User blog comment:Edwin Shade/Can Chess Ordinals Produce Functions With Uncountable Growth Rates ?/@comment-1605058-20171222153040/@comment-1605058-20171222185536

I'm not sure where you're getting at. \(\mathfrak{Ch}_n\) is a well-defined sequence whose supremum is not \(\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}\), which we can prove. It's not a matter of "considering" it to be a supremum or not.

Also, I've never said your function is not well-defined - it is, if we assume you did define all the other fundamental sequences. However, I don't see in what way it would have "uncountable growth rate".