I have not previously seen anything online that dives deeply into On:
In Conway's notation On denotes the ordinal numbers (and No denotes the set of all surreal Numbers). Basically the elements of On are just von Neumann ordinals. –Source
I would appreciate feedback on the following attempt to write large countable ordinals (& the functions that generate them) in Conway notation (my primary source of information in creating these constructions was Large Countable Ordinals):
Epsilon-Nought
$$\varepsilon_{0}=\{\omega,\omega^\omega,\omega^{\omega^\omega},…|\}$$
Feferman-Schutte Ordinal
$$\Gamma_0=\phi_{1,0}(0)=\{\phi_0(0),\phi_{\phi_0(0)}(0),\phi_{\phi_{\phi_0(0)}(0)}(0),…|\}$$
Small Veblen Ordinal
$$SVO=\{\phi_1(0), \phi_{1,0}(0), \phi_{1,0,0}(0),…|\}$$
Bachmann-Howard Ordinal
$$BHO=\{\psi(\Omega),\psi(\Omega^\Omega),\psi(\Omega^{\Omega^\Omega}),…|\}$$
Additionally, any online resources related to On would be greatly appreciated.
Best Answer
I have to say I know as little about large countable ordinals as I do about games. I actually didn't know the ordinal $\Gamma_0$ was thought of as the first impredicative ordinal, had a name and so on.
Regarding your definitions, the function $\phi_{\gamma}(\alpha)$ should also be greater than every ordinal $\phi_{\eta}^{\circ n}(\phi_{\gamma}(\beta)+1)$ for $\eta < \gamma$, $n \in \mathbb{N}$ and $\beta<\alpha$. I am not sure about what you mean by Vleben function, and I don't know about SVO, LVO, BHO.
Perhaps something you might find interesting is a phenomenon noticed by Conway and expended upon by Gonshor: the functions $\phi_{\gamma}$ can be extended to $\mathbf{No}$ in a natural way.
For $x=\{L\ | \ R\} \in \mathbf{No}$, you must know about $\omega^x=\phi_0(x)=\{0,\mathbb{N}\ \phi_0(L)\ | \ 2^{-\mathbb{N}} \ \phi_0(R)\}$. Then the class of numbers $e$ such that $\omega^e=e$ is parametrized by $\varepsilon_x=\phi_1(x):=\{\phi_0^{\circ \mathbb{N}}(0),\phi_0^{\circ \mathbb{N}}(\phi_1(L)+1)\ | \ \phi_0^{\circ \mathbb{N}}(\phi_1(R)-1)\}$, and one can keep going on. At every stage $0<\gamma$, the function $\phi_{\gamma}$ parametrizes the class of numbers $e$ with $\forall \eta < \gamma,\phi_{\eta}(e)=e$.
As for sources on $\mathbf{On}$, since this is just the class of ordinals, you can just look into this. I don't know that new insight on ordinal numbers has been gained by seeing them as surreal numbers, at least not in a significant way.
edit: to be more explicit, Conway's so-called $\omega$-map is defined inductively as $x \longmapsto \omega^x:=\{0,n \ \omega^{x_L}:n \in \mathbb{N} \wedge x' \in x_L \ | \ 2^{-n} \ \omega^{x''}:n \in \mathbb{N} \wedge x'' \in x_R\}$ where $x=\{x_L \ | \ x_R\}$. This is done so as to yield $r \omega^x < s \omega^y$ whenever $x<y$ and $r,s$ are strictly positive real numbers.
For $\phi_1$, this is $\phi_1(x):=\{\phi_0^{\circ n}(0),\phi_0^{\circ n}(\phi_1(x')+1): x' \in x_L \wedge n \in \mathbb{N} \ | \ \phi_0^{\circ n}(\phi_1(x'')-1): x' \in x_R \wedge n \in \mathbb{N}\}$, where $f^{\circ n}$ denotes the $n$-fold composition of a function $f$ with itself.
You can find both of those in Conway's On Numbers and Games, Chapter 3 and in Gonshor's An Introduction to the Theory of Surreal Numbers, Chapters 5 and 9. This is also discussed in some detail in Sections 5 and 6 of the pre-print Surreal Substructures (the formula for fixed points parametrizations is Remark 6.23).