User blog comment:MilkyWay90/Help with understanding Veblen array notation/@comment-30754445-20180811202716/@comment-30869823-20180818114019

@PsiCubed2 I understand OCFs now, it's just that it took a long time for me to grasp them (I am stuck not understanding Malhalo's now). The main thing I actually strugled with was how to practicaly compute these ordinals (like in a program). And it may just be a single tricky concept that when explained well is understandable in a few hours but finding such a good explantion isn't easy.

@Syst3ms I'm quite sure my uncountable veblen function reaches the BHO but it uses FS, I will give you the definitions here if you want to check:

Zero case: \( \phi_0(\beta)=\omega^\beta \)

Succesor case: \( \phi_{\alpha+1}(\beta)=\text{the $\beta^{th}$ fixed-point of $\phi_\alpha$} \)

Countable limit case: \( \phi_\alpha(\beta)=sup\{\phi_\gamma(\beta) : \gamma\in\alpha \} \)

Uncountable limit case: \( \phi_A(\beta)=\phi_{A[\beta]}(\beta) \)

\( \psi_{\varepsilon_{\Omega+1}+1}(0) \) should be the BHO

But this is fairly easy to extend upto (not including) the first inaccesible like (always use the first case which applies):

\( \phi_0(\beta)=\omega^\beta \)

\( \phi_{\alpha+1}(\beta)=\text{the $\beta^{th}$ weak-fixed-point of $\phi_\alpha$} \)

Limit cases:

If you can't diagonalise take the sup \( \beta\not\in \rm{cf}(\alpha) \): \( \phi_\alpha(\beta)=sup\{\phi_\gamma(\beta) : \gamma\in\alpha \} \)

If you don't waste cardinality of \( \alpha \) to diagonalise, diagonalise with \( \beta \) \( \rm{cardinality}(\beta)=\Omega_\gamma \and \rm{cf}(\alpha)=\Omega_{\gamma+1} \)  : \( \phi_\alpha(\beta)=\phi_{\alpha[\beta]}(\beta) \)

If you waste the cardinality of \( \alpha \) collapse \( \alpha \) first: \( \rm{cardinality}(\beta)<\Omega_\gamma \and \rm{cf}(\alpha)=\Omega_{\gamma+1} \) : \( \phi_\alpha(\beta)=\phi_{\phi_{\alpha}(\Omega_\gamma+1)}(\beta) \)

This defines for all ordinals wi​​​​th succesor cardinal cofanities so it defines upto the first regular limit cardinal which is the first inaccesible.

@PsiCubed2

I find this use of uncountable ordinals easier to understand if you are familiar with the fgh because it is the same principle, uncountable ordinals are used to diagonalise among countable ordinals.

This should also explain where the +1 came from, \( \psi_\Omega \) is a diagonilisation function and \( \psi_{\Omega+1} \) enumerates it's fixed points