User blog comment:P進大好きbot/Please Help me on study of Pair Sequence System (2-rowed Bashicu Matrix System)/@comment-35870936-20180813215517/@comment-35470197-20180814002848

Lemma 1.2 (a) in [Buc]: \(\Omega_2 = \psi_2(0)\).

Lemma 1.2 (c) in [Buc]: \(\psi_1(\psi_2(\psi_3(0))) < \Omega_2\).

Therefore we have \(\psi_1(\psi_2(\psi_3(0))) < \psi_2(0)\).

Lemma 1.2 (d) in [Buc]: \(\alpha \leq \beta\) impies \(\psi_0(\alpha) \leq \psi_0(\beta)\).

Therefore we have \(\psi_0(\psi_1(\psi_2(\psi_3(0)))) \leq \psi_0(\psi_2(0))\). Also we have \(\psi_0(\psi_1(\psi_2(0))) = \psi_0(\psi_2(0))\).

Therefore \(\psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\psi_2(0))\).

Lemma 1.2(a) in [Buc]: \(\Omega_3 = \psi_3(0)\)

Lemma 1.2 (c) in [Buc] \(\psi_2(0) + 1 < \Omega_3\).

Therefore we have \(\psi_0(\psi_2(0) + 1) \leq \psi_0(\psi_3(0))\) again by Lemma 1.2 (d) in [Buc].

On the other hand, we have \(G_0(\psi_0(\psi_2(0))) = \{0\} \in \psi_0(\psi_2(0))\) and \(\psi_0(\psi_2(0)) \in \varepsilon_{\Omega_{\omega}+1}\). Therefore we have \(\psi_0(\psi_2(0)) \in C_0(\psi_0(\psi_2(0)))\) by Lemma 1.9 in [Buc].

Lemma 1.6 (a) in [Buc]: \(\psi_0(\psi_2(0) + 1) = \psi_0(\psi_2(0)) \times \omega\).

Therefore we have \(\psi_0(\psi_2(0)) \leq \psi_0(\psi_2(0) + 1)\).

As a consequence, we obtain \begin{eqnarray*} \psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\psi_2(0)) < \psi_0(\psi_2(0) + 1) \leq \psi_0(\psi_3(0)). \end{eqnarray*} This contradicts what you state. Please tell me precisely which (in)equality is incorrect.

[Buc] W. Buchholz, A new system of proof-theoretic ordinal functions, Annals of Pure and Applied Logic, Volume 32, 1986, pp. 195--207.