Forum:Does this ordinal exist ?

Today I learned more about Madore's Psi function and wondered if there existed an ordinal $$\alpha$$ such that $$\alpha\mapsto\psi_{\alpha}(0)$$, or the ordinal $$\psi_{\psi_{\psi_{\psi_{._{._{._{\psi_0(0)}.}.}.}(0)}(0)}(0)}(0)$$. Is it even valid to speak of such an ordinal ? If so, what comes after this in ordinal notations ?

Help would be appreciated in answering this question. Thank you.Edwin Shade (talk) 04:25, November 20, 2017 (UTC)