Finite Notations for Infinite Terms

Abstract Buchholz (1991) presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive (not e 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 infinite (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. (1) give a new proof of a well-known trade-off theorem (Schwichtenberg, 1975), which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and 2. (2) construct a continuous normalization operator with an explicit modulus of continuity.