User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-24509095-20140507175906/@comment-5150073-20140508095225

Deedlit, $$\phi(1,I+1)$$ isn't good because we have to define additional set of rules for $$\psi(\alpha,\beta)$$. That gives unnecessary complexity to the notation. We can express the complicated one from the simple as:

$$\psi_I(\phi(1,I+1)) = \psi_I(\psi_{\Omega_{I+1}}(0))$$

$$\psi_I(\phi(2,I+1)) = \psi_I(\psi_{\Omega_{I+1}}(\Omega_{I+1}))$$

$$\psi_I(\phi(\omega,I+1)) = \psi_I(\psi_{\Omega_{I+1}}({\Omega_{I+1}}^\omega))$$

$$\psi_I(\Gamma_{I+1}) = \psi_I(\psi_{\Omega_{I+1}}({\Omega_{I+1}}^{\Omega_{I+1}}))$$

Moreover, they will catch on $$\psi_I(\psi_{\Omega_{I+1}}(\varepsilon_{\Omega_{I+1}})$$ by the same reason why original $$\vartheta$$ and $$\psi$$ catches at BHO.

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King, well, but I guess that then $$\psi_{\Omega_{I+1}$$ would be much much more powerful than $$\psi_{I+1}$$.