Finite investigations of transfinite derivations

Abstract

ω-derivations of arithmetic formulas are analyzed. A primitive recursive normalization operator E is constructed in the first part. It cut-eliminates not only recursively described derivations (i.e., well-founded proof-figures) but also arbitrary (not necessarily well-founded) proof-figures constructed from an axiom by derivation rules. This permits us to apply E in the theory of models, Its application in the theory of proofs is based on the formalizability of the fundamental properties of E in a primitive recursive arithmetic. Cut-eliminability in the Heyting arithmetic HAω+AC with the axiom of choice of all finite types is proved in the second part. The formulation allowing cut-elimination uses terms associated with the derivations by a method due to Carry, Howard, Girard, and Martin-Lof. These terms are included in the very formulation of the rules. The conservativity of HAω+AC over HA is obtained as one of the corollaries.