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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
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
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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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. 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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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
WHY THE THEORY R IS SPECIAL
, 2009
"... Abstract. Is it possible to give coordinatefree characterizations of salient theories? Such characterizations would always involve some notion of sameness of theories: we want to describe a theory modulo a notion of sameness, without having to give an axiomatization in a specific language. Such a c ..."
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Abstract. Is it possible to give coordinatefree characterizations of salient theories? Such characterizations would always involve some notion of sameness of theories: we want to describe a theory modulo a notion of sameness, without having to give an axiomatization in a specific language. Such a characterization could, e.g., be a first order formula in the language of partial preorderings that describes uniquely a degree in a particular structure of degrees of interpretability. Our theory would be contained in this degree. There are very few examples currently known along these lines, except some rather trivial ones. In this paper we provide a nontrivial characterization of TarskiMostowskiRobinson’s theory R. The characterization is in terms of the double degree structure of RE degrees of local and global interpretability. Consider the RE degrees of global interpretability that are in the minimal RE degree of local interpretability. These are the global degrees of the RE locally finitely satisfiable theories. We show that these degrees have a maximum and that R is in that maximum. In more mundane terms: an RE theory is locally finite iff it is globally interpretable in R. Dedicated to Harvey Friedman on the occasion of his 60th birthday. 1.
Cardinal Arithmetic in Weak Theories
, 2008
"... In this paper we develop the theory of cardinals in the theory COPY. This is the theory of two total, jointly injective binary predicates in a second order version, where we may quantify over binary relations. The only second order axioms of the theory are the axiom asserting the existence of an emp ..."
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In this paper we develop the theory of cardinals in the theory COPY. This is the theory of two total, jointly injective binary predicates in a second order version, where we may quantify over binary relations. The only second order axioms of the theory are the axiom asserting the existence of an empty relation and the adjunction axiom, which says that we may enrich any relation R with a pair x, y. The theory COPY is strictly weaker than the theory AS, adjunctive set theory. The relevant notion of weaker here is direct interpretability. We will explain and motivate this notion in the paper. A consequence is that our development of cardinals is inherited by stronger theories like AS. We will show that the cardinals satisfy (at least) Robinson’s Arithmetic Q. A curious aspect of our approach is that we develop cardinal multiplication using neither recursion nor pairing, thus diverging both from Frege’s paradigm and from the tradition in set theory. Our development directly uses the universal property characterizing the product that is familiar from category theory. The broader context of this paper is the study of a double degree structure: the degrees of (relative) interpretability and the finer degrees of direct interpretability. Most of the theories studied are in one of two degrees of interpretability: the bottom degree of predicate logic or the degree of Q. The theories will differ significantly if we compare them using direct interpretability.
CARDINAL ARITHMETIC IN THE STYLE OF
"... Abstract. In this paper we show how to interpret Robinson’s Arithmetic Q and the theory R of Tarski, Mostowski and Robinson as theories of cardinals in very weak theories of relations over a domain. Bei der Verfolgung eines Hasen wollte ich mit meinem Pferd über einen Morast setzen. Mitten im Sprung ..."
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Abstract. In this paper we show how to interpret Robinson’s Arithmetic Q and the theory R of Tarski, Mostowski and Robinson as theories of cardinals in very weak theories of relations over a domain. Bei der Verfolgung eines Hasen wollte ich mit meinem Pferd über einen Morast setzen. Mitten im Sprung musste ich erkennen, dass der Morast viel breiter war, als ich anfänglich eingeschätzt hatte. Schwebend in der Luft wendete ich daher wieder um, wo ich hergekommen war, um einen größeren Anlauf zu nehmen. Gleichwohl sprang ich zum zweiten Mal noch zu kurz und fiel nicht weit vom anderen Ufer bis an den Hals in den Morast. Hier hätte ich unfehlbar umkommen müssen, wenn nicht die Stärke meines Armes mich an meinem eigenen Haarzopf, samt dem Pferd, welches ich fest zwischen meine Knie schloss, wieder herausgezogen hätte. Baron von Münchhausen.
CONCATENATION AS A BASIS FOR Q AND THE INTUITIONISTIC VARIANT OF NELSON’S CLASSIC RESULT
, 2008
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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.