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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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Cited by 32 (9 self)
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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
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.
Faith & Falsity: a study of faithful interpretations and false Σ 0 1sentences
 Logic Group Preprint Series 216, Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS
, 2002
"... A theory T is trustworthy iff, whenever a theory U is interpretable in T, then it is faithfully interpretable. In this paper we provide a characterization of trustworthiness. We provide a simple proof of Friedman’s Theorem that finitely axiomatized, sequential, consistent theories are trustworthy. W ..."
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Cited by 6 (4 self)
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A theory T is trustworthy iff, whenever a theory U is interpretable in T, then it is faithfully interpretable. In this paper we provide a characterization of trustworthiness. We provide a simple proof of Friedman’s Theorem that finitely axiomatized, sequential, consistent theories are trustworthy. We provide an example of a theory whose schematic predicate logic is
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 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
Iterated Local Reflection vs Iterated Consistency
, 1995
"... For "natural enough" systems of ordinal notation we show that ff times iterated local reflection schema over a sufficiently strong arithmetic T proves the same \Pi 0 1 sentences as ! ff times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability ..."
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For "natural enough" systems of ordinal notation we show that ff times iterated local reflection schema over a sufficiently strong arithmetic T proves the same \Pi 0 1 sentences as ! ff times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at fflnumbers. We also derive the following more general "mixed" formulas estimating the consistency strength of iterated local reflection: for all ordinals ff 1 and all fi, (T ff ) fi j \Pi 0 1 T ! ff \Delta(1+fi) ; (T fi ) ff j \Pi 0 1 T fi+! ff : Here T ff stands for ff times iterated local reflection over T , T fi stands for fi times iterated consistency, and j \Pi 0 1 denotes (provable in T ) mutual \Pi 0 1 conservativity. In an appendix to this paper we develop our notion of "natural enough" system of ordinal notation and show that such systems do exist for every recursive ordinal. 1 Introduction Since the fundamental works of A.Turing [17] and S.Feferman [6...
Problems in the Logic of Provability
, 2005
"... 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 ..."
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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
WHAT IS THE RIGHT NOTION OF SEQUENTIALITY?
"... Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. ..."
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Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.
THE ARITHMETICS OF A THEORY
"... Abstract. In this paper we study the interpretations of a weak arithmetic, like Buss ’ theory S1 2, in a given theory U. We call these interpretations the arithmetics of U. We develop the basics of the structure of the arithmetics of U. We study the provability logic(s) of U from the standpoint of t ..."
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Abstract. In this paper we study the interpretations of a weak arithmetic, like Buss ’ theory S1 2, in a given theory U. We call these interpretations the arithmetics of U. We develop the basics of the structure of the arithmetics of U. We study the provability logic(s) of U from the standpoint of the framework of the arithmetics of U. Finally, we provide a deeper study of the arithmetics