User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-24509095-20140507175906/@comment-24509095-20140508090729

First, let $$\psi_{\Omega_{I+1}}(n)$$ be defined as in this blog post and let $$\psi_{I+1}(n)$$(Ikosarakt, if you're reading this, I prefer I+1) be defined WITHOUT $$\varphi$$. In the latter case, $$\psi_{I+1}(0)=\varepsilon_{I+1}$$, $$\psi_{I+1}(\Omega_{I+1})=\psi_{I+1}(\psi_{I+1}(\psi_{I+1}(...)))$$, and so on... But, $$\psi_{\Omega_{I+1}}(\Omega_{I+1})=\psi_{I+1}({\Omega_{I+1}}^{\Omega_{I+1}})$$. This looks very similar to the relationship between the $$\theta$$ function and the $$\psi$$ function because $$\theta(\Omega)=\psi(\Omega^{\Omega})$$.