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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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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
Can we make the Second Incompleteness Theorem coordinate free?
 DEPARTMENT OF PHILOSOPHY, UTRECHT UNIVERSITY, HEIDELBERGLAAN
"... Is it possible to give a coordinate free formulation of the Second Incompleteness Theorem? We pursue one possible approach to this question. We show that (i) cutfree consistency for finitely axiomatized theories can be uniquely characterized modulo EAprovable equivalence, (ii) consistency for fin ..."
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Is it possible to give a coordinate free formulation of the Second Incompleteness Theorem? We pursue one possible approach to this question. We show that (i) cutfree consistency for finitely axiomatized theories can be uniquely characterized modulo EAprovable equivalence, (ii) consistency for finitely axiomatized sequential theories can be uniquely characterized modulo EAprovable equivalence. The case of infinitely axiomatized ce theories is more delicate. We carefully discuss this in the paper.
A Version of the Second Incompleteness Theorem For Axiom Systems that Recognize Addition But Not Multiplication as a Total Function”, First Order Logic Revisited, Logos Verlag (Berlin) 2004
"... ABSTRACT: Let A(x; y; z) and M(x; y; z) denote predicates indicating x + y = z and x y = z respectively. Let us say an axiom system recognizes Addition and Multiplication both as Total Functions i it can prove: 8x8y9z A(x; y; z) AND 8x8y9z M(x; y; z) (1) We will introduce some new variations of t ..."
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ABSTRACT: Let A(x; y; z) and M(x; y; z) denote predicates indicating x + y = z and x y = z respectively. Let us say an axiom system recognizes Addition and Multiplication both as Total Functions i it can prove: 8x8y9z A(x; y; z) AND 8x8y9z M(x; y; z) (1) We will introduce some new variations of the Second Incompleteness Theorem for axiom systems which recognize Addition as a \total " function but which treat Multiplication as only a 3way relation. These generalizations of the Second Incompleteness Theorem are interesting because our prior work [30, 32, 34] has explored several types of boundarycase exceptions to the Second Incompleteness Theorem that occur when one weakens the the hypothesis for our main theorems only slightly further. 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
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.