User blog comment:Ubersketch/Collapsing the first 1-dinal/@comment-32697988-20190702175411/@comment-24725252-20190708233623

Extended Madore's psi is Madore's psi, but with higher cardinailites added and more psi functions, and in the definition of those functions is with W_n included. Why did you get that, and I belive not. \(\psi(\varepsilon_{\Omega + 2})\) is actaully much bigger than you think. What are ordinals like \(\psi(\varepsilon_{\Omega +  1} + 1)\), \(\psi(\varepsilon_{\Omega +  1} + \Omega)\), (\psi(\varepsilon_{\Omega +  1} *2)\), (\psi(\varepsilon_{\Omega +  1}^2)\), etc.

So, because the function is constant (or gets "stuck") between (\varepsilon_{\Omega + 1}\) and \(\overline{\Omega}\), \(\overline{\psi}(\overline{\Omega} + 1)\) is no larger than \(\psi(\varepsilon_{\Omega + 1} + 1)\). Also, \(C(\overline{\Omega} + \Omega)\) cannot generate the first gixed point of n-> \(\psi(\varepsilon_{\Omega + 1} + n)\), so (\overline{\psi}(\overline{\Omega} + \Omega)\) is the fixed point of that. This continues, and so (\overline{\psi}(\overline{\Omega} + n)\ = \(\psi(\varepsilon_{\Omega +  1} + n)\), when n =< \(\varepsilon_{\Omega + 1}\). So, then \overline{\Omega} acts like (\varepsilon_{\Omega +  1}\), resulting in the limit is what is said it is.