Finite Notations for Infinite Terms

In 1] Buchholz presented a method to build notation systems for innnite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read oo by a primitive (not "0-) recursive function its n-th predecessor and e.g. the last rule applied. Here we extend the method to the more general setting of innnite (typed) terms, in order to make it applicable in other proof-theoretic contexts as well as in recursion theory. As examples, we use the method to (1) give a new proof of a well-known trade-oo theorem 6], which says that detours through higher types can be eliminated by the use of transsnite recursion along higher ordinals, and (2) construct a continuous normalization operator with an explicit modulus of continuity. It is well known that in order to study primitive recursion in higher types it is useful to unfold the primitive recursion operators into innnite terms. A similar phenomenon occurs in proof theory, where one expands induction axioms. For applications it then is often necessary to code these innnite objects by natural numbers. A standard method to design such a coding is to proceed as in Kleene's system O of ordinal notations; cf. 6] for a recursion theoretic and 7] for a proof theoretic application of this method. However, working with such codes is not easy. For instance, to prove that the standard operation reducing the cut rank by one can be represented by a primitive recursive operation on codes requires some careful applications of Kleene's recursion theorem for primitive recursive functions. Now Buchholz in 1] proposed to use a method familiar from systems of ordinal notations to construct codings of innnite derivations. The basic idea is to introduce a primitive recursive notation system for well-founded !-derivations in the same way as one usually introduces an ordinal notation system as a system of terms generated from constants for particular ordinals by function symbols corresponding to certain ordinal functions. A less-than relation between ordinal notations is then derived from the properties of the denoted ordinals and ordinal functions. Instead of ordinals Buchholz considers well-founded derivations in the standard system Z 1 of !-arithmetic, with an unrestricted !-rule. Each derivation in Peano-arithmetic Z can be viewed as a notation for a particular Z 1-derivation; so Z-derivations can play the role of constants here. The role of ordinal functions is taken over by …