Bar Induction and Restricted Classical Logic

Abstract

Bar induction is originally discussed by L. E. J. Brouwer under the name of “bar theorem” in his intuitionistic mathematics. Nowadays, there are several formulations of bar induction. Over a well-known classical subsystem \(\mathsf{RCA_0}\) of second-order arithmetic, they are equivalent to the full second-order comprehension axiom. However, their interrelation from the purely constructive point of view (in the sense of Bishop) is still unknown. In this paper, we investigate the interrelation between decidable bar induction, monotone bar induction, and bar induction with neither the decidable condition nor the monotonicity condition in the assumptions over an intuitionistic fragment of \(\mathsf{RCA_0}\), and show that the third one is equivalent to the second one plus the numerical constant domain axiom which comes from the study of intermediate predicate logics. In addition, we consider the restrictions of bar induction where the side-predicates are of the form \(\exists z\, \mathrm{Q_{qf}}(z)\) where \(\mathrm{Q_{qf}}(z)\) is quantifier-free. Then we show the close relation between the restrictions of bar induction and the negative translation of a principle classically equivalent to the arithmetical comprehension axiom.