I recently came across Feferman's original $\theta$ function. I understood the idea behind it. But to get a better idea of the growth I compared it with the Veblen function. When the multi-variable-Veblen-function "ran out". I thought why can't you name the Veblen functions using uncountable ordinals?
So I defined the following extension
Definition
$f'$ is the function naming the fixed points of $f$.
$f'(\alpha)=\text{the $\alpha^\text{th}$ ordinal in }\{y|y=f(y)\}$
(assuming $\Omega[\beta]=\beta$ and $\beta<\Omega$)
zero: $\varphi_0(\beta)=\omega^\beta$
succesor: $\varphi_{\alpha+1}=\varphi^{'}_\alpha$
countable: $\varphi_{\alpha}(\beta)[n]=\varphi_{\alpha[n]}(\beta)$
uncountable: $\varphi_A(\beta)=\varphi_{A[\beta]}(\beta)$
Examples
For countable ordinals $\varphi_\alpha$ is the single argument Veblen function.
$\varphi_\Omega(\alpha)=\varphi_\alpha(\alpha)$
$\varphi_{\Omega+1}(\beta)=\Gamma_\beta=\varphi(1,0,\beta)$
$\varphi_{\Omega+1+\alpha}(\beta)=\varphi(1,\alpha,\beta)$
$\varphi_{\Omega\cdot\alpha+1}(\beta)=\varphi(\alpha,0,\beta)$
$\varphi_{\Omega^2+1}(\beta)=\varphi(1,0,0,\beta)$
$\varphi_{\Omega^\alpha+1}(\beta)=\varphi(1,0,\dots,\beta)\text{ α zeros}$
$\varphi_{\Omega^\Omega+1}(0)=\text{LVO}$
Questions
These examples suggest $\varphi_{\alpha+1}(\beta)=\theta(\alpha,\beta)$
Is this the case? I feel like there is something very similar in both definitions (the next function names the fixed points of the previous).
Can it reach bigger ordinals than The Bachmann–Howard ordinal?
Does $\psi$ get stuck at a certain uncountable ordinals like $\theta$? Which?
Is it practical to extend this further, to ordinals of larger coffanity?