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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) recursively 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.) Of course in the proof of(I) and (II) we cannot completely dispense with coding. But we only need the comparatively trivial coding of syntactic objects (such as formulas, sequents, finite derivations, etc.) and even this plays a rather marginal rrle, while the central part of our proof is coding-free. Content

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APALD7 32(3) (1986) 195-299 CONTENTS W. BUCHHOLZ, A new System of proof-theoretic ordinal functions J. CARTMELL, Generalized algebraic theories and contextual categories J. C. E. DEKKER, Isols and Burnside's lemma Y. GUREVICH and S. SHELAH, Fixed-point extensions of first-order logic P. S. MULRY, Adjointness in recursion A. SCEDRÖV, Ön the impossibility of explicit upper bounds on lengths of some provably finite algorithms in computable analysis AUTHOR INDEX 195 209 245 265 281 291 299In this paper we present a family of ordinal functions xp v (v ^ a>), which seems to provide the so far simplest method for denoting large constructive ordinals. These functions are a simplified version of the 0-functions, but nevertheless have the same strength as those. This will be shown at the end of the paper (Theorem 3.7) by using proof-theoretic results from [1], [2], [5]. -In Section 1 we define the functions ip v and prove their main properties. In Section 2 we define a primitive recursive notation System (OT, <) based on the functions ip v . This System has the great advantage that its ordering relation < is very simple and can be defined without reference to sets of coefficients or any similar concept. OT is introduced as a subset of a larger set T of terms, which plays an important role in Section 3. There we show that the Statement PRWO(^0ßo))) which says that there exist no primitive recursive infinite descending sequences in ({xeOT: x 0 Q ( O } y <), is not provable in J7{-CA 0 . This result is essentially used in Simpson [6] to establish the unprovability of a certain theorem of finite combinatorics. The proof of n}-CA 0 1 PRWO(^0ßo>) is based on the following results from [1]:ID v KVn3A:c:(A:) = 0 (V^Ö>) 0168-0072/86/$3.50