 where \(\varepsilon_{\Omega_{\omega}+1} \) is the least epsilon number after \(\Omega_{\omega}\), i.e., the least ordinal \(\eta&gt;\Omega_{\omega}\) such that \(\omega^{\eta}=\eta\). The proof-theoretic ordinal of \((\Pi^1_1{\Hy}\bCA)+{{\BI}}\) is \({\psi_{\sOmega_{1}}(\varepsilon_{\Omega_{\omega}+1})}\). Although the definition of the set \(C^{{\Omega_{\omega}}}(\varepsilon_{\Omega_{\omega}+1},0)\) and its ordering is set-theoretic, it turns that it also has a purely primitive recursive definition which can be given in a fragment of PRA. Thus the set-theoretic presentation mainly serves the purpose of a “visualization” of an elementary well-ordering.