Notation systems for infinitary derivations

It is one of Kurt Schiitte's great merits to have established cut-elimination on infinitary derivations as a powerful and elegant tool for proof-theoretic investigations. Compared to the Gentzen-Takeuti approach where ordinals are assigned to finite derivations in a rather cryptic way, the use of infinitary derivations together with the canonical assignment of ordinals as lengths of derivations provides a very perspicious and conceptually clear-cut method which has proved successful even with respect to the strongest systems analyzed till now. But on the other side something is lost when passing from finite to (unrestricted) infinite derivations, in so far as along these lines one only obtains information on the provable/-/~-sentences of a formal theory, while Gentzen's method-if successfully applied-yields stronger results, e.g. bounds for provable//~ (provably recursive functions) or the unprovability of primitive recursive wellfoundedness PRWO. Of course, as pointed out by Kreisel [7] such stronger results can be recaptured by arithmetizing the cut-elimination procedure for (primitive) recur-sively represented infinite derivations via the (Primitive) Recursion Theorem (cf. Schwichtenberg [15], Girard [5]). But this requires a lot of cumbersome and boring coding machinery which on the other side is not completely trivial, and it seems to me that all presentations of this subject in the existing literature are more or less unsatisfactory. Our purpose here is to provide a technically smooth method for the finitary treatment of infinite derivations in ~-arithmetic Zoo, where we don't need numerical codes but instead are working with natural notations for infinite derivations. These notations are finite terms generated from finite derivations (considered as constants) by certain function symbols Ik, a, Rc, E corresponding to the operations Jk, A : Zoo ~Zoo, ~c : Zoo x Zoo ~Zoo, 6 ~ : Z oo ~Zoo which make up the cut-elimination procedure for Z ~ developed by Schtitte [12] and Minc [10]. (Minc' contribution was to modify Schiitte's cut-elimination procedure by incorporating the so-called repetition-rule, which is crucial for the subsequent work.) 278 w. Buchholz In order to demonstrate the working of our method we will prove two wellknown results of classical proof theory for the system Z + TI.<r O.e. Peano-Arithmetic together with the scheme of transfinite induction along any proper segment of some prim. rec. wellordering ~(). These results are (I) If Z+ TI.<t~-Vx3yR(x,y) (R~Z ~ then there are prim. rec. (II) PRA~PRWO(-<)~II~ (Z + TI.<t). Of course in the proof of(I) and (II) we cannot completely …