In this paper we expose a method for building models for interpretability logics. The method can be compared to the method of taking unions of chains in classical model theory. Many applications of the method share a common part. We isolate this common part in a main lemma. Doing so, many of our results become applications of this main lemma. We also briefly describe how our method can be generalized to modal logics with a different signature. With the general method, we prove completeness for the interpretability logics IL, ILM, ILM0 and ILW ∗. We also apply our method to obtain a classification of the essential Σ1-sentences of essentially reflexive theories. We briefly comment on such a classification for finitely axiomatizable theories. As a digression we proof some results on self-provers. Towards the end of the paper we concentrate on modal matters concerning IL(All), the interpretability logic of all reasonable arithmetical theories. We prove the modal incompleteness of the logic ILW ∗ P0. We put forward a new principle R, and show it to be arithmetically sound in any reasonable arithmetical theory. Finally we make some general remarks on the logics ILRW and IL(All).

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Secondly, we show that a firstorder theory of finite signature is sequential (is a theory of sequences) iff it is definitionally equivalent to an extension in the same language of a

In the first part of the paper we discuss some conceptual problems related to the notion of proof. In the second part we survey five major open problems in Provability Logic as well as possible directions for future research in this area. 1

In this paper we formulate a logic #ILM. This logic extends ILM and contains a new unary modal operator #1 . The formulas of this logic can be evaluated on Veltman frames. We show that #ILM is modally sound and complete with respect to a certain class of Veltman frames. An arithmetical interpretation of the modal formulas can be obtained by reading the #1 operator as formalized #1-ness in PA and # as formalized #1-conservativity between finite extensions of PA. We show that under this arithmetically interpretation #ILM is sound and complete. The main motivation for formulating #ILM at all is that one counterexample for interpolation in ILM seems to emerge because of the lack of ILM to express #1 -ness. We show that #ILM does not have interpolation either. Our counterexample seems to emerge because of the inability of #ILM to express #-interpolation[7]. (A formula # # has a #1 -interpolant if there exist some # #1 such that PA # and #.) The text of this paper formed the master's thesis of the author at the ILLC, June 2003, under supervision of Prof. Dr. D.H.J. de Jongh. Contents 1

Abstract. In this paper we give an informally semi-rigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion m-sequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.