User blog comment:KthulhuHimself/Norminals Revisited./@comment-1605058-20151130200835

If you want all these to be completely well-defined, then you certainly have to specify exactly how your syntax looks like. What I have in mind in particular is the fact that you don't say anything about how you introduce quantifications over predicates of earlier languages (i.e. what new characters you add, what are the well-founded formulas in new language).