User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20191006023842/@comment-35470197-20191009093825

Taranovsky just proposed the canonicity, and assigned canonical ones to sepecific examples, as far as I understand. The canonical assignements are not unique, and if you extend it without any reasoning based on set theories, they will be actually ambiguous and non-canonical. Then Taranovsky's assumption of the consistency with the existence of the assignment is irrelevant to the results. It is not an assumption that ensures the existence of any functions given by filling a finite table with undefined values. It is not even formalised by the Skolem normal form.

Is there no precise definition of the correspondence in your and Scorcher007‎'s context? If so, how could you state "this correspondence is right" or "it is incorrect because it has a gap"?