User blog comment:Pellucidar12/Attempt at a FGH related notation/@comment-29915175-20170416151157/@comment-1605058-20170418171952

It is impossible to define FSes for uncomputable ordinals algorithmically, but you can define them explicitly by other means. See here for an example on how to define it (note that for higher admissibles it doesn't exactly work because of some admissible ordinal misunderstandings some years ago).