User blog comment:Edwin Shade/The Grand List Of Transfinite Ordinals/@comment-11227630-20171222030119/@comment-1605058-20171222100700

\(\omega^{\mathfrak{Ch}_{\!\!\!\!\sim}}_1\) has not been proven to be equal to \(\omega_1\), though it is conjectured to be.

Beyond that, \(0^\#\) and \(0^\dagger\) are not ordinals - they are reals.