Intuitionistic Choice and Restricted Classical Logic

Abstract

Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic
in all 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 mu-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 K¨onig'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.