User blog:Fejfo/Uncountable indexed veblen function

This is the 2 argument Veblen function: It is usually extend to mutliple arguments, then to transfinitly many arguements and dropped to be replaced by unintuitive OCFs.
 * 1) \( \phi_0(\beta)= \omega^\beta \)
 * 2) \( \phi_{\alpha+1}(\beta)=\text{the $\beta^{th}$ ordinal in } \{ \gamma\mid\gamma=\phi_{\alpha}(\gamma)\} \)
 * 3) \( \phi_{\alpha}(\beta)[n]=\phi_{\alpha[n]}(\beta) \)

But I defined:
 * 1) \( \phi_0(\beta)= \omega^\beta \)
 * 2) \( \phi_{\alpha+1}(\beta)=\text{the $\beta^{th}$ ordinal in } \{ \gamma\mid\gamma=\phi_{\alpha}(\gamma)\} \)
 * 3) \( \phi_{\alpha}(\beta)[n]=\phi_{\alpha[n]}(\beta) \) if \( \beta\not\in{\rm cf}(\alpha) \)
 * 4) \( \phi_{\alpha}(\beta)=\phi_{\alpha[\beta]}(\beta) \) if \( \beta\in{\rm cf}(\alpha) \)

Analasis
For countable α this will be the normal veblen function.

\( \phi_{\Omega}(\beta)=\phi_\beta(\beta) \) so \( \phi_{\Omega+1} \) enumerates the fixed points of the normal veblen function (the Γ-function)

This continues \( \phi_{\Omega+1+\alpha}(\beta)=\phi(1,\alpha,\beta) \) (since rule 4 never applies). And even \( \phi_{\Omega\cdot\alpha+1}(\beta)=\phi(\alpha,0,\beta) \) Or \( \phi_{\Omega^2+1}(\beta)=\phi(1,0,0,\beta) \) Until \( \phi_{\Omega^\alpha+1}(\beta)=\phi(1@\alpha,\beta) \)

So \( \phi_{\Omega^\Omega+1}(0) \) is the larger veblen ordinal but this notation doesn't end there. I hope you can see this yields very simular results to Feferman's original \( \theta \) function. \[ \phi_{\alpha+1}(\beta)= \theta(\alpha,\beta) \]

Using \(  \Omega_2 \)
The current defintion wastes \( \Omega_2 \) but we can fixed that by adding one simple rule: This applies the veblen function if you have cardinalities to spare so \( \phi_{\Omega_2}(\beta)=\phi_{\phi_{\Omega_2}(\Omega+1)}=\phi_{\Gamma_{\Omega+1}}(\beta) \) \( \phi_{\Omega_3}(\beta)=\phi_{\phi_{\Omega_3}(\Omega_2+1)}(\beta)=\phi_{\Gamma_{\Omega_2+1}}(\beta) \)
 * 1) \( \phi_0(\beta)= \omega^\beta \)
 * 2) \( \phi_{\alpha+1}(\beta)=\text{the $\beta^{th}$ ordinal in } \{ \gamma\mid\gamma=\phi_{\alpha}(\gamma)\} \)
 * 3) \( \phi_{\alpha}(\beta)[n]=\phi_{\alpha[n]}(\beta) \) if \( \beta\not\in{\rm cf}(\alpha) \)
 * 4) \( \phi_{\alpha}(\beta)=\phi_{\alpha[\beta]}(\beta) \) if \( \exists\gamma\mid\Omega_\gamma\le\beta<\Omega_{\gamma+1}={\rm cf}(\alpha) \) with other words only collapse with β if you can't collapse with the next cardinal
 * 5) \( \phi_{\alpha}(\beta)=\phi_{\phi_{\alpha}(\Omega_\gamma+1)}(\beta) \) if \( \exists\gamma\mid {\rm cf}(\alpha)=\Omega_{\gamma+1} \)

This cascade of veblen functions continues until \( \Omega_\omega \) at which point rule 3 applies again: \( \phi_{\Omega_\omega}(\beta)[n]=\phi_{\Omega_n}(\beta) \)

So this is defined upto the first regular non-succesor cardinal, which is the first inaccesible cardinal.