Extending ILM with an operator for $\Sigma_1$-ness Evan Goris Abstract: In this paper we formulate a logic $\Sigma$ILM. This logic extends ILM and contains a new unary modal operator $\Sigma_1$. The formulas of this logic can be evaluated on Veltman frames. We show that $\Sigma$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 $\Sigma_1$ operator as formalized $\Sigma_1$-ness in PA and |> as formalized $\Pi_1$-conservativity between finite extensions of PA. We show that under this arithmetically interpretation $\Sigma$ILM is sound and complete. The main motivation for formulating $\Sigma$ILM at all is that one counterexample for interpolation in ILM seems to emerge because of the lack of ILM to express $\Sigma_1$-ness. We show that $\Sigma$ILM does not have interpolation either. Our counterexample seems to emerge because of the inability of $\Sigma$ILM to express $\Sigma$-interpolation. (A formula A -> B has a $\Sigma_1$-interpolant if there exist some $\Sigma_1$ formula S such that PA |- A -> S and PA |- S -> B.) Keywords: Interpolation, Interpretability logic