is a fixed point of}\ \Gamma_\alpha \rightarrow \ \alpha[n]\\ \end{cases} $$