Comparing Constructive Arithmetical Theories Based on N P-PIND and coN P-PIND

摘要

In this note we show that the intuitionistic theory of polynomial induction on Π_1~(b+)-formulas does not imply the intuitionistic theory I S_2~1 of polynomial induction on Σ_1~(b+)-formulas. We also show the converse assuming the Polynomial Hierarchy does not collapse. Similar results hold also for length induction in place of polynomial induction. We also investigate the relation between various other intuitionistic first-order theories of hounded arithmetic. Our method is mostly semantical, we use Kripke models of the theories.