An Independence Result for Intuitionistic Bounded Arithmetic

It is shown that the intuitionistic theory of polynomial induction on positive Π1 (coNP) formulas does not prove the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y| → x = |z|). This implies the unprovability of the scheme ¬¬PIND(Σ 1 ) in the mentioned theory. However, this theory contains the sentence ∀x, y¬¬∃z ≤ y(x ≤ |y| → x = |z|). The above independence result is proved by constructing an ω-chain of submodels of a countable model of S2+Ω3+¬exp such that none of the worlds in the chain satisfies the sentence, and interpreting the chain as a Kripke model.