User blog comment:Wythagoras/Catching function in normal ordinals/@comment-11227630-20140203095754/@comment-32697988-20171105190831

Weak catching function

\(c(0)=\varepsilon_0\)

\(c(1)=\varepsilon_1\)

\(c(2)=\varepsilon_2\)

\(c(\omega)=\varepsilon_\omega\)

\(c(\omega+1)=\varepsilon_{\omega+1}\)

\(c(\omega2)=\varepsilon_{\omega2}\)

\(c(\omega^2)=\varepsilon_{\omega^2}\)

\(c(\omega^\omega)=\varepsilon_{\omega^\omega}\)

\(c(c(0))=\varepsilon_{\varepsilon_0}\)

\(c(c(\omega))=\varepsilon_{\varepsilon_\omega}\)

\(c(c(c(0)))=\varepsilon_{\varepsilon_{\varepsilon_0}}\)

\(c(\Omega)=\zeta_0\)

\(c(\Omega+1)=\varepsilon_{\zeta_0+1}\)

\(c(\Omega+2)=\varepsilon_{\zeta_0+2}\)

\(c(\Omega+\omega)=\varepsilon_{\zeta_0+\omega}\)

\(c(\Omega+c(\Omega))=\varepsilon_{\zeta_02}\)

\(c(\Omega+c(\Omega)+1)=\varepsilon_{\zeta_02+1}\)

\(c(\Omega+c(\Omega)\times\omega)=\varepsilon_{\zeta_0\omega}\)

\(c(\Omega+c(\Omega)^2)=\varepsilon_{\zeta_0^2}\)

\(c(\Omega+c(\Omega+1))=\varepsilon_{\varepsilon_{\zeta_0+1}}\)

\(c(\Omega2)=\zeta_1\)

\(c(\Omega3)=\zeta_2\)

\(c(\Omega\omega)=\zeta_\omega\)

The c function is the same as psi

\(c(\varepsilon_{\Omega+1})=BHO=\psi_2(0)\)

\(\varepsilon_{\Omega+1}=c_1(0)\)

\(c(c_1(0))=\psi(\varepsilon_{\Omega+1})\)

\(c_1(1)=\psi_2(1)=\varepsilon_{\Omega+2}\)

\(c(c_1(1))=\psi(\varepsilon_{\Omega+2})\)

\(c(c_1(2))=\psi(\varepsilon_{\Omega+3})\)

\(c(c_1(\omega))=\psi(\varepsilon_{\Omega+\omega})\)

\(c(c_1(c(0)))=\psi(\varepsilon_{\Omega+\varepsilon_0})\)

\(c(c_1(c(\Omega)))=\psi(\varepsilon_{\Omega+\zeta_0})\)

\(c(c_1(c(c_1(0))))=\psi(\varepsilon_{\Omega+BHO})\)

\(c(c_1(\Omega))=\psi(\varepsilon_{\Omega2}\)

\(c(c_1(\Omega+1))=\psi(\varepsilon_{\Omega2+1})\)

\(c(c_1(\Omega+2))=\psi(\varepsilon_{\Omega2+2})\)

\(c(c_1(\Omega2))=\psi(\varepsilon_{\Omega3})\)

\(c(c_1(\Omega\omega))=\psi(\varepsilon_{\Omega\omega})\)

\(c(c_1(\Omega^2))=\psi(\varepsilon_{\Omega^2})\)

\(c(c_1(\Omega^\omega))=\psi(\varepsilon_{\Omega^\omega})\)

\(c(c_1(\Omega^\Omega))=\psi(\varepsilon_{\Omega^\Omega})\)

\(c(c_1(c_1(0)))=\psi(\varepsilon_{\varepsilon_{\Omega+1}})\)

\(c(\Omega_2)=\psi(\zeta_{\Omega+1})\)