User blog:Hyp cos/Catching hierarchy fails

Catching hierarchy fails beyond \(C_1(\Omega)\) (or \(C_{\Omega_2}(\Omega)\)).

Recall the definition: But what's \(C_1(\Omega)\)? The \(\Omega\) works as the diagonalizer of C, which is outside, so \(C_1(\Omega)\) refers to the outside C.
 * 1) \(C_\pi(0)=\psi_\pi(\Omega_\omega)\).
 * 2) If \(C_\pi(\alpha)=\psi_\pi(\beta)\), then \(C_\pi(\alpha+1)=\psi_\pi(\gamma)\) where \(\psi(\gamma)\) is the least ordinal that \(g_{\psi(\gamma)}\) is comparable to \(f_{\psi(\gamma)}\) and \(\gamma>\beta\), and both \(\psi_\pi(\beta)\) and \(\psi_\pi(\gamma)\) are full-simplified.
 * 3) \(\pi\) is the diagonalizer of \(C_\pi\) function.
 * 4) For limit \(\alpha\), \(C_\pi(\alpha)[n]=C_\pi(\alpha[n])\).

Now consider \(C_1(\Omega+1)\). \(C_1(\Omega+1)=\psi_1(\gamma)\) where \(\psi(\gamma)\) is the least ordinal that \(g_{\psi(\gamma)}\approx f_{\psi(\gamma)}\) and \(\gamma>\beta\) where \(C_1(\Omega)=\psi_1(\beta)\). However, \(\beta\) is undetermined, so \(C_1(\Omega+1)\) gets undefined.

As a result, in the comparisons, only \(C(C_1(\Omega)+1)\), \(C(C_1(\Omega)2)\), \(C(C_1(\Omega)^2)\), etc. are used.