Constructively accessible ordinal numbers

In [11] the constructive ordinals were extended to constructive finite number classes by using systems of notations where mappings at limit ordinals are just partial recursive functions. It turned out that these systems are equivalent, both in terms of ordinals represented and the forms of the sets of notations, to extensions obtained by using mappings at limit ordinals which are partial recursive in (sets of notations for) previously defined number classes. In this article these results are extended to constructive transfinite number classes. We present a system (F, ||) which, in terms of our analogy with the classical ordinals, provides notations for the ordinals less than the first “constructively inaccessible” ordinal, and show that the above equivalence holds at least this far.