The Veblen Hierarchy named with uncountable ordinals vs ordinal collapsing functions

The Veblen Hierarchy named with uncountable ordinals vs ordinal collapsing functions

I recently came across Fefermans 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 cant 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. falphatextthe alphatextth ordinal in yyfy assuming Omegabetabeta and betaOmegazero: varphi_0betaomegabetasuccesor: varphi_alpha1varphi_alphacountable: varphi_alphabetanvarphi_alphanbetauncountable: varphi_Abetavarphi_Abetabeta Examples For countable ordinals varphi_alpha is the single argument Veblen function. varphi_Omegaalphavarphi_alphaalpha varphi_Omega1betaGamma_betavarphi1,0,beta varphi_Omega1alphabetavarphi1,alpha,beta varphi_Omegacdotalpha1betavarphialpha,0,beta varphi_Omega21betavarphi1,0,0,beta varphi_Omegaalpha1betavarphi1,0,dots,betatext zeros varphi_OmegaOmega10textLVO Questions These examples suggest varphi_alpha1betathetaalpha,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 BachmannHoward 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