Disclaimer: I am not a professional mathematician.
Background: I have been researching large countable ordinals for awhile & I think the Veblen function is particularly eloquent. My understanding is that $\Gamma_0$, the small Veblen ordinal& the large Veblen ordinal are all significantly smaller than the first uncountable ordinal$\omega_1$. Having some extra time during quarantine, I had an idea to extend the Veblen function to the domain of uncountable ordinals & created the following notation. I would like to know how far this notation reaches & if anything similar already exists.
Note: For the sake of brevity I have omitted numerous steps from the hand written derivation of this notation.
Consider $\phi_0'(\alpha)=\omega_\alpha$ such that:$$\phi_0'(0)=\omega_0=\omega$$$$\phi_0'(1)=\omega_1$$
Nesting these functions results in:$$\phi_0'(\phi_0'(0))=\omega_\omega$$$$\phi_0'(\phi_0'(\phi_0'(0)))=\omega_{\omega_\omega}$$
Next, consider the supremum of the previous nestings:$$\phi_1'(0)=\sup\{\omega, \omega_\omega, \omega_{\omega_\omega},...\}$$
$\phi_1'(0)$ is then the first fixed point of $\phi_0'(\alpha)$ which correlates to $\phi_1(0)=\varepsilon_0$ being the first fixed point of $\phi_0(\alpha)=\omega^\alpha$ in the original Veblen function.
Continuing as in the original case, we eventually hit the limit of our single variable function. At this point ($\Gamma_0$ in the original), we turn to the multivariable function:$$\phi_{1,0}'(0)=\phi'(1,0,0)=\sup\{\phi_1'(0),\phi_{\phi_1'(0)}'(0),\phi_{\phi_{\phi_1'(0)}'(0)}'(0),...\}$$
Again, like in the original case with the small Veblen ordinal, we eventually get stuck. At this point we move to the version of the Veblen function with a transfinite number of variables.
$$\phi'(1@\omega)=\sup\{\phi'(1,0),\phi'(1,0,0),\phi'(1,0,0,0)\}$$
Eventually this notation reaches as cap as well. In the orginal case, this is called the large Veblen ordinal & is the cap of the original Veblen function. In the expansion, we simply iterate our 'jump' operator:$$\phi_0''(0)=\sup\{\phi'(1@0),\phi'(1@\omega),\phi'(1@\varepsilon_0),...\}$$
We can keep going by iterating the base function such that:
$$\Phi_0(0)=\sup\{\phi_{0}'(0), \phi_0''(0), \phi_0'''(0),...\}$$
Given the general form $\alpha_\gamma^\beta(\delta)$ we are essentially:
- maxing out $\delta \leadsto$ incrementing $\gamma$
- maxing out single variable $\gamma \leadsto$ multivariable $\gamma$
- maxing out multivariable $\gamma \leadsto$ incrementing $\beta$
- maxing out $\beta \leadsto$ incrementing $\alpha$
Repeating the process a couple more times results in:$$\sup\{\Phi_0(0),\Phi_0'(0),\Phi_0''(0),...\}=\psi_0(0)$$$$\sup\{\psi_0(0),\psi_0'(0),\psi_0''(0),...\}=\Psi_0(0)$$
Looping repeatedly reminded me of the original Veblen function process & so I created the following function:$$\Xi(\alpha, \beta, \gamma, \delta)=\alpha_\gamma^\beta(\delta)$$
Such that:$$\Xi(0,0,0,0)=\phi_0(0)=1$$$$\Xi(0,0,0,1)=\phi_0(1)=\omega$$$$\Xi(0,0,1,0)=\phi_1(0)=\varepsilon_0$$$$\Xi(0,1,0,0)=\phi_0'(0)=\omega$$$$\Xi(0,1,0,1)=\phi_0'(1)=\omega_1$$$$\Xi(1,0,0,0)=\Phi_0(0)$$$$\Xi(2,0,0,0)=\psi_0(0)$$$$\Xi(3,0,0,0)=\Psi_0(0)$$
If you made it this far, thank you for taking the time. To reiterate, how far does this notation reach& does anything like this already exist?