On the Provability Logic of Bounded Arithmetic

In this paper we develop techniques to build various sets of highly undecidable sentences in Ido + i2,. Our results stem from an attempt to prove that the modal logic of provability in Ido + Ql, here called PLSZ, is the same as the modal logic L of provability in PA. It is already known that L s PLSZ. We prove here some strict containments of the form PLSZ c T/z(%) where %’ is a class of Kripke frames. Stated informally the problem is whether the provability predicates of Ido + Q2, and PA share the same modal properties. It turns out that while Ido + !Sl certainly satisfies all the properties needed to carry out the proof of Godel’s second incompleteness theorem (namely L E PLQ), the question whether L = PLQ might depend on difficult issues of computational complexity. In fact if