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Explicit Provability And Constructive Semantics
 Bulletin of Symbolic Logic
, 2001
"... In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is noth ..."
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In 1933 G odel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that G odel's provability calculus is nothing but the forgetful projection of LP. This also achieves G odel's objective of defining intuitionistic propositional logic Int via classical proofs and provides a BrouwerHeytingKolmogorov style provability semantics for Int which resisted formalization since the early 1930s. LP may be regarded as a unified underlying structure for intuitionistic, modal logics, typed combinatory logic and #calculus.
Elementary explicit types and polynomial time operations
, 2008
"... This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polyn ..."
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Cited by 6 (5 self)
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This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polynomial time. The systems considered are natural extensions of the firstorder applicative theories introduced in
Three Processes in Natural Language Interpretation
 Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman. Natick, Mass.: Association for Symbolic Logic
, 2000
"... . To address complications involving ambiguity, presupposition and implicature, three processes underlying natural language interpretation are isolated: translation, entailment and attunement. A metalanguage integrating these processes is outlined, elaborating on a prooftheoretic approach to pr ..."
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. To address complications involving ambiguity, presupposition and implicature, three processes underlying natural language interpretation are isolated: translation, entailment and attunement. A metalanguage integrating these processes is outlined, elaborating on a prooftheoretic approach to presupposition. To appear: Festschrift for Solomon Feferman (ASL Lecture Notes series) x1. Introduction. However outrageous Montague's slogan "English as a formal language" [24] may sound, the pressure to push the claim as far as it can go is, for many, irresistible. Basic to Montague's understanding of a formal language is the possibility of a modeltheoretic interpretation  of obvious interest in various applications (e.g. databases) that employ models. But formulas of predicate logic (firstorder or higherorder, modal or otherwise) differ significantly from English sentences marked with ambiguity and presupposition. Consider, for instance, (s) If Sylvester gets holds of a canary, the...
The Suslin operator in applicative theories: its prooftheoretic analysis via ordinal theories
"... The Suslin operator E1 is a type2 functional testing for the wellfoundedness of binary relations on the natural numbers. In the context of applicative theories, its prooftheoretic strength has been analyzed in Jäger and Strahm [18]. This article provides a more direct approach to the computation o ..."
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The Suslin operator E1 is a type2 functional testing for the wellfoundedness of binary relations on the natural numbers. In the context of applicative theories, its prooftheoretic strength has been analyzed in Jäger and Strahm [18]. This article provides a more direct approach to the computation of the upper bounds in question. Several theories featuring the Suslin operator are embedded into ordinal theories tailored for dealing with nonmonotone inductive definitions that enable a smooth definition of the application relation. 1
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Reflections on reflections in explicit mathematics
 Ann. Pure Appl. Logic
, 2005
"... We give a broad discussion of reflection principles in explicit mathematics, thereby addressing various kinds of universe existence principles. The prooftheoretic strength of the relevant systems of explicit mathematics is couched in terms of suitable extensions of KripkePlatek set theory. 1 ..."
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We give a broad discussion of reflection principles in explicit mathematics, thereby addressing various kinds of universe existence principles. The prooftheoretic strength of the relevant systems of explicit mathematics is couched in terms of suitable extensions of KripkePlatek set theory. 1
Collections, Sets and Types
, 1995
"... We give a first order formulation of Church's type theory in which types are mere sets. This formulation is obtained by replacing λcalculus by a language of combinators (skolemized comprehension schemes), introducing a distinction between propositions and their contents, relativizing quantifie ..."
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We give a first order formulation of Church's type theory in which types are mere sets. This formulation is obtained by replacing λcalculus by a language of combinators (skolemized comprehension schemes), introducing a distinction between propositions and their contents, relativizing quantifiers and at last replacing typing predicates by membership to some sets. The theory obtained this way has both a type theoretical flavor and a set theoretical one. Like set theory, it is a first order theory, and it uses only one notion of collection. Like type theory, it gives an explicit notation for objects, a primitive notion of function and propositions are objects.
Operations, sets and classes
"... Operational set theory, in the form described below, is an enterprise which consolidates classical set theory with some central concepts of Feferman’s explicit mathematics. It provides for a careful distinction between operations and settheoretic functions and as such reconciles set theory with nee ..."
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Operational set theory, in the form described below, is an enterprise which consolidates classical set theory with some central concepts of Feferman’s explicit mathematics. It provides for a careful distinction between operations and settheoretic functions and as such reconciles set theory with needs arising in constructive environments and even in those enhanced by computer science. In the following we consider, primarily from a prooftheoretic perspective, the theory OST and some of its most important extensions and determine their consistency strengths by exhibiting equivalent systems in the realm of traditional theories of sets and classes.
The Suslin operator in applicative theories reconsidered
"... The Suslin operator E1 is a type2 functional testing for the wellfoundedness of binary relations on the natural numbers. In the context of applicative theories, its prooftheoretic strength has been analyzed in Jäger and Strahm [18]. This article provides a more direct approach to the computation o ..."
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The Suslin operator E1 is a type2 functional testing for the wellfoundedness of binary relations on the natural numbers. In the context of applicative theories, its prooftheoretic strength has been analyzed in Jäger and Strahm [18]. This article provides a more direct approach to the computation of the upper bounds in question. Several theories featuring the Suslin operator are embedded into ordinal theories tailored for dealing with nonmonotone inductive definitions that enable a smooth definition of the application relation. 1