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16
An Equivalence between Second Order Bounded Domain Bounded Arithmetic and First Order Bounded Arithmetic
, 1993
"... We introduce a bounded domain version V 2 (BD) of Buss's second order theory V 2 of bounded arithmetic and show that this version is equivalent to the rst order theory S 3 : More precisely, we construct two natural interpretations V 3 and S 2 (BD) which are inverse to each other and pr ..."
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Cited by 28 (4 self)
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We introduce a bounded domain version V 2 (BD) of Buss's second order theory V 2 of bounded arithmetic and show that this version is equivalent to the rst order theory S 3 : More precisely, we construct two natural interpretations V 3 and S 2 (BD) which are inverse to each other and preserve the syntactic structure of bounded formulae. As a corollary, for the bounded domain case we obtain Buss's result concerning 1 expressibility in V 2 as a direct consequence of his main result for rst order theories. Using only plain corollaries of the cut elimination theorem, we show that V 2 (BD) prove the same formulae where 8 stand for rst order quanti ers. Combined with the above mentioned result this gives an alternative proof of Buss's characterization of 2 functions. All this readily extends to the case V k (BD) vs. S k+1 (i; k 1).
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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Cited by 5 (3 self)
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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.
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
SOME THEOREMS ON THE LATTlCE OF LOCAL INTERPRETABILITY TYPES by JAN KRAJÍÈEK in Prague (Czechoslova,kia)
"... In [4] J. MYCIELSKI introduced a very general notion of multidimensional local interpretability of first ordet theories. If we definethe relation ~ between theories T, S by T ~ S iff Tis multidimensionaly locally interpretable in S, then ~is a. preordering. The induced partia.l ordering is a la.ttic ..."
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In [4] J. MYCIELSKI introduced a very general notion of multidimensional local interpretability of first ordet theories. If we definethe relation ~ between theories T, S by T ~ S iff Tis multidimensionaly locally interpretable in S, then ~is a. preordering. The induced partia.l ordering is a la.ttice ordering, it is called tke lattice of
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
"... ..."
Sequence encoding without induction Emil Jeˇrábek ∗
, 2012
"... We show that the universally axiomatized, inductionfree theory PA − is a sequential theory in the sense of Pudlák [5], in contrast to the closely related Robinson’s arithmetic. Ever since Gödel’s [1] arithmetization of syntax in the proof of his incompleteness theorem, sequence encoding has been an ..."
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We show that the universally axiomatized, inductionfree theory PA − is a sequential theory in the sense of Pudlák [5], in contrast to the closely related Robinson’s arithmetic. Ever since Gödel’s [1] arithmetization of syntax in the proof of his incompleteness theorem, sequence encoding has been an indispensable tool in the study of arithmetical theories and related areas of mathematical logic. While a common approach is to develop a particular sequence encoding in a suitable base theory and work with that, a general concept of theories supporting encoding of sequences of their elements, called sequential theories, was isolated by Pudlák [4, 5] during his work on interpretability. A similar but weaker notion was defined earlier by Vaught [7]. More generally, theories with “containers ” of various kind were studied by Visser [8], who includes a discussion of variants of the notion of sequentiality and some historical remarks. It is known that fairly weak arithmetical theories can be sequential (e.g., fragments of bounded arithmetic such as Buss ’ S1 2, cf. Krajíček [3]), nevertheless sequential theories described in the literature so far generally involve some form of the induction schema. Indeed,