Reduction of higher type levels by means of an ordinal analysis of finite terms

Abstract

In his Habilitationsschrift [6] Helmut Schwichtenberg has proved the following result about definitions of ordinal recursive functionals (a more precise formulation will be given in subsection 3.3 hereafter): if such a definition, say of functional Qi, contains subdefinitions in which auxiliary functionals of higher type levels than Qi are introduced by primitive or transfinite recursion, then these “detours through higher type levels” can be eliminated by means of transfinite recursion over a new, canonicly constructed wellordering, which has, roughly spoken, an exponentially bounded order type. Together with results of Kreisel and Tait in the other direction (see also [9]), this reveals an interesting connection between (definition by detours through) higher type levels on the one hand and transfinite recursion on the other hand. The purpose of the present paper is to contribute to a deeper insight into this rather fundamental connection, namely by means of an alternative, conceptually more simple proof of Schwichtenberg’s result. This alternative proof makes, just as Schwichtenberg’s original proof (see [6] or [7]), also use of a representation of the relevant functionals by terms, but now these terms are ordinary finite terms, which are (technically as well as mentally) much easier to be managed than the infinite terms that play a central role in [6] and [7]. For one thing: coding finite terms by numbers is an almost trivial affair, whereas this is considerably more complicated for infinite terms (and, moreover, in order to manipulate codes of the latter one needs additional technical tools like, e.g., the primitive recursion theorem for indices of primitive recursive functions). What now comes instead of the manipulation of infinite terms is an analysis of a certain successor relation, in the style of San&is [5] and Howard [2], between finite terms. This successor relation looks partly like a reduction relation of the