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The Interpretability Logic of all Reasonable Arithmetical Theories
 ERKENNTNIS
, 1999
"... This paper is a presentation of a status quaestionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question. ..."
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Cited by 9 (5 self)
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This paper is a presentation of a status quaestionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.
E.: Modal Matters in Interpretability Logics
, 2004
"... 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 res ..."
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Cited by 4 (2 self)
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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 Σ1sentences of essentially reflexive theories. We briefly comment on such a classification for finitely axiomatizable theories. As a digression we proof some results on selfprovers. 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).
The Modal Completeness of ILW
"... This paper contains a completeness proof for the system ILW, a rather bewildering axiom system belonging to the family of interpretability logics. ..."
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This paper contains a completeness proof for the system ILW, a rather bewildering axiom system belonging to the family of interpretability logics.
Prooftheoretic analysis by iterated reflection
 Arch. Math. Logic
"... Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omegarule. We compare the information obtained by this kind of analysis with the results obtained by the more usual prooftheoretic techni ..."
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Cited by 3 (1 self)
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Progressions of iterated reflection principles can be used as a tool for ordinal analysis of formal systems. Technically, in some sense, they replace the use of omegarule. We compare the information obtained by this kind of analysis with the results obtained by the more usual prooftheoretic techniques. In some cases the techniques of iterated reflection principles allows to obtain sharper results, e.g., to define prooftheoretic ordinals relevant to logical complexity Π 0 1. We provide a more general version of the fine structure formulas for iterated reflection principles (due to U. Schmerl [24]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣn, IΣ − n, IΠ − n and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform Σ1reflection principle for T is Σ2conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl’s theorem. 1
Growing commas –a study of sequentiality and concatenation. Logic Group Preprint Series 257
 Department of Philosophy, Utrecht University
, 2007
"... In his paper [Grz05], Andrzej Grzegorczyk introduces a theory of concatenation TC. We show that TC does not define pairing. We determine a reasonable extension of TC that is sequential, i.e., has a good sequence coding. ..."
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In his paper [Grz05], Andrzej Grzegorczyk introduces a theory of concatenation TC. We show that TC does not define pairing. We determine a reasonable extension of TC that is sequential, i.e., has a good sequence coding.
The Interpolation Theorem for IL and ILP
 Uppsala University
, 1998
"... In this article we establish interpolation for the minimal system of interpretability logic IL. We prove that arrow interpolation holds for IL and that turnstile interpolation and interpolation for the modality easily follow from this. Furthermore, these properties are extended to the system ILP. ..."
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In this article we establish interpolation for the minimal system of interpretability logic IL. We prove that arrow interpolation holds for IL and that turnstile interpolation and interpolation for the modality easily follow from this. Furthermore, these properties are extended to the system ILP. The related issue of Beth Definability is also addressed. As usual, the arrow interpolation property implies the Beth property. From the latter it follows via an argumentation which is standard in provability logic, that IL has the fixed point property. Finally we observe that a general result of Maksimova [11] for provability logics can be extended to interpretability logics, implying that all extensions of IL have the Beth property. Keywords Interpretability Logic, Interpolation Properties, Beth Property, Fixed Point Property. 1 Introduction 1.1 Some History Interpretability logics are extensions of provability logics introduced by Visser in [15]. There the modal logics IL, ILM and ILP a...
Pairs, sets and sequences in first order theories. forthcoming
, 2007
"... 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 nonsurjective ordered pairing are definiti ..."
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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 nonsurjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of nonsurjective 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
The Predicative Frege Hierarchy
, 2006
"... In this paper, we characterize the strength of the predicative Frege hierarchy, P n+1 V, introduced by John Burgess in his book [Bur05]. We show that P n+1 V and Q + con n (Q) are mutually interpretable. It follows that PV: = P 1 V is mutually interpretable with Q. This fact was proved earlier by Mi ..."
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In this paper, we characterize the strength of the predicative Frege hierarchy, P n+1 V, introduced by John Burgess in his book [Bur05]. We show that P n+1 V and Q + con n (Q) are mutually interpretable. It follows that PV: = P 1 V is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [Gan06] using a different proof. Another consequence of the our main result is that P 2 V is mutually interpretable with Kalmar Arithmetic (a.k.a. EA, EFA, I∆0+EXP, Q3). The fact that P 2 V interprets EA, was proved earlier by Burgess. We provide a different proof. Each of the theories P n+1 V is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, P ω V, is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable
BiUnary Interpretability Logic
"... Introduction In recent years several modal systems have been introduced to study the relation of relative interpretability between arithmetical theories. The intepretability principles of several important classes of arithmetical theories have been axiomatised. In [6] the system ILP is shown to be ..."
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Introduction In recent years several modal systems have been introduced to study the relation of relative interpretability between arithmetical theories. The intepretability principles of several important classes of arithmetical theories have been axiomatised. In [6] the system ILP is shown to be the interpretability logic of all \Sigma 0 1 sound finitely axiomatised sequential theories that extend I\Delta 0 +SupExp; in [1] it is shown that ILM is the interpretability logic of PA. Montagna and H'ajek [2] show that ILM is also the logic of \Pi 0 1 conservativity of all \Sigma 0 1 sound extensions of I\Sigma 1 . (As is wellknown, in the case of PA the two relations of relative interpretability and of \Pi 0 1 conservativity coincide). Given the above results it is only natural to consider a modal