User blog comment:Edwin Shade/Ordinal Questions/@comment-1605058-20171128082743/@comment-2602:306:CD4B:DFF0:2CAF:DADB:1D03:6475-20171129014610

No, $$\psi_I(0)$$ is the first omega fixed point. Trivially, $$\psi(\psi_I(0))<\omega_1$$, so that's probably not quite what you want.