Computable functionals and arithmetic of ordinal types (Q1105588)

The author defines, for each countable ordinal \(\gamma\), the set \(T_{\gamma}\) of functionals of type \(\gamma\): \(T_ 0\) is the set \({\mathbb{N}}\) of natural numbers, for each countable ordinal \(\gamma\), \(T_{\gamma+1}\) is the set of all (total) functions from \(T_{\gamma}\) to \({\mathbb{N}}\), and for each limit ordinal \(\gamma\), \(T_{\gamma}=\cup_{\gamma'<\gamma}T_{\gamma'}\). He then starts from an arbitrary numeration \(\nu\) of an initial segment of the countable ordinals. The first ordinal not numerated by \(\nu\), is called \(| \nu |\). He assumes that for each \(\tau <| \nu |\) a subclass \(\Phi_{\tau}\) of \(T_{\tau}\) is given. He describes a construction for extending each one of the classes \(\Phi_{\tau}\) to a class \(\rho(\Phi_{\tau})\) in such a way that the system \((\rho(\Phi_{\tau}))_{\tau <| \nu |}\) has the following nice property: it is a natural model for a formal system \(L_{\nu}\), introduced by the author, which might be described as \(\nu\)-th order arithmetic. The definition of the extending operator is not very easy. The guiding idea seems to be, very loosely speaking, that if some functional belongs to the system \(\rho(\Phi) = \cup_{\tau <| \nu |}\rho(\Phi_{\tau})\), then also a functional which can be defined recursively, with the first functional as an oracle, must belong to \(\rho(\Phi)\). The article builds upon earlier work by \textit{N. V. Belyakin} [Algebra Logika 13, 132-144 (1974; Zbl 0296.02023)], which treats the same problem up to level three. It seems that the translation from the Russia original has not been done very carefully. I found the expression ``contraction axiom'' and ``axiom scheme of contradiction'' where I would expect ``comprehension axiom'' and ``axiom scheme of comprehension'', respectively. When constructing his extension operator, the author assumes that the sets \(\Phi_{\tau}\) are ``totally ordered'', but it seems to me that he has to assume that they are well-ordered. Perhaps this is also a translator's slip.