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Interpretability logic
 Mathematical Logic, Proceedings of the 1988 Heyting Conference
, 1990
"... Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle 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 ..."
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Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle 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
Extending ILM with an operator for Σ_1ness
, 2003
"... 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 interpretatio ..."
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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 #1ness in PA and # as formalized #1conservativity 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