Independence results for weak systems of intuitionistic arithmetic Morteza Moniri Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran email: ezmoniri@ipm.ir Abstract This paper proves some independence results for weak fragments of Heyting arithmetic by using Kripke models. We present a necessary condition for linear Kripke models of arithmetical theories which are closed under the negative trans- lation and use it to show that the union of the worlds in any linear Kripke model of HA satisfies PA. We construct a two-node PA-normal Kripke structure which does not force iΣ 2 . We prove i∀ 1  i∃ 1 , i∃ 1  i∀ 1 , iΠ 2  iΣ 2 and iΣ 2  iΠ 2 . We use Smorynski’s operation Σ ′ to show HA  lΠ 1 . 2000 Mathematics Subject Classification: 03F30, 03F55, 03H15. Key words and phrases: Heyting Arithmetic, Kripke Models, Weak Fragments of HA, Least Number Principle. 0. Introduction In this note we use Kripke models to prove some independence results for weak frag- ments of Heyting arithmetic. Kripke semantics for intuitionistic logic was introduced by Kripke in 1965. The meta-logic of Kripke model theory is classical logic. Any intuitionis- tic theory is complete with respect to its Kripke models. The first comprehensive study of Kripke models of Heyting arithmetic HA (the intuitionistic counterpart of first order Peano arithmetic PA) was done by Smorynski in his PhD thesis which appeared as [S]. In [S], Smorynski introduced his method for constructing Kripke models of HA (Smorynski’s operations Σ ′ and Σ ∗ ) and used them to prove several independence results for HA. Even now, Smorynski’s method for constructing non-trivial Kripke models of HA is the only method which is known. Kripke models of weak fragments of HA are usually more accessible. In [W1], [MM], [M1] and [M2], Kripke models are used to prove results on certain weak fragments of HA. Here, we continue the line. 1