User blog comment:WaxPlanck/Chess Ordinals and Uncountable Growth/@comment-5029411-20180123010330

We can construct branches that is analogous to 3-dimensional but higher-dimensional. So $$\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_4}$$ = $$\omega_1^{{\mathfrak{Ch}_{\!\!\!\!\sim}}_3}$$ = $$\omega_1$$.