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14
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 33 (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
Categories of Theories and Interpretations
 Logic in Tehran. Proceedings of the workshop and conference on Logic, Algebra and Arithmetic, held October 18–22
"... In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, biinterpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We ..."
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Cited by 5 (3 self)
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In this paper we study categories of theories and interpretations. In these categories, notions of sameness of theories, like synonymy, biinterpretability and mutual interpretability, take the form of isomorphism. We study the usual notions like monomorphism and product in the various theories. We provide some examples to separate notions across categories. In contrast, we show that, in some cases, notions in different categories do coincide. E.g., we can, under suchandsuch conditions, infer synonymity of two theories from their being equivalent in the sense of a coarser equivalence relation. We illustrate that the categories offer an appropriate framework for conceptual analysis of notions. For example, we provide a ‘coordinate free ’ explication of the notion of axiom scheme. Also we give a closer analysis of the objectlanguage / metalanguage distinction. Our basic category can be enriched with a form of 2structure. We use
On the monadic secondorder transduction hierarchy
 HAL Archive
"... Vol. 6 (2:2) 2010, pp. 1–28 www.lmcsonline.org ..."
Growing commas  a study of sequentiality and concatenation
 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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Cited by 2 (1 self)
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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.
Pairs, sets and sequences in first order theories
, 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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Cited by 2 (1 self)
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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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Cited by 2 (0 self)
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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
Strong logics of first and second order
 Bulletin of Symbolic Logic
"... In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ωlogic and βlogic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns o ..."
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In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ωlogic and βlogic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the firstorder side, this leads to a new presentation of Woodin’s Ωlogic. On the secondorder side, we compare the strongest logic with full secondorder logic and argue that the comparison lends support to Quine’s claim that secondorder
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
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"... A note on noninterpretability in ominimal structures by Ricardo B i a n c o n i (São Paulo) Abstract. We prove that if M is an ominimal structure whose underlying order is dense then Th(M) does not interpret the theory of an infinite discretely ordered structure. We also make a conjecture concerni ..."
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A note on noninterpretability in ominimal structures by Ricardo B i a n c o n i (São Paulo) Abstract. We prove that if M is an ominimal structure whose underlying order is dense then Th(M) does not interpret the theory of an infinite discretely ordered structure. We also make a conjecture concerning the class of the theory of an infinite discretely ordered ominimal structure. Introduction. In [9], ´ Swierczkowski proves that Th(〈ω, <〉) is not interpretable (with parameters) in RCF (the theory of real closed fields) by showing that a preordering with successors is not definable in R. (We recall that a preordering with successors is a reflexive and transitive binary relation ≪, satisfying ∀x∀y (x ≪ y ∨ y ≪ x) and ∀x∃y Succ(x, y),