Weak Arithmetics and Kripke Models

In the first section of this paper we show that i Π₁ ≡ W??lΠ₁ and that a Kripke model which decides bounded formulas forces iΠ₁ if and only if the union of the worlds in any (complete) path in it satisflies IΠ₁. In particular, the union of the worlds in any path of a Kripke model of HA models IΠ₁. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Π_m‐formulas in a linear Kripke model deciding Δ₀‐formulas it is necessary and sufficient that the model be Σ_m‐elementary. This implies that if a linear Kripke model forces PEM_prenex, then it forces PEM. We also show that, for each n ≥ 1, iΦ_n does not prove ??(IΠ_n's are Burr's fragments of HA.