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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
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
No Escape from Vardanyan’s Theorem
, 2003
"... Vardanyan’s Theorem states that the set of PAvalid principles of Quantified Modal Logic, QML, is complete Π 0 2. We generalize this result to a wide class of theories. The crucial step in the generalization is avoiding the use of Tennenbaum’s Theorem. ..."
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Vardanyan’s Theorem states that the set of PAvalid principles of Quantified Modal Logic, QML, is complete Π 0 2. We generalize this result to a wide class of theories. The crucial step in the generalization is avoiding the use of Tennenbaum’s Theorem.
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.
Towards the Interpretability Logic of x,arianne B. Kalsbeek Logic Group
, 1991
"... Among the different interpretability logics corresponding to (classes of) arthmetical theories, the interpretability logic of Itp+EXP (to which we will refer as ILeXp), takes a special place. Though we have no explicit axiomatization for ILeXp, we do have a complete description of the theory. Visser ..."
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Among the different interpretability logics corresponding to (classes of) arthmetical theories, the interpretability logic of Itp+EXP (to which we will refer as ILeXp), takes a special place. Though we have no explicit axiomatization for ILeXp, we do have a complete description of the theory. Visser shows, in [VIS], that relative interpretability
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