On Nonuniqueness in Transfinite Progressions

In the literature, there have been several attempts to classify various types of classes of sets, functions, theories, etc. by means of transfinite progressions in a manner similar to Kleene's classification of hyperarithmetic sets. (See e.g. [1], [2], [3], [5], [10]). In [6] Kreisel considers two rather general types of recursive, transfinite, progressions and shows that nonuniqueness (see Def. 3) must set in such progressions. He, however, leaves open the question whether the bounds for nonuniqueness that he obtains are the best possible.