Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbert-style programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove

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