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 ε 0-) recursive function its nth 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.