Intuitionistic Choice and Restricted Classical Logic

Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in a finite types together with various forms of the axiom of choice and a numerical omniscience schema (NOS) which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theoretic strength of such systems can be determined by functional interpretation based on a non‐constructive μ ‐operator and his well‐known results on the strength of this operator from the 70's. In this note we consider a weaker form LNOS (lesser numerical omniscience schema) of NOS which suffices to derive the strong form of binary König's lemma studied by Coquand/Palmgren and gives rise to a new and mathematically strong semi‐classical system which, nevertheless, can proof theoretically be reduced to primitive recursive arithmetic PRA. The proof of this fact relies on functional interpretation and a majorization technique developed in a previous paper.